List of prime numbers

Prime numbers are the fundamental building blocks of arithmetic, indivisible except by one and themselves. But these unique numbers come in many fascinating variations, each defined by peculiar properties or intricate patterns. Join us as we explore a cabinet of curiosities, uncovering the diverse 'personalities' hidden within the infinite sequence of primes. Prime numbers are not a monolithic group, but rather form numerous distinct categories based on their structure, relationship to other numbers, or mathematical properties. The search for new primes and the classification of existing ones is an ongoing journey that reveals deeper connections within number theory. Many types of primes are linked to specific mathematical formulas or constants, highlighting the intricate beauty of number sequences.

Source: Wikipedia

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Prime numbers are the fundamental building blocks of arithmetic, indivisible except by one and themselves. But these unique numbers come in many fascinating variations, each defined by peculiar properties or intricate patterns. Join us as we explore a cabinet of curiosities, uncovering the diverse 'personalities' hidden within the infinite sequence of primes.

The Many Faces of Prime Numbers

From the very first prime, 2, to numbers with millions of digits, primes have captivated mathematicians for millennia. But not all primes are created equal; many belong to special families, defined by unique characteristics or curious relationships to other numbers. Let's delve into some of these remarkable categories.

Fermat Primes: A Quest for Patterns

Named after the brilliant French mathematician Pierre de Fermat, Fermat primes are numbers of a very specific form. Fermat conjectured that all numbers of this form would be prime. While the first five fit the bill, he was later proven wrong for the sixth and beyond. This highlights the challenge of predicting prime behavior!

The formula for a Fermat number is surprisingly elegant: Fn equals two raised to the power of two raised to 'n', plus one. The first few Fermat primes are 3, 5, 17, 257, and 65537. To this day, these are the only known Fermat primes, making them quite rare.

F_n = 2^{2^n} + 1

Mersenne Primes: Giants of Number Theory

These primes are named after the 17th-century French monk Marin Mersenne, who studied them extensively. Mersenne primes are numbers that are one less than a power of two. They are particularly famous because the largest known prime numbers are almost always Mersenne primes.

A Mersenne prime follows the form Mp = 2^p - 1, where 'p' itself must also be a prime number. The first few examples are 3, 7, 31, and 127. The Great Internet Mersenne Prime Search (GIMPS) project constantly hunts for new record-breaking Mersenne primes, using distributed computing power from volunteers worldwide.

M_p = 2^p - 1

Double Mersenne Primes: Primes Within Primes

Taking the concept a step further, a Double Mersenne prime is a Mersenne prime where the exponent 'p' is itself a Mersenne prime. It's like a prime nested inside another prime, adding another layer of mathematical intrigue. As you can imagine, these are exceedingly rare.

The formula reveals this nesting: M{Mp} = 2 to the power of (2^p - 1), all minus 1. Only four such primes are currently known: 7, 127, 2,147,483,647, and the enormous 170,141,183,460,469,231,731,687,303,715,884,105,727. Each discovery is a significant computational feat.

M_{M_p} = 2^{(2^p - 1)} - 1

Fibonacci Primes: From Nature to Number Theory

The Fibonacci sequence, where each number is the sum of the two preceding ones (1, 1, 2, 3, 5, 8...), appears everywhere in nature. A Fibonacci prime is simply a number in this famous sequence that is also prime. These numbers bridge the worlds of additive and multiplicative patterns.

The sequence starts with 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89... The Fibonacci primes are those terms that are also prime, like 2, 3, 5, 13, and 89. Discovering large Fibonacci primes involves complex primality tests due to their rapid growth.

2 3 5 13 89 233 1597 28657 514229 433494437

Chen Primes: Neighboring Perfection

Named after Chinese mathematician Chen Jingrun, these primes are closely related to the famous Twin Prime Conjecture. A Chen prime 'p' is a prime number such that 'p + 2' is either also prime (making it a twin prime) or a semiprime—a number that is the product of two primes.

This definition helps us categorize primes that are 'close' to another prime. Examples include 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Many of the primes we commonly think of are actually Chen primes, showing how frequently primes appear near other special numbers.

Emirps: Primes That Read Backwards

An 'emirp' is a prime number that yields a different prime when its digits are reversed. (The word 'emirp' is 'prime' spelled backwards, fittingly enough!) Numbers like 11, 101, and other palindromic primes are excluded from this definition, as they remain the same when reversed.

Consider 13, which reverses to 31—both are prime. Another pair is 17 and 71. Other emirps include 37, 73, 79, 97, 107, 113, 149, and 157. These primes offer a playful glimpse into the properties of numbers based on their digits.

Palindromic Primes: Symmetrical Beauty

If a prime number reads the same forwards and backwards, it's called a palindromic prime. These numbers possess a beautiful symmetry, making them a favorite among recreational mathematicians. The simplest examples are single-digit primes and 11.

Beyond these, larger examples like 101, 131, and 151 show the elegance of these symmetrical primes. While there are infinitely many palindromic numbers, whether there are infinitely many palindromic primes remains an open question in mathematics.

2 3 5 7 11 101 131 151 181 191 313 353 373 383

Mills Primes: A Constant's Gift

Mills primes are generated by a remarkable mathematical constant, known as Mills' constant, denoted by theta (θ). When you raise theta to the power of 3 raised to 'n' and then take the floor (round down to the nearest integer), you always get a prime number. This is a very rare property!

The sequence starts with 2, 11, 1361, 2,521,008,887, and continues with incredibly large primes. Mills' constant itself is not precisely known, as its value depends on the exact sequence of primes it generates. This makes Mills primes a fascinating theoretical construct.

\lfloor \theta^{3^n} \rfloor

Home Primes: The Iterative Journey

Home primes are found through an iterative process: starting with a number, you write its prime factorization, concatenate the prime factors (in base 10), and then repeat the process with the new number. You keep going until you eventually reach a prime number. That final prime is the 'home prime' for your starting number.

For example, starting with 4: its prime factorization is 2 x 2. Concatenated, that's 22. 22's factors are 2 x 11, so that's 211. 211 is a prime, so the home prime of 4 is 211. It's a journey through prime factors that always ends with a prime!

Beyond the Lists: An Infinite Wonderland

These are just a handful of the many types of prime numbers that mathematicians categorize and study. Each type offers a unique lens through which to explore the endless, mysterious landscape of prime numbers. The quest for new primes and a deeper understanding of their distribution continues to drive mathematical discovery.

Article

List of prime numbers

This is a list of articles about prime numbers. A prime number (or prime) is a natural number greater than 1 that has no divisors other than 1 and itself. By Euclid's theorem, there are an infinite number of prime numbers. Subsets of the prime numbers may be generated with various formulas for primes.

The first 1,000 primes are listed below, followed by lists of notable types of prime numbers in alphabetical order, giving their respective first terms. The number 1 is neither prime nor composite.

The first 1,000 prime numbers

List of prime numbers

The following table lists the first 1,000 primes, with 20 columns of consecutive primes in each of the 50 rows.

<table><thead><tr><th></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</th><th>13</th><th>14</th><th>15</th><th>16</th><th>17</th><th>18</th><th>19</th><th>20</th></tr></thead><tbody><tr><td>1–20</td><td>2</td><td>3</td><td>5</td><td>7</td><td>11</td><td>13</td><td>17</td><td>19</td><td>23</td><td>29</td><td>31</td><td>37</td><td>41</td><td>43</td><td>47</td><td>53</td><td>59</td><td>61</td><td>67</td><td>71</td></tr><tr><td>21–40</td><td>73</td><td>79</td><td>83</td><td>89</td><td>97</td><td>101</td><td>103</td><td>107</td><td>109</td><td>113</td><td>127</td><td>131</td><td>137</td><td>139</td><td>149</td><td>151</td><td>157</td><td>163</td><td>167</td><td>173</td></tr><tr><td>41–60</td><td>179</td><td>181</td><td>191</td><td>193</td><td>197</td><td>199</td><td>211</td><td>223</td><td>227</td><td>229</td><td>233</td><td>239</td><td>241</td><td>251</td><td>257</td><td>263</td><td>269</td><td>271</td><td>277</td><td>281</td></tr><tr><td>61–80</td><td>283</td><td>293</td><td>307</td><td>311</td><td>313</td><td>317</td><td>331</td><td>337</td><td>347</td><td>349</td><td>353</td><td>359</td><td>367</td><td>373</td><td>379</td><td>383</td><td>389</td><td>397</td><td>401</td><td>409</td></tr><tr><td>81–100</td><td>419</td><td>421</td><td>431</td><td>433</td><td>439</td><td>443</td><td>449</td><td>457</td><td>461</td><td>463</td><td>467</td><td>479</td><td>487</td><td>491</td><td>499</td><td>503</td><td>509</td><td>521</td><td>523</td><td>541</td></tr><tr><td>101–120</td><td>547</td><td>557</td><td>563</td><td>569</td><td>571</td><td>577</td><td>587</td><td>593</td><td>599</td><td>601</td><td>607</td><td>613</td><td>617</td><td>619</td><td>631</td><td>641</td><td>643</td><td>647</td><td>653</td><td>659</td></tr><tr><td>121–140</td><td>661</td><td>673</td><td>677</td><td>683</td><td>691</td><td>701</td><td>709</td><td>719</td><td>727</td><td>733</td><td>739</td><td>743</td><td>751</td><td>757</td><td>761</td><td>769</td><td>773</td><td>787</td><td>797</td><td>809</td></tr><tr><td>141–160</td><td>811</td><td>821</td><td>823</td><td>827</td><td>829</td><td>839</td><td>853</td><td>857</td><td>859</td><td>863</td><td>877</td><td>881</td><td>883</td><td>887</td><td>907</td><td>911</td><td>919</td><td>929</td><td>937</td><td>941</td></tr><tr><td>161–180</td><td>947</td><td>953</td><td>967</td><td>971</td><td>977</td><td>983</td><td>991</td><td>997</td><td>1009</td><td>1013</td><td>1019</td><td>1021</td><td>1031</td><td>1033</td><td>1039</td><td>1049</td><td>1051</td><td>1061</td><td>1063</td><td>1069</td></tr><tr><td>181–200</td><td>1087</td><td>1091</td><td>1093</td><td>1097</td><td>1103</td><td>1109</td><td>1117</td><td>1123</td><td>1129</td><td>1151</td><td>1153</td><td>1163</td><td>1171</td><td>1181</td><td>1187</td><td>1193</td><td>1201</td><td>1213</td><td>1217</td><td>1223</td></tr><tr><td>201–220</td><td>1229</td><td>1231</td><td>1237</td><td>1249</td><td>1259</td><td>1277</td><td>1279</td><td>1283</td><td>1289</td><td>1291</td><td>1297</td><td>1301</td><td>1303</td><td>1307</td><td>1319</td><td>1321</td><td>1327</td><td>1361</td><td>1367</td><td>1373</td></tr><tr><td>221–240</td><td>1381</td><td>1399</td><td>1409</td><td>1423</td><td>1427</td><td>1429</td><td>1433</td><td>1439</td><td>1447</td><td>1451</td><td>1453</td><td>1459</td><td>1471</td><td>1481</td><td>1483</td><td>1487</td><td>1489</td><td>1493</td><td>1499</td><td>1511</td></tr><tr><td>241–260</td><td>1523</td><td>1531</td><td>1543</td><td>1549</td><td>1553</td><td>1559</td><td>1567</td><td>1571</td><td>1579</td><td>1583</td><td>1597</td><td>1601</td><td>1607</td><td>1609</td><td>1613</td><td>1619</td><td>1621</td><td>1627</td><td>1637</td><td>1657</td></tr><tr><td>261–280</td><td>1663</td><td>1667</td><td>1669</td><td>1693</td><td>1697</td><td>1699</td><td>1709</td><td>1721</td><td>1723</td><td>1733</td><td>1741</td><td>1747</td><td>1753</td><td>1759</td><td>1777</td><td>1783</td><td>1787</td><td>1789</td><td>1801</td><td>1811</td></tr><tr><td>281–300</td><td>1823</td><td>1831</td><td>1847</td><td>1861</td><td>1867</td><td>1871</td><td>1873</td><td>1877</td><td>1879</td><td>1889</td><td>1901</td><td>1907</td><td>1913</td><td>1931</td><td>1933</td><td>1949</td><td>1951</td><td>1973</td><td>1979</td><td>1987</td></tr><tr><td>301–320</td><td>1993</td><td>1997</td><td>1999</td><td>2003</td><td>2011</td><td>2017</td><td>2027</td><td>2029</td><td>2039</td><td>2053</td><td>2063</td><td>2069</td><td>2081</td><td>2083</td><td>2087</td><td>2089</td><td>2099</td><td>2111</td><td>2113</td><td>2129</td></tr><tr><td>321–340</td><td>2131</td><td>2137</td><td>2141</td><td>2143</td><td>2153</td><td>2161</td><td>2179</td><td>2203</td><td>2207</td><td>2213</td><td>2221</td><td>2237</td><td>2239</td><td>2243</td><td>2251</td><td>2267</td><td>2269</td><td>2273</td><td>2281</td><td>2287</td></tr><tr><td>341–360</td><td>2293</td><td>2297</td><td>2309</td><td>2311</td><td>2333</td><td>2339</td><td>2341</td><td>2347</td><td>2351</td><td>2357</td><td>2371</td><td>2377</td><td>2381</td><td>2383</td><td>2389</td><td>2393</td><td>2399</td><td>2411</td><td>2417</td><td>2423</td></tr><tr><td>361–380</td><td>2437</td><td>2441</td><td>2447</td><td>2459</td><td>2467</td><td>2473</td><td>2477</td><td>2503</td><td>2521</td><td>2531</td><td>2539</td><td>2543</td><td>2549</td><td>2551</td><td>2557</td><td>2579</td><td>2591</td><td>2593</td><td>2609</td><td>2617</td></tr><tr><td>381–400</td><td>2621</td><td>2633</td><td>2647</td><td>2657</td><td>2659</td><td>2663</td><td>2671</td><td>2677</td><td>2683</td><td>2687</td><td>2689</td><td>2693</td><td>2699</td><td>2707</td><td>2711</td><td>2713</td><td>2719</td><td>2729</td><td>2731</td><td>2741</td></tr><tr><td>401–420</td><td>2749</td><td>2753</td><td>2767</td><td>2777</td><td>2789</td><td>2791</td><td>2797</td><td>2801</td><td>2803</td><td>2819</td><td>2833</td><td>2837</td><td>2843</td><td>2851</td><td>2857</td><td>2861</td><td>2879</td><td>2887</td><td>2897</td><td>2903</td></tr><tr><td>421–440</td><td>2909</td><td>2917</td><td>2927</td><td>2939</td><td>2953</td><td>2957</td><td>2963</td><td>2969</td><td>2971</td><td>2999</td><td>3001</td><td>3011</td><td>3019</td><td>3023</td><td>3037</td><td>3041</td><td>3049</td><td>3061</td><td>3067</td><td>3079</td></tr><tr><td>441–460</td><td>3083</td><td>3089</td><td>3109</td><td>3119</td><td>3121</td><td>3137</td><td>3163</td><td>3167</td><td>3169</td><td>3181</td><td>3187</td><td>3191</td><td>3203</td><td>3209</td><td>3217</td><td>3221</td><td>3229</td><td>3251</td><td>3253</td><td>3257</td></tr><tr><td>461–480</td><td>3259</td><td>3271</td><td>3299</td><td>3301</td><td>3307</td><td>3313</td><td>3319</td><td>3323</td><td>3329</td><td>3331</td><td>3343</td><td>3347</td><td>3359</td><td>3361</td><td>3371</td><td>3373</td><td>3389</td><td>3391</td><td>3407</td><td>3413</td></tr><tr><td>481–500</td><td>3433</td><td>3449</td><td>3457</td><td>3461</td><td>3463</td><td>3467</td><td>3469</td><td>3491</td><td>3499</td><td>3511</td><td>3517</td><td>3527</td><td>3529</td><td>3533</td><td>3539</td><td>3541</td><td>3547</td><td>3557</td><td>3559</td><td>3571</td></tr><tr><td>501–520</td><td>3581</td><td>3583</td><td>3593</td><td>3607</td><td>3613</td><td>3617</td><td>3623</td><td>3631</td><td>3637</td><td>3643</td><td>3659</td><td>3671</td><td>3673</td><td>3677</td><td>3691</td><td>3697</td><td>3701</td><td>3709</td><td>3719</td><td>3727</td></tr><tr><td>521–540</td><td>3733</td><td>3739</td><td>3761</td><td>3767</td><td>3769</td><td>3779</td><td>3793</td><td>3797</td><td>3803</td><td>3821</td><td>3823</td><td>3833</td><td>3847</td><td>3851</td><td>3853</td><td>3863</td><td>3877</td><td>3881</td><td>3889</td><td>3907</td></tr><tr><td>541–560</td><td>3911</td><td>3917</td><td>3919</td><td>3923</td><td>3929</td><td>3931</td><td>3943</td><td>3947</td><td>3967</td><td>3989</td><td>4001</td><td>4003</td><td>4007</td><td>4013</td><td>4019</td><td>4021</td><td>4027</td><td>4049</td><td>4051</td><td>4057</td></tr><tr><td>561–580</td><td>4073</td><td>4079</td><td>4091</td><td>4093</td><td>4099</td><td>4111</td><td>4127</td><td>4129</td><td>4133</td><td>4139</td><td>4153</td><td>4157</td><td>4159</td><td>4177</td><td>4201</td><td>4211</td><td>4217</td><td>4219</td><td>4229</td><td>4231</td></tr><tr><td>581–600</td><td>4241</td><td>4243</td><td>4253</td><td>4259</td><td>4261</td><td>4271</td><td>4273</td><td>4283</td><td>4289</td><td>4297</td><td>4327</td><td>4337</td><td>4339</td><td>4349</td><td>4357</td><td>4363</td><td>4373</td><td>4391</td><td>4397</td><td>4409</td></tr><tr><td>601–620</td><td>4421</td><td>4423</td><td>4441</td><td>4447</td><td>4451</td><td>4457</td><td>4463</td><td>4481</td><td>4483</td><td>4493</td><td>4507</td><td>4513</td><td>4517</td><td>4519</td><td>4523</td><td>4547</td><td>4549</td><td>4561</td><td>4567</td><td>4583</td></tr><tr><td>621–640</td><td>4591</td><td>4597</td><td>4603</td><td>4621</td><td>4637</td><td>4639</td><td>4643</td><td>4649</td><td>4651</td><td>4657</td><td>4663</td><td>4673</td><td>4679</td><td>4691</td><td>4703</td><td>4721</td><td>4723</td><td>4729</td><td>4733</td><td>4751</td></tr><tr><td>641–660</td><td>4759</td><td>4783</td><td>4787</td><td>4789</td><td>4793</td><td>4799</td><td>4801</td><td>4813</td><td>4817</td><td>4831</td><td>4861</td><td>4871</td><td>4877</td><td>4889</td><td>4903</td><td>4909</td><td>4919</td><td>4931</td><td>4933</td><td>4937</td></tr><tr><td>661–680</td><td>4943</td><td>4951</td><td>4957</td><td>4967</td><td>4969</td><td>4973</td><td>4987</td><td>4993</td><td>4999</td><td>5003</td><td>5009</td><td>5011</td><td>5021</td><td>5023</td><td>5039</td><td>5051</td><td>5059</td><td>5077</td><td>5081</td><td>5087</td></tr><tr><td>681–700</td><td>5099</td><td>5101</td><td>5107</td><td>5113</td><td>5119</td><td>5147</td><td>5153</td><td>5167</td><td>5171</td><td>5179</td><td>5189</td><td>5197</td><td>5209</td><td>5227</td><td>5231</td><td>5233</td><td>5237</td><td>5261</td><td>5273</td><td>5279</td></tr><tr><td>701–720</td><td>5281</td><td>5297</td><td>5303</td><td>5309</td><td>5323</td><td>5333</td><td>5347</td><td>5351</td><td>5381</td><td>5387</td><td>5393</td><td>5399</td><td>5407</td><td>5413</td><td>5417</td><td>5419</td><td>5431</td><td>5437</td><td>5441</td><td>5443</td></tr><tr><td>721–740</td><td>5449</td><td>5471</td><td>5477</td><td>5479</td><td>5483</td><td>5501</td><td>5503</td><td>5507</td><td>5519</td><td>5521</td><td>5527</td><td>5531</td><td>5557</td><td>5563</td><td>5569</td><td>5573</td><td>5581</td><td>5591</td><td>5623</td><td>5639</td></tr><tr><td>741–760</td><td>5641</td><td>5647</td><td>5651</td><td>5653</td><td>5657</td><td>5659</td><td>5669</td><td>5683</td><td>5689</td><td>5693</td><td>5701</td><td>5711</td><td>5717</td><td>5737</td><td>5741</td><td>5743</td><td>5749</td><td>5779</td><td>5783</td><td>5791</td></tr><tr><td>761–780</td><td>5801</td><td>5807</td><td>5813</td><td>5821</td><td>5827</td><td>5839</td><td>5843</td><td>5849</td><td>5851</td><td>5857</td><td>5861</td><td>5867</td><td>5869</td><td>5879</td><td>5881</td><td>5897</td><td>5903</td><td>5923</td><td>5927</td><td>5939</td></tr><tr><td>781–800</td><td>5953</td><td>5981</td><td>5987</td><td>6007</td><td>6011</td><td>6029</td><td>6037</td><td>6043</td><td>6047</td><td>6053</td><td>6067</td><td>6073</td><td>6079</td><td>6089</td><td>6091</td><td>6101</td><td>6113</td><td>6121</td><td>6131</td><td>6133</td></tr><tr><td>801–820</td><td>6143</td><td>6151</td><td>6163</td><td>6173</td><td>6197</td><td>6199</td><td>6203</td><td>6211</td><td>6217</td><td>6221</td><td>6229</td><td>6247</td><td>6257</td><td>6263</td><td>6269</td><td>6271</td><td>6277</td><td>6287</td><td>6299</td><td>6301</td></tr><tr><td>821–840</td><td>6311</td><td>6317</td><td>6323</td><td>6329</td><td>6337</td><td>6343</td><td>6353</td><td>6359</td><td>6361</td><td>6367</td><td>6373</td><td>6379</td><td>6389</td><td>6397</td><td>6421</td><td>6427</td><td>6449</td><td>6451</td><td>6469</td><td>6473</td></tr><tr><td>841–860</td><td>6481</td><td>6491</td><td>6521</td><td>6529</td><td>6547</td><td>6551</td><td>6553</td><td>6563</td><td>6569</td><td>6571</td><td>6577</td><td>6581</td><td>6599</td><td>6607</td><td>6619</td><td>6637</td><td>6653</td><td>6659</td><td>6661</td><td>6673</td></tr><tr><td>861–880</td><td>6679</td><td>6689</td><td>6691</td><td>6701</td><td>6703</td><td>6709</td><td>6719</td><td>6733</td><td>6737</td><td>6761</td><td>6763</td><td>6779</td><td>6781</td><td>6791</td><td>6793</td><td>6803</td><td>6823</td><td>6827</td><td>6829</td><td>6833</td></tr><tr><td>881–900</td><td>6841</td><td>6857</td><td>6863</td><td>6869</td><td>6871</td><td>6883</td><td>6899</td><td>6907</td><td>6911</td><td>6917</td><td>6947</td><td>6949</td><td>6959</td><td>6961</td><td>6967</td><td>6971</td><td>6977</td><td>6983</td><td>6991</td><td>6997</td></tr><tr><td>901–920</td><td>7001</td><td>7013</td><td>7019</td><td>7027</td><td>7039</td><td>7043</td><td>7057</td><td>7069</td><td>7079</td><td>7103</td><td>7109</td><td>7121</td><td>7127</td><td>7129</td><td>7151</td><td>7159</td><td>7177</td><td>7187</td><td>7193</td><td>7207</td></tr><tr><td>921–940</td><td>7211</td><td>7213</td><td>7219</td><td>7229</td><td>7237</td><td>7243</td><td>7247</td><td>7253</td><td>7283</td><td>7297</td><td>7307</td><td>7309</td><td>7321</td><td>7331</td><td>7333</td><td>7349</td><td>7351</td><td>7369</td><td>7393</td><td>7411</td></tr><tr><td>941–960</td><td>7417</td><td>7433</td><td>7451</td><td>7457</td><td>7459</td><td>7477</td><td>7481</td><td>7487</td><td>7489</td><td>7499</td><td>7507</td><td>7517</td><td>7523</td><td>7529</td><td>7537</td><td>7541</td><td>7547</td><td>7549</td><td>7559</td><td>7561</td></tr><tr><td>961–980</td><td>7573</td><td>7577</td><td>7583</td><td>7589</td><td>7591</td><td>7603</td><td>7607</td><td>7621</td><td>7639</td><td>7643</td><td>7649</td><td>7669</td><td>7673</td><td>7681</td><td>7687</td><td>7691</td><td>7699</td><td>7703</td><td>7717</td><td>7723</td></tr><tr><td>981–1000</td><td>7727</td><td>7741</td><td>7753</td><td>7757</td><td>7759</td><td>7789</td><td>7793</td><td>7817</td><td>7823</td><td>7829</td><td>7841</td><td>7853</td><td>7867</td><td>7873</td><td>7877</td><td>7879</td><td>7883</td><td>7901</td><td>7907</td><td>7919</td></tr></tbody></table>

(sequence A000040 in the OEIS).

The Goldbach conjecture verification project reports that it has computed all primes smaller than 4×1018. That means 95,676,260,903,887,607 primes (nearly 1017), but they were not stored. There are known formulae to evaluate the prime-counting function (the number of primes smaller than a given value) faster than computing the primes. This has been used to compute that there are 1,925,320,391,606,803,968,923 primes (roughly 2×1021) smaller than 1023. A different computation found that there are 18,435,599,767,349,200,867,866 primes (roughly 2×1022) smaller than 1024, if the Riemann hypothesis is true.

Lists of primes by type

List of prime numbers

Below are listed the first prime numbers of many named forms and types. More details are in the article for the name. n is a natural number (including 0) in the definitions.

Balanced primes

Balanced primes are primes with equal-sized prime gaps before and after them, making them the arithmetic mean of their next larger and next smaller prime.

• (/wiki/5(number)), (/wiki/53(number)), (/wiki/157(number)), (/wiki/173(number)), 211, (/wiki/257(number)), 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903, 2963, 3307, 3313, 3637, 3733, 4013, 4409, 4457, 4597, 4657, 4691, 4993, 5107, 5113, 5303, 5387, 5393 (OEIS: A006562).

Bell primes

Bell primes are primes that are also the number of partitions of some finite set.

(/wiki/2), (/wiki/5), (/wiki/877(number)), 27644437, 35742549198872617291353508656626642567, 359334085968622831041960188598043661065388726959079837. The next term has 6,539 digits. (OEIS: A051131)

Chen primes

Chen primes are primes p such that p+2 is either a prime or semiprime.

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/11(number)), (/wiki/13(number)), (/wiki/17(number)), (/wiki/19(number)), (/wiki/23(number)), (/wiki/29(number)), (/wiki/31(number)), (/wiki/37(number)), (/wiki/41(number)), (/wiki/47(number)), (/wiki/53(number)), (/wiki/59(number)), (/wiki/67(number)), (/wiki/71(number)), (/wiki/83(number)), (/wiki/89(number)), (/wiki/101(number)), (/wiki/107(number)), (/wiki/109(number)), (/wiki/113(number)), (/wiki/127(number)), (/wiki/131(number)), (/wiki/137(number)), (/wiki/139(number)), (/wiki/149(number)), (/wiki/157(number)), (/wiki/167(number)), (/wiki/179(number)), (/wiki/181(number)), (/wiki/191(number)), (/wiki/197(number)), (/wiki/199(number)), (/wiki/211(number)), (/wiki/227(number)), (/wiki/233(number)), (/wiki/239(number)), (/wiki/251(number)), (/wiki/257(number)), (/wiki/263(number)), (/wiki/269(number)), (/wiki/281(number)), (/wiki/293(number)), (/wiki/307(number)), (/wiki/311(number)), (/wiki/317(number)), (/wiki/337(number)), (/wiki/347(number)), (/wiki/353(number)), (/wiki/359(number)), (/wiki/379(number)), (/wiki/389(number)), (/wiki/401(number)), (/wiki/409(number)) (OEIS: A109611)

Circular primes

A circular prime is a number that remains prime on any cyclic rotation of its base 10 digits.

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/11(number)), (/wiki/13(number)), (/wiki/17(number)), (/wiki/31(number)), (/wiki/37(number)), (/wiki/71(number)), (/wiki/73(number)), (/wiki/79(number)), (/wiki/97(number)), (/wiki/113(number)), (/wiki/131(number)), (/wiki/197(number)), (/wiki/199(number)), (/wiki/311(number)), (/wiki/337(number)), (/wiki/373(number)), (/wiki/719(number)), (/wiki/733(number)), (/wiki/919(number)), (/wiki/971(number)), (/wiki/991(number)), (/wiki/1193(number)), (/wiki/1931(number)), (/wiki/3119(number)), (/wiki/3779(number)), (/wiki/7000(number)#7700to7799), (/wiki/7000(number)#7900to7999), (/wiki/9000(number)#9300to9399), (/wiki/9000(number)#9300to9399), (/wiki/10000(number)), (/wiki/10000(number)), (/wiki/10000(number)), (/wiki/30000(number)), (/wiki/30000(number)), (/wiki/70000(number)), (/wiki/90000(number)), (/wiki/90000(number)), (/wiki/90000(number)), (/wiki/90000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)) (OEIS: A068652)

Some sources only include the smallest prime in each cycle. For example, listing 13, but omitting 31.

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/11(number)), (/wiki/13(number)), (/wiki/17(number)), (/wiki/37(number)), (/wiki/79(number)), (/wiki/113(number)), (/wiki/197(number)), (/wiki/199(number)), (/wiki/337(number)), (/wiki/1193(number)), (/wiki/3779(number)), (/wiki/10000(number)), (/wiki/10000(number)), (/wiki/100000(number)), (/wiki/100000(number)), 1111111111111111111, 11111111111111111111111 (OEIS: A016114)

Cluster primes

A cluster prime is a prime p such that every even natural number k ≤ p − 3 is the difference of two primes not exceeding p.

(/wiki/3), (/wiki/5), (/wiki/7), (/wiki/11(number)), (/wiki/13(number)), (/wiki/17(number)), (/wiki/19(number)), (/wiki/23(number)), ... (OEIS: A038134)

All primes between 3 and 89, inclusive, are cluster primes. The first 10 primes that are not cluster primes are:

(/wiki/2), (/wiki/97(number)), (/wiki/127(number)), (/wiki/149(number)), (/wiki/191(number)), (/wiki/211(number)), (/wiki/223(number)), (/wiki/227(number)), (/wiki/229(number)), (/wiki/251(number)).

Cousin primes

Cousin primes are pairs of primes that differ by four.

((/wiki/3), (/wiki/7)), ((/wiki/7), (/wiki/11(number))), ((/wiki/13(number)), (/wiki/17(number))), ((/wiki/19(number)), (/wiki/23(number))), ((/wiki/37(number)), (/wiki/41(number))), ((/wiki/43(number)), (/wiki/47(number))), ((/wiki/67(number)), (/wiki/71(number))), ((/wiki/79(number)), (/wiki/83(number))), ((/wiki/97(number)), (/wiki/101(number))), ((/wiki/103(number)), (/wiki/107(number))), ((/wiki/109(number)), (/wiki/113(number))), ((/wiki/127(number)), (/wiki/131(number))), ((/wiki/163(number)), (/wiki/167(number))), ((/wiki/193(number)), (/wiki/197(number))), ((/wiki/223(number)), (/wiki/227(number))), ((/wiki/229(number)), (/wiki/233(number))), ((/wiki/277(number)), (/wiki/281(number))) (OEIS: A023200, OEIS: A046132)

Cuban primes

Cuban primes are primes ${\displaystyle p}$ of the form ${\displaystyle p=k^{3}-(k-1)^{3},}$ where ${\displaystyle k}$ is a natural number.

(/wiki/7), (/wiki/19(number)), (/wiki/37(number)), (/wiki/61(number)), (/wiki/127(number)), (/wiki/271(number)), (/wiki/331(number)), (/wiki/397(number)), (/wiki/547(number)), (/wiki/631(number)), (/wiki/919(number)), (/wiki/1657(number)), (/wiki/1801(number)), (/wiki/1951(number)), (/wiki/2269(number)), (/wiki/2437(number)), (/wiki/2791(number)), (/wiki/3169(number)), (/wiki/3571(number)), (/wiki/4000(number)#4200to4299), (/wiki/4000(number)#4400to4499), (/wiki/5000(number)#5100to5199), (/wiki/5000(number)#5400to5499), (/wiki/6000(number)#6200to6299), (/wiki/7000(number)#7001to7099), (/wiki/7000(number)#7300to7399), (/wiki/8000(number)#8200to8299), (/wiki/9000(number)#9200to9299), (/wiki/10000(number)), (/wiki/10000(number)), (/wiki/10000(number)), (/wiki/10000(number)), (/wiki/10000(number)), (/wiki/10000(number)), (/wiki/10000(number)), (/wiki/10000(number)), (/wiki/20000(number)), (/wiki/20000(number)), (/wiki/20000(number)), (/wiki/20000(number)), (/wiki/20000(number)), (/wiki/20000(number)), (/wiki/30000(number)), (/wiki/30000(number)) (OEIS: A002407)

The term is also used to refer to primes ${\displaystyle p}$ of the form ${\displaystyle p=(k^{3}-(k-2)^{3})/2,}$ where ${\displaystyle k}$ is a natural number.

(/wiki/13(number)), (/wiki/109(number)), (/wiki/193(number)), (/wiki/433(number)), (/wiki/769(number)), (/wiki/1201(number)), (/wiki/1453(number)), (/wiki/2029(number)), (/wiki/3469(number)), (/wiki/3889(number)), (/wiki/4000(number)#4800to4899), (/wiki/10000(number)), (/wiki/10000(number)), (/wiki/10000(number)), (/wiki/10000(number)), (/wiki/20000(number)), (/wiki/20000(number)), (/wiki/20000(number)), (/wiki/20000(number)), (/wiki/30000(number)), (/wiki/40000(number)), (/wiki/40000(number)), (/wiki/60000(number)), (/wiki/60000(number)), (/wiki/60000(number)), (/wiki/60000(number)), (/wiki/70000(number)), (/wiki/70000(number)), (/wiki/80000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)) (OEIS: A002648)

Cullen primes

Cullen primes are primes p of the form p=k2k + 1, for some natural number k.

(/wiki/3), 393050634124102232869567034555427371542904833 (OEIS: A050920)

Delicate primes

Delicate primes are those primes that always become a composite number when any of their base 10 digit is changed.

294001, 505447, 584141, 604171, 971767, 1062599, 1282529, 1524181, 2017963, 2474431, 2690201, 3085553, 3326489, 4393139 (OEIS: A050249)

Dihedral primes

Dihedral primes are primes that satisfy 180° rotational symmetry and mirror symmetry on a seven-segment display.

(/wiki/2), (/wiki/5), (/wiki/11(number)), (/wiki/101(number)), (/wiki/181(number)), (/wiki/1181(number)), (/wiki/1000(number)#1800to1899), (/wiki/10000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)), (/wiki/100000(number)) (OEIS: A134996)

Real Eisenstein primes

Real Eisenstein primes are real Eisenstein integers that are irreducible. Equivalently, they are primes of the form 3k − 1, for a positive integer k.

(/wiki/2), (/wiki/5), (/wiki/11(number)), (/wiki/17(number)), (/wiki/23(number)), (/wiki/29(number)), (/wiki/41(number)), (/wiki/47(number)), (/wiki/53(number)), (/wiki/59(number)), (/wiki/71(number)), (/wiki/83(number)), (/wiki/89(number)), (/wiki/101(number)), (/wiki/107(number)), (/wiki/113(number)), (/wiki/131(number)), (/wiki/137(number)), (/wiki/149(number)), (/wiki/167(number)), (/wiki/173(number)), (/wiki/179(number)), (/wiki/191(number)), (/wiki/197(number)), (/wiki/227(number)), (/wiki/233(number)), (/wiki/239(number)), (/wiki/251(number)), (/wiki/257(number)), (/wiki/263(number)), (/wiki/269(number)), (/wiki/281(number)), (/wiki/293(number)), (/wiki/311(number)), (/wiki/317(number)), (/wiki/347(number)), (/wiki/353(number)), (/wiki/359(number)), (/wiki/383(number)), (/wiki/389(number)), (/wiki/401(number)) (OEIS: A003627)

Emirps

Emirps are primes that become a different prime after their base 10 digits have been reversed. The name "emirp" is the reverse of the word "prime".

(/wiki/13(number)), (/wiki/17(number)), (/wiki/31(number)), (/wiki/37(number)), (/wiki/71(number)), (/wiki/73(number)), (/wiki/79(number)), (/wiki/97(number)), (/wiki/107(number)), (/wiki/113(number)), (/wiki/149(number)), (/wiki/157(number)), (/wiki/167(number)), (/wiki/179(number)), (/wiki/199(number)), (/wiki/311(number)), (/wiki/337(number)), (/wiki/347(number)), (/wiki/359(number)), (/wiki/389(number)), (/wiki/701(number)), (/wiki/709(number)), (/wiki/733(number)), (/wiki/739(number)), (/wiki/743(number)), (/wiki/751(number)), (/wiki/761(number)), (/wiki/769(number)), (/wiki/907(number)), (/wiki/937(number)), (/wiki/941(number)), (/wiki/953(number)), (/wiki/967(number)), (/wiki/971(number)), (/wiki/983(number)), (/wiki/991(number)) (OEIS: A006567)

Euclid primes

Euclid primes are primes p such that p−1 is a primorial.

(/wiki/3), (/wiki/7), (/wiki/31(number)), (/wiki/211(number)), (/wiki/2311(number)), 200560490131 (OEIS: A018239)

Euler irregular primes

Euler irregular primes are primes ${\displaystyle p}$ that divide an Euler number ${\displaystyle E{2n},}$ for some ${\displaystyle 0\leq 2n\leq p-3.}$

(/wiki/19(number)), (/wiki/31(number)), (/wiki/43(number)), (/wiki/47(number)), (/wiki/61(number)), (/wiki/67(number)), (/wiki/71(number)), (/wiki/79(number)), (/wiki/101(number)), (/wiki/137(number)), (/wiki/139(number)), (/wiki/149(number)), (/wiki/193(number)), (/wiki/223(number)), (/wiki/241(number)), (/wiki/251(number)), (/wiki/263(number)), (/wiki/277(number)), (/wiki/307(number)), (/wiki/311(number)), (/wiki/349(number)), (/wiki/353(number)), (/wiki/359(number)), (/wiki/373(number)), (/wiki/379(number)), (/wiki/419(number)), (/wiki/433(number)), (/wiki/461(number)), (/wiki/463(number)), (/wiki/491(number)), (/wiki/509(number)), (/wiki/541(number)), (/wiki/563(number)), (/wiki/571(number)), (/wiki/577(number)), (/wiki/587(number)) (OEIS: A120337)

Euler (p, p − 3) irregular primes

Euler (p, p - 3) irregular primes are primes p that divide the (p + 3)rd Euler number.

(/wiki/149(number)), (/wiki/241(number)), (/wiki/1000000(number)) (OEIS: A198245)

Factorial primes

Factorial primes are of the form n! ± 1.

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/23(number)), (/wiki/719(number)), (/wiki/5000(number)#5001to5000), (/wiki/10000000(number)), (/wiki/100000000(number)), 87178291199, 10888869450418352160768000001, 265252859812191058636308479999999, 263130836933693530167218012159999999, 8683317618811886495518194401279999999 (OEIS: A088054)

Fermat primes

Fermat primes are primes p of the form p = 22k + 1, for a non-negative integer k. As of June 2024 only five Fermat primes have been discovered.

(/wiki/3), (/wiki/5), (/wiki/17(number)), (/wiki/257(number)), (/wiki/65537(number)) (OEIS: A019434)

Generalized Fermat primes

Generalized Fermat primes are primes p of the form p = a2k + 1, for a non-negative integer k and even natural number a.

<table><thead><tr><th>{\displaystyle a}</th><th>Generalized Fermat primes with base a</th></tr></thead><tbody><tr><td>2</td><td>3, 5, 17, 257, 65537, ... (OEIS: A019434)</td></tr><tr><td>4</td><td>5, 17, 257, 65537, ...</td></tr><tr><td>6</td><td>7, 37, 1297, ...</td></tr><tr><td>8</td><td>(none exist)</td></tr><tr><td>10</td><td>11, 101, ...</td></tr><tr><td>12</td><td>13, ...</td></tr><tr><td>14</td><td>197, ...</td></tr><tr><td>16</td><td>17, 257, 65537, ...</td></tr><tr><td>18</td><td>19, ...</td></tr><tr><td>20</td><td>401, 160001, ...</td></tr><tr><td>22</td><td>23, ...</td></tr><tr><td>24</td><td>577, 331777, ...</td></tr></tbody></table>

Fibonacci primes

Fibonacci primes are primes that appear in the Fibonacci sequence.

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/13(number)), (/wiki/89(number)), (/wiki/233(number)), (/wiki/1597(number)), 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917 (OEIS: A005478)

Fortunate primes

Fortunate primes are primes that are also Fortunate numbers. There are no known composite Fortunate numbers.

(/wiki/3), (/wiki/5), (/wiki/7), (/wiki/13(number)), (/wiki/17(number)), (/wiki/19(number)), (/wiki/23(number)), (/wiki/37(number)), (/wiki/47(number)), (/wiki/59(number)), (/wiki/61(number)), (/wiki/67(number)), (/wiki/71(number)), (/wiki/79(number)), (/wiki/89(number)), (/wiki/101(number)), (/wiki/103(number)), (/wiki/107(number)), (/wiki/109(number)), (/wiki/127(number)), (/wiki/151(number)), (/wiki/157(number)), (/wiki/163(number)), (/wiki/167(number)), (/wiki/191(number)), (/wiki/197(number)), (/wiki/199(number)), (/wiki/223(number)), (/wiki/229(number)), (/wiki/233(number)), (/wiki/239(number)), (/wiki/271(number)), (/wiki/277(number)), (/wiki/283(number)), (/wiki/293(number)), (/wiki/307(number)), (/wiki/311(number)), (/wiki/313(number)), (/wiki/331(number)), (/wiki/353(number)), (/wiki/373(number)), (/wiki/379(number)), (/wiki/383(number)), (/wiki/397(number)) (OEIS: A046066)

Gaussian primes

Gaussian primes are primes p of the form p = 4k + 3, for a non-negative integer k.

(/wiki/3), (/wiki/7), (/wiki/11(number)), (/wiki/19(number)), (/wiki/23(number)), (/wiki/31(number)), (/wiki/43(number)), (/wiki/47(number)), (/wiki/59(number)), (/wiki/67(number)), (/wiki/71(number)), (/wiki/79(number)), (/wiki/83(number)), (/wiki/103(number)), (/wiki/107(number)), (/wiki/127(number)), (/wiki/131(number)), (/wiki/139(number)), (/wiki/151(number)), (/wiki/163(number)), (/wiki/167(number)), (/wiki/179(number)), (/wiki/191(number)), (/wiki/199(number)), (/wiki/211(number)), (/wiki/223(number)), (/wiki/227(number)), (/wiki/239(number)), (/wiki/251(number)), (/wiki/263(number)), (/wiki/271(number)), (/wiki/283(number)), (/wiki/307(number)), (/wiki/311(number)), (/wiki/331(number)), (/wiki/347(number)), (/wiki/359(number)), (/wiki/367(number)), (/wiki/379(number)), (/wiki/383(number)), (/wiki/419(number)), (/wiki/431(number)), (/wiki/439(number)), (/wiki/443(number)), (/wiki/463(number)), (/wiki/467(number)), (/wiki/479(number)), (/wiki/487(number)), (/wiki/491(number)), (/wiki/499(number)), (/wiki/503(number)) (OEIS: A002145)

Good primes

Good primes are primes p satisfying ab < p2, for all primes a and b such that a,b < p

(/wiki/5), (/wiki/11(number)), (/wiki/17(number)), (/wiki/29(number)), (/wiki/37(number)), (/wiki/41(number)), (/wiki/53(number)), (/wiki/59(number)), (/wiki/67(number)), (/wiki/71(number)), (/wiki/97(number)), (/wiki/101(number)), (/wiki/127(number)), (/wiki/149(number)), (/wiki/179(number)), (/wiki/191(number)), (/wiki/223(number)), (/wiki/227(number)), (/wiki/251(number)), (/wiki/257(number)), (/wiki/269(number)), (/wiki/307(number)) (OEIS: A028388)

Happy primes

Happy primes are primes that are also happy numbers.

(/wiki/7), (/wiki/13(number)), (/wiki/19(number)), (/wiki/23(number)), (/wiki/31(number)), (/wiki/79(number)), (/wiki/97(number)), (/wiki/103(number)), (/wiki/109(number)), (/wiki/139(number)), (/wiki/167(number)), (/wiki/193(number)), (/wiki/239(number)), (/wiki/263(number)), (/wiki/293(number)), (/wiki/313(number)), (/wiki/331(number)), (/wiki/367(number)), (/wiki/379(number)), (/wiki/383(number)), (/wiki/397(number)), (/wiki/409(number)), (/wiki/487(number)), (/wiki/563(number)), (/wiki/617(number)), (/wiki/653(number)), (/wiki/673(number)), (/wiki/683(number)), (/wiki/709(number)), (/wiki/739(number)), (/wiki/761(number)), (/wiki/863(number)), (/wiki/881(number)), (/wiki/907(number)), (/wiki/937(number)), (/wiki/1009(number)), (/wiki/1033(number)), (/wiki/1039(number)), (/wiki/1093(number)) (OEIS: A035497)

Harmonic primes

Harmonic primes are primes p for which there are no solutions to Hk ≡ 0 (mod p) and Hk ≡ −ωp (mod p), for 1 ≤ k ≤ p−2, where Hk denotes the k-th harmonic number and ωp denotes the Wolstenholme quotient.

(/wiki/5), (/wiki/13(number)), (/wiki/17(number)), (/wiki/23(number)), (/wiki/41(number)), (/wiki/67(number)), (/wiki/73(number)), (/wiki/79(number)), (/wiki/107(number)), (/wiki/113(number)), (/wiki/139(number)), (/wiki/149(number)), (/wiki/157(number)), (/wiki/179(number)), (/wiki/191(number)), (/wiki/193(number)), (/wiki/223(number)), (/wiki/239(number)), (/wiki/241(number)), (/wiki/251(number)), (/wiki/263(number)), (/wiki/277(number)), (/wiki/281(number)), (/wiki/293(number)), (/wiki/307(number)), (/wiki/311(number)), (/wiki/317(number)), (/wiki/331(number)), (/wiki/337(number)), (/wiki/349(number)) (OEIS: A092101)

Higgs primes

Higgs primes are primes p for which p − 1 divides the square of the product of all smaller Higgs primes.

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/11(number)), (/wiki/13(number)), (/wiki/19(number)), (/wiki/23(number)), (/wiki/29(number)), (/wiki/31(number)), (/wiki/37(number)), (/wiki/43(number)), (/wiki/47(number)), (/wiki/53(number)), (/wiki/59(number)), (/wiki/61(number)), (/wiki/67(number)), (/wiki/71(number)), (/wiki/79(number)), (/wiki/101(number)), (/wiki/107(number)), (/wiki/127(number)), (/wiki/131(number)), (/wiki/139(number)), (/wiki/149(number)), (/wiki/151(number)), (/wiki/157(number)), (/wiki/173(number)), (/wiki/181(number)), (/wiki/191(number)), (/wiki/197(number)), (/wiki/199(number)), (/wiki/211(number)), (/wiki/223(number)), (/wiki/229(number)), (/wiki/263(number)), (/wiki/269(number)), (/wiki/277(number)), (/wiki/283(number)), (/wiki/311(number)), (/wiki/317(number)), (/wiki/331(number)), (/wiki/347(number)), (/wiki/349(number)) (OEIS: A007459)

Highly cototient primes

Highly cototient primes are primes that are a cototient more often than any integer below it except 1.

(/wiki/2), (/wiki/23(number)), (/wiki/47(number)), (/wiki/59(number)), (/wiki/83(number)), (/wiki/89(number)), (/wiki/113(number)), (/wiki/167(number)), (/wiki/269(number)), (/wiki/389(number)), (/wiki/419(number)), (/wiki/509(number)), (/wiki/659(number)), (/wiki/839(number)), (/wiki/1049(number)), (/wiki/1259(number)), (/wiki/1889(number)) (OEIS: A105440)

Home primes

For n ≥ 2, write the prime factorization of n in base 10 and concatenate the factors; iterate until a prime is reached.

For a non-negative integer, its home prime is obtained by concatenating its prime factors in increasing order repeatedly, until a prime is achieved.

2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, 11, 223, 13, 13367, 1129, 31636373, 17, 233, 19, 3318308475676071413, 37, 211, 23, 331319, 773, 3251, 13367, 227, 29, 547, 31, 241271, 311, 31397, 1129, 71129, 37, 373, 313, 3314192745739, 41, 379, 43, 22815088913, 3411949, 223, 47, 6161791591356884791277 (OEIS: A037274)

Irregular primes

Irregular primes are odd primes p that divide the class number of the p-th cyclotomic field.

(/wiki/37(number)), (/wiki/59(number)), (/wiki/67(number)), (/wiki/101(number)), (/wiki/103(number)), (/wiki/131(number)), (/wiki/149(number)), (/wiki/157(number)), (/wiki/233(number)), (/wiki/257(number)), (/wiki/263(number)), (/wiki/271(number)), (/wiki/283(number)), (/wiki/293(number)), (/wiki/307(number)), (/wiki/311(number)), (/wiki/347(number)), (/wiki/353(number)), (/wiki/379(number)), (/wiki/389(number)), (/wiki/401(number)), (/wiki/409(number)), (/wiki/421(number)), (/wiki/433(number)), (/wiki/461(number)), (/wiki/463(number)), (/wiki/467(number)), (/wiki/491(number)), (/wiki/523(number)), (/wiki/541(number)), (/wiki/547(number)), (/wiki/557(number)), (/wiki/577(number)), (/wiki/587(number)), (/wiki/593(number)), (/wiki/607(number)), (/wiki/613(number)) (OEIS: A000928)

(p, p − 3) irregular primes

The (p, p - 3) irregular primes are primes p such that (p, p − 3) is an irregular pair.

(/wiki/16843(number)), 2124679 (OEIS: A088164)

(p, p − 5) irregular primes

The (p, p - 5) irregular primes are primes p such that (p, p − 5) is an irregular pair.

(p, p − 9) irregular primes

The (p, p - 9) irregular primes are primes p such that (p, p − 9) is an irregular pair.

(/wiki/67(number)), (/wiki/877(number)) (OEIS: A212557)

Isolated primes

Isolated primes are primes p such that both p − 2 and p + 2 are both composite.

(/wiki/2), (/wiki/23(number)), (/wiki/37(number)), (/wiki/47(number)), (/wiki/53(number)), (/wiki/67(number)), (/wiki/79(number)), (/wiki/83(number)), (/wiki/89(number)), (/wiki/97(number)), (/wiki/113(number)), (/wiki/127(number)), (/wiki/131(number)), (/wiki/157(number)), (/wiki/163(number)), (/wiki/167(number)), (/wiki/173(number)), (/wiki/211(number)), (/wiki/223(number)), (/wiki/233(number)), (/wiki/251(number)), (/wiki/257(number)), (/wiki/263(number)), (/wiki/277(number)), (/wiki/293(number)), (/wiki/307(number)), (/wiki/317(number)), (/wiki/331(number)), (/wiki/337(number)), (/wiki/353(number)), (/wiki/359(number)), (/wiki/367(number)), (/wiki/373(number)), (/wiki/379(number)), (/wiki/383(number)), (/wiki/389(number)), (/wiki/397(number)), (/wiki/401(number)), (/wiki/409(number)), (/wiki/439(number)), (/wiki/443(number)), (/wiki/449(number)), (/wiki/457(number)), (/wiki/467(number)), (/wiki/479(number)), (/wiki/487(number)), (/wiki/491(number)), (/wiki/499(number)), (/wiki/503(number)), (/wiki/509(number)), (/wiki/541(number)), (/wiki/547(number)), (/wiki/557(number)), (/wiki/563(number)), (/wiki/577(number)), (/wiki/587(number)), (/wiki/593(number)), (/wiki/607(number)), (/wiki/613(number)), (/wiki/631(number)), (/wiki/647(number)), (/wiki/653(number)), (/wiki/673(number)), (/wiki/677(number)), (/wiki/683(number)), (/wiki/691(number)), (/wiki/701(number)), (/wiki/709(number)), (/wiki/719(number)), (/wiki/727(number)), (/wiki/733(number)), (/wiki/739(number)), (/wiki/743(number)), (/wiki/751(number)), (/wiki/757(number)), (/wiki/761(number)), (/wiki/769(number)), (/wiki/773(number)), (/wiki/787(number)), (/wiki/797(number)), (/wiki/839(number)), (/wiki/853(number)), (/wiki/863(number)), (/wiki/877(number)), (/wiki/887(number)), (/wiki/907(number)), (/wiki/911(number)), (/wiki/919(number)), (/wiki/929(number)), (/wiki/937(number)), (/wiki/941(number)), (/wiki/947(number)), (/wiki/953(number)), (/wiki/967(number)), (/wiki/971(number)), (/wiki/977(number)), (/wiki/983(number)), (/wiki/991(number)), (/wiki/997(number)) (OEIS: A007510)

Leyland primes

Leyland primes are primes p of the form p = ab + ba, where a and b are integers larger than one.

(/wiki/17(number)), (/wiki/593(number)), 32993, 2097593, 8589935681, 59604644783353249, 523347633027360537213687137, 43143988327398957279342419750374600193 (OEIS: A094133)

Long primes

Long primes, or full reptend primes, are odd primes p for which ${\displaystyle (10^{p-1}-1)/p}$ is a cyclic number. Bases other than 10 are also used.

(/wiki/7), (/wiki/17(number)), (/wiki/19(number)), (/wiki/23(number)), (/wiki/29(number)), (/wiki/47(number)), (/wiki/59(number)), (/wiki/61(number)), (/wiki/97(number)), (/wiki/109(number)), (/wiki/113(number)), (/wiki/131(number)), (/wiki/149(number)), (/wiki/167(number)), (/wiki/179(number)), (/wiki/181(number)), (/wiki/193(number)), (/wiki/223(number)), (/wiki/229(number)), (/wiki/233(number)), (/wiki/257(number)), (/wiki/263(number)), (/wiki/269(number)), (/wiki/313(number)), (/wiki/337(number)), (/wiki/367(number)), (/wiki/379(number)), (/wiki/383(number)), (/wiki/389(number)), (/wiki/419(number)), (/wiki/433(number)), (/wiki/461(number)), (/wiki/487(number)), (/wiki/491(number)), (/wiki/499(number)), (/wiki/503(number)), (/wiki/509(number)), (/wiki/541(number)), (/wiki/571(number)), (/wiki/577(number)), (/wiki/593(number)) (OEIS: A001913)

Lucas primes

Lucas primes are primes that appear in the Lucas sequence.

(/wiki/2), (/wiki/3), (/wiki/7), (/wiki/11(number)), (/wiki/29(number)), (/wiki/47(number)), (/wiki/199(number)), (/wiki/521(number)), (/wiki/2207(number)), (/wiki/3571(number)), 9349, 3010349, 54018521, 370248451, 6643838879, 119218851371, 5600748293801, 688846502588399, 32361122672259149 (OEIS: A005479)

Lucky primes

Lucky primes are primes that are also lucky numbers.

(/wiki/3), (/wiki/7), (/wiki/13(number)), (/wiki/31(number)), (/wiki/37(number)), (/wiki/43(number)), (/wiki/67(number)), (/wiki/73(number)), (/wiki/79(number)), (/wiki/127(number)), (/wiki/151(number)), (/wiki/163(number)), (/wiki/193(number)), (/wiki/211(number)), (/wiki/223(number)), (/wiki/241(number)), (/wiki/283(number)), (/wiki/307(number)), (/wiki/331(number)), (/wiki/349(number)), (/wiki/367(number)), (/wiki/409(number)), (/wiki/421(number)), (/wiki/433(number)), (/wiki/463(number)), (/wiki/487(number)), (/wiki/541(number)), (/wiki/577(number)), (/wiki/601(number)), (/wiki/613(number)), (/wiki/619(number)), (/wiki/631(number)), (/wiki/643(number)), (/wiki/673(number)), (/wiki/727(number)), (/wiki/739(number)), (/wiki/769(number)), (/wiki/787(number)), (/wiki/823(number)), (/wiki/883(number)), (/wiki/937(number)), (/wiki/991(number)), (/wiki/997(number)) (OEIS: A031157)

Mersenne primes

Mersenne primes are primes p of the form p = 2k − 1, for some non-negative integer k.

(/wiki/3), (/wiki/7), (/wiki/31(number)), (/wiki/127(number)), (/wiki/8191(number)), (/wiki/131071(number)), (/wiki/524287(number)), (/wiki/2147483647), 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727 (OEIS: A000668)

As of 2024, there are 52 known Mersenne primes. The 13th, 14th, and 52nd have respectively 157, 183, and 41,024,320 digits. The largest known primes since 1992 have all been Mersenne primes, with the largest as of 2026 being 2136,279,841−1, the 52nd Mersenne prime.

Mersenne divisors

Mersenne divisors are primes that divide 2k − 1, for some prime k. Every Mersenne prime p is also a Mersenne divisor, with k = p.

3, 7, 23, 31, 47, 89, 127, 167, 223, 233, 263, 359, 383, 431, 439, 479, 503, 719, 839, 863, 887, 983, 1103, 1319, 1367, 1399, 1433, 1439, 1487, 1823, 1913, 2039, 2063, 2089, 2207, 2351, 2383, 2447, 2687, 2767, 2879, 2903, 2999, 3023, 3119, 3167, 3343 (OEIS: A122094)

Mersenne prime exponents

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/13(number)), (/wiki/17(number)), (/wiki/19(number)), (/wiki/31(number)), (/wiki/61(number)), (/wiki/89(number)), (/wiki/107(number)), (/wiki/127(number)), (/wiki/521(number)), (/wiki/607(number)), (/wiki/1279(number)), (/wiki/2203(number)), (/wiki/2281(number)), (/wiki/3217(number)), 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, (/wiki/43,112,609(number)), 57885161, 74207281, 77232917 (OEIS: A000043)

As of September 2025, two more are known to be in the sequence, but it is not known whether they are the next: 82589933, 136279841

Double Mersenne primes

A subset of Mersenne primes of the form 22p−1 − 1 for prime p.

(/wiki/7), (/wiki/127(number)), (/wiki/2147483647), 170141183460469231731687303715884105727 (primes in OEIS: A077586)

Generalized repunit primes

Of the form (an − 1) / (a − 1) for fixed integer a.

For a = 2, these are the Mersenne primes, while for a = 10 they are the repunit primes. For other small a, they are given below:

a = 3: (/wiki/13(number)), (/wiki/1093(number)), 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 (OEIS: A076481)

a = 5: (/wiki/31(number)), 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531 (OEIS: A086122)

a = 6: (/wiki/7), (/wiki/43(number)), 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371 (OEIS: A165210)

a = 7: 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457

a = 8: (/wiki/73(number)) (the only prime for a = 8)

Other generalizations and variations

Many generalizations of Mersenne primes have been defined. This include the following:

• Primes of the form bn − (b − 1)n, including the Mersenne primes and the cuban primes as special cases • Williams primes, of the form (b − 1)·bn − 1

Mills primes

Of the form ⌊θ3n⌋, where θ is Mills' constant. This form is prime for all positive integers n.

(/wiki/2), (/wiki/11(number)), (/wiki/1361(number)), 2521008887, 16022236204009818131831320183 (OEIS: A051254)

Minimal primes

Primes for which there is no shorter sub-sequence of the decimal digits that form a prime. There are exactly 26 minimal primes:

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/11(number)), (/wiki/19(number)), (/wiki/41(number)), (/wiki/61(number)), (/wiki/89(number)), (/wiki/409(number)), (/wiki/449(number)), (/wiki/499(number)), (/wiki/881(number)), (/wiki/991(number)), 6469, 6949, (/wiki/9001(number)), 9049, 9649, 9949, 60649, 666649, 946669, 60000049, 66000049, 66600049 (OEIS: A071062)

Newman–Shanks–Williams primes

(/wiki/7), (/wiki/41(number)), (/wiki/239(number)), 9369319, 63018038201, 489133282872437279, 19175002942688032928599 (OEIS: A088165)

Non-generous primes

Primes p for which the least positive primitive root is not a primitive root of p2. Three such primes are known; it is not known whether there are more.

(/wiki/2), 40487, 6692367337 (OEIS: A055578)

Palindromic primes

Primes that remain the same when their decimal digits are read backwards.

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/11(number)), (/wiki/101(number)), (/wiki/131(number)), (/wiki/151(number)), (/wiki/181(number)), (/wiki/191(number)), (/wiki/313(number)), (/wiki/353(number)), (/wiki/373(number)), (/wiki/383(number)), (/wiki/727(number)), (/wiki/757(number)), (/wiki/787(number)), (/wiki/797(number)), (/wiki/919(number)), (/wiki/929(number)), 10301, 10501, 10601, 11311, 11411, 12421, 12721, 12821, 13331, 13831, 13931, 14341, 14741 (OEIS: A002385)

Palindromic wing primes

Primes of the form ${\displaystyle {\frac {a{\big (}10^{m}-1{\big )}}{9}}\pm b\times 10^{\frac {m-1}{2}}}$ with ${\displaystyle 0\leq a\pm b<10}$. This means all digits except the middle digit are equal.

(/wiki/101(number)), (/wiki/131(number)), (/wiki/151(number)), (/wiki/181(number)), (/wiki/191(number)), (/wiki/313(number)), (/wiki/353(number)), (/wiki/373(number)), (/wiki/383(number)), (/wiki/727(number)), (/wiki/757(number)), (/wiki/787(number)), (/wiki/797(number)), (/wiki/919(number)), (/wiki/929(number)), 11311, 11411, 33533, 77377, 77477, 77977, 1114111, 1117111, 3331333, 3337333, 7772777, 7774777, 7778777, 111181111, 111191111, 777767777, 77777677777, 99999199999 (OEIS: A077798)

Partition primes

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/11(number)), (/wiki/101(number)), 17977, 10619863, 6620830889, 80630964769, 228204732751, 1171432692373, 1398341745571, 10963707205259, 15285151248481, 10657331232548839, 790738119649411319, 18987964267331664557 (OEIS: A049575)

Pell primes

Primes in the Pell number sequence P0 = 0, P1 = 1, Pn = 2Pn−1 + Pn−2.

(/wiki/2), (/wiki/5), (/wiki/29(number)), 5741, 33461, 44560482149, 1746860020068409, 68480406462161287469, 13558774610046711780701, 4125636888562548868221559797461449 (OEIS: A086383)

Permutable primes

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/11(number)), (/wiki/13(number)), (/wiki/17(number)), (/wiki/31(number)), (/wiki/37(number)), (/wiki/71(number)), (/wiki/73(number)), (/wiki/79(number)), (/wiki/97(number)), (/wiki/113(number)), (/wiki/131(number)), (/wiki/199(number)), (/wiki/311(number)), (/wiki/337(number)), (/wiki/373(number)), (/wiki/733(number)), (/wiki/919(number)), (/wiki/991(number)), 1111111111111111111, 11111111111111111111111 (OEIS: A003459)

Perrin primes

Primes in the Perrin number sequence P(0) = 3, P(1) = 0, P(2) = 2, P(n) = P(n−2) + P(n−3).

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/17(number)), (/wiki/29(number)), (/wiki/277(number)), (/wiki/367(number)), (/wiki/853(number)), 14197, 43721, 1442968193, 792606555396977, 187278659180417234321, 66241160488780141071579864797 (OEIS: A074788)

Pierpont primes

Of the form 2u3v + 1 for some integers u,v ≥ 0.

These are also class 1- primes.

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/13(number)), (/wiki/17(number)), (/wiki/19(number)), (/wiki/37(number)), (/wiki/73(number)), (/wiki/97(number)), (/wiki/109(number)), (/wiki/163(number)), (/wiki/193(number)), (/wiki/257(number)), (/wiki/433(number)), (/wiki/487(number)), (/wiki/577(number)), (/wiki/769(number)), (/wiki/1153(number)), (/wiki/1297(number)), (/wiki/1459(number)), (/wiki/2593(number)), (/wiki/2917(number)), (/wiki/3457(number)), (/wiki/3889(number)), 10369, 12289, 17497, 18433, 39367, 52489, (/wiki/65537(number)), 139969, 147457 (OEIS: A005109)

Pillai primes

Primes p for which there exist n > 0 such that p divides n! + 1 and n does not divide p − 1.

(/wiki/23(number)), (/wiki/29(number)), (/wiki/59(number)), (/wiki/61(number)), (/wiki/67(number)), (/wiki/71(number)), (/wiki/79(number)), (/wiki/83(number)), (/wiki/109(number)), (/wiki/137(number)), (/wiki/139(number)), (/wiki/149(number)), (/wiki/193(number)), (/wiki/227(number)), (/wiki/233(number)), (/wiki/239(number)), (/wiki/251(number)), (/wiki/257(number)), (/wiki/269(number)), (/wiki/271(number)), (/wiki/277(number)), (/wiki/293(number)), (/wiki/307(number)), (/wiki/311(number)), (/wiki/317(number)), (/wiki/359(number)), (/wiki/379(number)), (/wiki/383(number)), (/wiki/389(number)), (/wiki/397(number)), (/wiki/401(number)), (/wiki/419(number)), (/wiki/431(number)), (/wiki/449(number)), (/wiki/461(number)), (/wiki/463(number)), (/wiki/467(number)), (/wiki/479(number)), (/wiki/499(number)) (OEIS: A063980)

Primes of the form n4 + 1

(/wiki/2), (/wiki/17(number)), (/wiki/257(number)), (/wiki/1297(number)), (/wiki/65537(number)), 160001, 331777, 614657, 1336337, 4477457, 5308417, 8503057, 9834497, 29986577, 40960001, 45212177, 59969537, 65610001, 126247697, 193877777, 303595777, 384160001, 406586897, 562448657, 655360001 (OEIS: A037896)

Primeval primes

Primes for which there are more prime permutations of some or all the decimal digits than for any smaller number.

(/wiki/2), (/wiki/13(number)), (/wiki/37(number)), (/wiki/107(number)), (/wiki/113(number)), (/wiki/137(number)), (/wiki/1013(number)), (/wiki/1237(number)), (/wiki/1367(number)), 10079 (OEIS: A119535)

Primorial primes

(/wiki/3), (/wiki/5), (/wiki/7), (/wiki/29(number)), (/wiki/31(number)), (/wiki/211(number)), (/wiki/2309(number)), (/wiki/2311(number)), 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (union of OEIS: A057705 and OEIS: A018239)

Proth primes

(/wiki/3), (/wiki/5), (/wiki/13(number)), (/wiki/17(number)), (/wiki/41(number)), (/wiki/97(number)), (/wiki/113(number)), (/wiki/193(number)), (/wiki/241(number)), (/wiki/257(number)), (/wiki/353(number)), (/wiki/449(number)), (/wiki/577(number)), (/wiki/641(number)), (/wiki/673(number)), (/wiki/769(number)), (/wiki/929(number)), (/wiki/1153(number)), (/wiki/1217(number)), (/wiki/1409(number)), (/wiki/1601(number)), (/wiki/2113(number)), (/wiki/2689(number)), (/wiki/2753(number)), (/wiki/3137(number)), (/wiki/3329(number)), (/wiki/3457(number)), 4481, 4993, 6529, 7297, 7681, 7937, 9473, 9601, 9857 (OEIS: A080076)

Pythagorean primes

(/wiki/5), (/wiki/13(number)), (/wiki/17(number)), (/wiki/29(number)), (/wiki/37(number)), (/wiki/41(number)), (/wiki/53(number)), (/wiki/61(number)), (/wiki/73(number)), (/wiki/89(number)), (/wiki/97(number)), (/wiki/101(number)), (/wiki/109(number)), (/wiki/113(number)), (/wiki/137(number)), (/wiki/149(number)), (/wiki/157(number)), (/wiki/173(number)), (/wiki/181(number)), (/wiki/193(number)), (/wiki/197(number)), (/wiki/229(number)), (/wiki/233(number)), (/wiki/241(number)), (/wiki/257(number)), (/wiki/269(number)), (/wiki/277(number)), (/wiki/281(number)), (/wiki/293(number)), (/wiki/313(number)), (/wiki/317(number)), (/wiki/337(number)), (/wiki/349(number)), (/wiki/353(number)), (/wiki/373(number)), (/wiki/389(number)), (/wiki/397(number)), (/wiki/401(number)), (/wiki/409(number)), (/wiki/421(number)), (/wiki/433(number)), (/wiki/449(number)) (OEIS: A002144)

Prime quadruplets

((/wiki/5), (/wiki/7), (/wiki/11(number)), (/wiki/13(number))), (11, 13, (/wiki/17(number)), (/wiki/19(number))), ((/wiki/101(number)), (/wiki/103(number)), (/wiki/107(number)), (/wiki/109(number))), ((/wiki/191(number)), (/wiki/193(number)), (/wiki/197(number)), (/wiki/199(number))), ((/wiki/821(number)), (/wiki/823(number)), (/wiki/827(number)), (/wiki/829(number))), ((/wiki/1481(number)), (/wiki/1483(number)), (/wiki/1487(number)), (/wiki/1489(number))), ((/wiki/1871(number)), (/wiki/1873(number)), (/wiki/1877(number)), (/wiki/1879(number))), ((/wiki/2081(number)), (/wiki/2083(number)), (/wiki/2087(number)), (/wiki/2089(number))), ((/wiki/3251(number)), (/wiki/3253(number)), (/wiki/3257(number)), (/wiki/3259(number))), ((/wiki/3461(number)), (/wiki/3463(number)), (/wiki/3467(number)), (/wiki/3469(number))), (5651, 5653, 5657, 5659), (9431, 9433, 9437, 9439) (OEIS: A007530, OEIS: A136720, OEIS: A136721, OEIS: A090258)

Quartan primes

(/wiki/2), (/wiki/17(number)), (/wiki/97(number)), (/wiki/257(number)), (/wiki/337(number)), (/wiki/641(number)), (/wiki/881(number)) (OEIS: A002645)

Ramanujan primes

Integers Rn that are the smallest to give at least n primes from x/2 to x for all x ≥ Rn (all such integers are primes).

(/wiki/2), (/wiki/11(number)), (/wiki/17(number)), (/wiki/29(number)), (/wiki/41(number)), (/wiki/47(number)), (/wiki/59(number)), (/wiki/67(number)), (/wiki/71(number)), (/wiki/97(number)), (/wiki/101(number)), (/wiki/107(number)), (/wiki/127(number)), (/wiki/149(number)), (/wiki/151(number)), (/wiki/167(number)), (/wiki/179(number)), (/wiki/181(number)), (/wiki/227(number)), (/wiki/229(number)), (/wiki/233(number)), (/wiki/239(number)), (/wiki/241(number)), (/wiki/263(number)), (/wiki/269(number)), (/wiki/281(number)), (/wiki/307(number)), (/wiki/311(number)), (/wiki/347(number)), (/wiki/349(number)), (/wiki/367(number)), (/wiki/373(number)), (/wiki/401(number)), (/wiki/409(number)), (/wiki/419(number)), (/wiki/431(number)), (/wiki/433(number)), (/wiki/439(number)), (/wiki/461(number)), (/wiki/487(number)), (/wiki/491(number)) (OEIS: A104272)

Regular primes

Primes p that do not divide the class number of the p-th cyclotomic field.

(/wiki/3), (/wiki/5), (/wiki/7), (/wiki/11(number)), (/wiki/13(number)), (/wiki/17(number)), (/wiki/19(number)), (/wiki/23(number)), (/wiki/29(number)), (/wiki/31(number)), (/wiki/41(number)), (/wiki/43(number)), (/wiki/47(number)), (/wiki/53(number)), (/wiki/61(number)), (/wiki/71(number)), (/wiki/73(number)), (/wiki/79(number)), (/wiki/83(number)), (/wiki/89(number)), (/wiki/97(number)), (/wiki/107(number)), (/wiki/109(number)), (/wiki/113(number)), (/wiki/127(number)), (/wiki/137(number)), (/wiki/139(number)), (/wiki/151(number)), (/wiki/163(number)), (/wiki/167(number)), (/wiki/173(number)), (/wiki/179(number)), (/wiki/181(number)), (/wiki/191(number)), (/wiki/193(number)), (/wiki/197(number)), (/wiki/199(number)), (/wiki/211(number)), (/wiki/223(number)), (/wiki/227(number)), (/wiki/229(number)), (/wiki/239(number)), (/wiki/241(number)), (/wiki/251(number)), (/wiki/269(number)), (/wiki/277(number)), (/wiki/281(number)) (OEIS: A007703)

Repunit primes

(/wiki/11(number)), 1111111111111111111 (19 digits), 11111111111111111111111 (23 digits) (OEIS: A004022)

The next have 317, 1031, 49081, 86453, 109297, and 270343 digits, respectively (OEIS: A004023).

Residue classes of primes

Of the form an + d for fixed integers a and d. Also called primes congruent to d modulo a.

The primes of the form 2n+1 are the odd primes, including all primes other than 2. Some sequences have alternate names: 4n+1 are the Pythagorean primes, 4n+3 are the integer Gaussian primes, and 6n+5 are the Eisenstein primes (with 2 omitted). The classes 10n+d (d = 1, 3, 7, 9) are primes ending in the decimal digit d.

If a and d are relatively prime, the arithmetic progression contains infinitely many primes.

2n+1: (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/11(number)), (/wiki/13(number)), (/wiki/17(number)), (/wiki/19(number)), (/wiki/23(number)), (/wiki/29(number)), (/wiki/31(number)), (/wiki/37(number)), (/wiki/41(number)), (/wiki/43(number)), (/wiki/47(number)), (/wiki/53(number)) (OEIS: A065091) 4n+1: 5, 13, 17, 29, 37, 41, 53, (/wiki/61(number)), (/wiki/73(number)), (/wiki/89(number)), (/wiki/97(number)), (/wiki/101(number)), (/wiki/109(number)), (/wiki/113(number)), (/wiki/137(number)) (OEIS: A002144) 4n+3: 3, 7, 11, 19, 23, 31, 43, 47, (/wiki/59(number)), (/wiki/67(number)), (/wiki/71(number)), (/wiki/79(number)), (/wiki/83(number)), (/wiki/103(number)), (/wiki/107(number)) (OEIS: A002145) 6n+1: 7, 13, 19, 31, 37, 43, 61, 67, 73, 79, 97, 103, 109, (/wiki/127(number)), (/wiki/139(number)) (OEIS: A002476) 6n+5: 5, 11, 17, 23, 29, 41, 47, 53, 59, 71, 83, 89, 101, 107, 113 (OEIS: A007528) 8n+1: 17, 41, 73, 89, 97, 113, 137, (/wiki/193(number)), (/wiki/233(number)), (/wiki/241(number)), (/wiki/257(number)), (/wiki/281(number)), (/wiki/313(number)), (/wiki/337(number)), (/wiki/353(number)) (OEIS: A007519) 8n+3: 3, 11, 19, 43, 59, 67, 83, 107, (/wiki/131(number)), 139, (/wiki/163(number)), (/wiki/179(number)), (/wiki/211(number)), (/wiki/227(number)), (/wiki/251(number)) (OEIS: A007520) 8n+5: 5, 13, 29, 37, 53, 61, 101, 109, (/wiki/149(number)), (/wiki/157(number)), (/wiki/173(number)), (/wiki/181(number)), (/wiki/197(number)), (/wiki/229(number)), (/wiki/269(number)) (OEIS: A007521) 8n+7: 7, 23, 31, 47, 71, 79, 103, 127, (/wiki/151(number)), (/wiki/167(number)), (/wiki/191(number)), (/wiki/199(number)), (/wiki/223(number)), (/wiki/239(number)), (/wiki/263(number)) (OEIS: A007522) 10n+1: 11, 31, 41, 61, 71, 101, 131, 151, 181, 191, 211, 241, 251, (/wiki/271(number)), 281 (OEIS: A030430) 10n+3: 3, 13, 23, 43, 53, 73, 83, 103, 113, 163, 173, 193, 223, 233, 263 (OEIS: A030431) 10n+7: 7, 17, 37, 47, 67, 97, 107, 127, 137, 157, 167, 197, 227, 257, (/wiki/277(number)) (OEIS: A030432) 10n+9: 19, 29, 59, 79, 89, 109, 139, 149, 179, 199, 229, 239, 269, (/wiki/349(number)), (/wiki/359(number)) (OEIS: A030433) 12n+1: 13, 37, 61, 73, 97, 109, 157, 181, 193, 229, 241, 277, 313, 337, 349 (OEIS: A068228) 12n+5: 5, 17, 29, 41, 53, 89, 101, 113, 137, 149, 173, 197, 233, 257, 269 (OEIS: A040117) 12n+7: 7, 19, 31, 43, 67, 79, 103, 127, 139, 151, 163, 199, 211, 223, 271 (OEIS: A068229) 12n+11: 11, 23, 47, 59, 71, 83, 107, 131, 167, 179, 191, 227, 239, 251, 263 (OEIS: A068231)

Safe primes

(/wiki/5), (/wiki/7), (/wiki/11(number)), (/wiki/23(number)), (/wiki/47(number)), (/wiki/59(number)), (/wiki/83(number)), (/wiki/107(number)), (/wiki/167(number)), (/wiki/179(number)), (/wiki/227(number)), (/wiki/263(number)), (/wiki/347(number)), (/wiki/359(number)), (/wiki/383(number)), (/wiki/467(number)), (/wiki/479(number)), (/wiki/503(number)), (/wiki/563(number)), (/wiki/587(number)), (/wiki/719(number)), (/wiki/839(number)), (/wiki/863(number)), (/wiki/887(number)), (/wiki/983(number)), (/wiki/1019(number)), (/wiki/1187(number)), (/wiki/1283(number)), (/wiki/1307(number)), (/wiki/1319(number)), (/wiki/1367(number)), (/wiki/1439(number)), (/wiki/1487(number)), (/wiki/1523(number)), (/wiki/1619(number)), (/wiki/1823(number)), (/wiki/1907(number)) (OEIS: A005385)

Self primes in base 10

Primes that cannot be generated by any integer added to the sum of its decimal digits.

(/wiki/3), (/wiki/5), (/wiki/7), (/wiki/31(number)), (/wiki/53(number)), (/wiki/97(number)), (/wiki/211(number)), (/wiki/233(number)), (/wiki/277(number)), (/wiki/367(number)), (/wiki/389(number)), (/wiki/457(number)), (/wiki/479(number)), (/wiki/547(number)), (/wiki/569(number)), (/wiki/613(number)), (/wiki/659(number)), (/wiki/727(number)), (/wiki/839(number)), (/wiki/883(number)), (/wiki/929(number)), (/wiki/1021(number)), (/wiki/1087(number)), (/wiki/1109(number)), (/wiki/1223(number)), (/wiki/1289(number)), (/wiki/1447(number)), (/wiki/1559(number)), (/wiki/1627(number)), (/wiki/1693(number)), (/wiki/1783(number)), (/wiki/1873(number)) (OEIS: A006378)

Sexy primes

((/wiki/5), (/wiki/11(number))), ((/wiki/7), (/wiki/13(number))), (11, (/wiki/17(number))), (13, (/wiki/19(number))), (17, (/wiki/23(number))), (23, (/wiki/29(number))), ((/wiki/31(number)), (/wiki/37(number))), (37, (/wiki/43(number))), ((/wiki/41(number)), (/wiki/47(number))), (47, (/wiki/53(number))), (53, (/wiki/59(number))), ((/wiki/61(number)), (/wiki/67(number))), (67, (/wiki/73(number))), (73, (/wiki/79(number))), ((/wiki/83(number)), (/wiki/89(number))), ((/wiki/97(number)), (/wiki/103(number))), ((/wiki/101(number)), (/wiki/107(number))), (103, (/wiki/109(number))), (107, (/wiki/113(number))), ((/wiki/131(number)), (/wiki/137(number))), ((/wiki/151(number)), (/wiki/157(number))), (157, (/wiki/163(number))), ((/wiki/167(number)), (/wiki/173(number))), (173, (/wiki/179(number))), ((/wiki/191(number)), (/wiki/197(number))), ((/wiki/193(number)), (/wiki/199(number))) (OEIS: A023201, OEIS: A046117)

Smarandache–Wellin primes

Primes that are the concatenation of the first n primes written in decimal.

(/wiki/2), (/wiki/23(number)), (/wiki/2357(number)) (OEIS: A069151)

The fourth Smarandache-Wellin prime is the 355-digit concatenation of the first 128 primes that end with 719.

Solinas primes

• (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/11(number)), (/wiki/13(number)) (OEIS: A165255) • 232 − 5, the largest prime that fits into 32 bits of memory. • 264 − 59, the largest prime that fits into 64 bits of memory.

Sophie Germain primes

Where p and 2p + 1 are both prime. A Sophie Germain prime has a corresponding safe prime.

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/11(number)), (/wiki/23(number)), (/wiki/29(number)), (/wiki/41(number)), (/wiki/53(number)), (/wiki/83(number)), (/wiki/89(number)), (/wiki/113(number)), (/wiki/131(number)), (/wiki/173(number)), (/wiki/179(number)), (/wiki/191(number)), (/wiki/233(number)), (/wiki/239(number)), (/wiki/251(number)), (/wiki/281(number)), (/wiki/293(number)), (/wiki/359(number)), (/wiki/419(number)), (/wiki/431(number)), (/wiki/443(number)), (/wiki/491(number)), (/wiki/509(number)), (/wiki/593(number)), (/wiki/641(number)), (/wiki/653(number)), (/wiki/659(number)), (/wiki/683(number)), (/wiki/719(number)), (/wiki/743(number)), (/wiki/761(number)), (/wiki/809(number)), (/wiki/911(number)), (/wiki/953(number)) (OEIS: A005384)

Stern primes

Primes that are not the sum of a smaller prime and twice the square of a nonzero integer.

(/wiki/2), (/wiki/3), (/wiki/17(number)), (/wiki/137(number)), (/wiki/227(number)), (/wiki/977(number)), (/wiki/1187(number)), (/wiki/1493(number)) (OEIS: A042978)

As of 2011, these are the only known Stern primes, and possibly the only existing.

Super-primes

Primes with prime-numbered indexes in the sequence of prime numbers (the 2nd, 3rd, 5th, ... prime).

(/wiki/3), (/wiki/5), (/wiki/11(number)), (/wiki/17(number)), (/wiki/31(number)), (/wiki/41(number)), (/wiki/59(number)), (/wiki/67(number)), (/wiki/83(number)), (/wiki/109(number)), (/wiki/127(number)), (/wiki/157(number)), (/wiki/179(number)), (/wiki/191(number)), (/wiki/211(number)), (/wiki/241(number)), (/wiki/277(number)), (/wiki/283(number)), (/wiki/331(number)), (/wiki/353(number)), (/wiki/367(number)), (/wiki/401(number)), (/wiki/431(number)), (/wiki/461(number)), (/wiki/509(number)), (/wiki/547(number)), (/wiki/563(number)), (/wiki/587(number)), (/wiki/599(number)), (/wiki/617(number)), (/wiki/709(number)), (/wiki/739(number)), (/wiki/773(number)), (/wiki/797(number)), (/wiki/859(number)), (/wiki/877(number)), (/wiki/919(number)), (/wiki/967(number)), (/wiki/991(number)) (OEIS: A006450)

Supersingular primes

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/11(number)), (/wiki/13(number)), (/wiki/17(number)), (/wiki/19(number)), (/wiki/23(number)), (/wiki/29(number)), (/wiki/31(number)), (/wiki/41(number)), (/wiki/47(number)), (/wiki/59(number)), (/wiki/71(number)) (OEIS: A002267)

Thabit primes

(/wiki/2), (/wiki/5), (/wiki/11(number)), (/wiki/23(number)), (/wiki/47(number)), (/wiki/191(number)), (/wiki/383(number)), 6143, 786431, 51539607551, 824633720831, 26388279066623, 108086391056891903, 55340232221128654847, 226673591177742970257407 (OEIS: A007505)

(/wiki/7), (/wiki/13(number)), (/wiki/97(number)), (/wiki/193(number)), (/wiki/769(number)), 12289, 786433, 3221225473, 206158430209, 6597069766657 (OEIS: A039687)

Prime triplets

Where (p, p+2, p+6) or (p, p+4, p+6) are all prime.

((/wiki/5), (/wiki/7), (/wiki/11(number))), (7, 11, (/wiki/13(number))), (11, 13, (/wiki/17(number))), (13, 17, (/wiki/19(number))), (17, 19, (/wiki/23(number))), ((/wiki/37(number)), (/wiki/41(number)), (/wiki/43(number))), (41, 43, (/wiki/47(number))), ((/wiki/67(number)), (/wiki/71(number)), (/wiki/73(number))), ((/wiki/97(number)), (/wiki/101(number)), (/wiki/103(number))), (101, 103, (/wiki/107(number))), (103, 107, (/wiki/109(number))), (107, 109, (/wiki/113(number))), ((/wiki/191(number)), (/wiki/193(number)), (/wiki/197(number))), (193, 197, (/wiki/199(number))), ((/wiki/223(number)), (/wiki/227(number)), (/wiki/229(number))), (227, 229, (/wiki/233(number))), ((/wiki/277(number)), (/wiki/281(number)), (/wiki/283(number))), ((/wiki/307(number)), (/wiki/311(number)), (/wiki/313(number))), (311, 313, (/wiki/317(number))), ((/wiki/347(number)), (/wiki/349(number)), (/wiki/353(number))) (OEIS: A007529, OEIS: A098414, OEIS: A098415)

Left-truncatable

Primes that remain prime when the leading decimal digit is successively removed.

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/13(number)), (/wiki/17(number)), (/wiki/23(number)), (/wiki/37(number)), (/wiki/43(number)), (/wiki/47(number)), (/wiki/53(number)), (/wiki/67(number)), (/wiki/73(number)), (/wiki/83(number)), (/wiki/97(number)), (/wiki/113(number)), (/wiki/137(number)), (/wiki/167(number)), (/wiki/173(number)), (/wiki/197(number)), (/wiki/223(number)), (/wiki/283(number)), (/wiki/313(number)), (/wiki/317(number)), (/wiki/337(number)), (/wiki/347(number)), (/wiki/353(number)), (/wiki/367(number)), (/wiki/373(number)), (/wiki/383(number)), (/wiki/397(number)), (/wiki/443(number)), (/wiki/467(number)), (/wiki/523(number)), (/wiki/547(number)), (/wiki/613(number)), (/wiki/617(number)), (/wiki/643(number)), (/wiki/647(number)), (/wiki/653(number)), (/wiki/673(number)), (/wiki/683(number)) (OEIS: A024785)

Right-truncatable

Primes that remain prime when the least significant decimal digit is successively removed.

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/23(number)), (/wiki/29(number)), (/wiki/31(number)), (/wiki/37(number)), (/wiki/53(number)), (/wiki/59(number)), (/wiki/71(number)), (/wiki/73(number)), (/wiki/79(number)), (/wiki/233(number)), (/wiki/239(number)), (/wiki/293(number)), (/wiki/311(number)), (/wiki/313(number)), (/wiki/317(number)), (/wiki/373(number)), (/wiki/379(number)), (/wiki/593(number)), (/wiki/599(number)), (/wiki/719(number)), (/wiki/733(number)), (/wiki/739(number)), (/wiki/797(number)), (/wiki/2333(number)), (/wiki/2339(number)), (/wiki/2393(number)), (/wiki/2399(number)), (/wiki/2939(number)), (/wiki/3119(number)), (/wiki/3137(number)), (/wiki/3733(number)), (/wiki/3739(number)), (/wiki/3793(number)), (/wiki/3797(number)) (OEIS: A024770)

Two-sided

Primes that are both left-truncatable and right-truncatable. There are exactly fifteen two-sided primes:

(/wiki/2), (/wiki/3), (/wiki/5), (/wiki/7), (/wiki/23(number)), (/wiki/37(number)), (/wiki/53(number)), (/wiki/73(number)), (/wiki/313(number)), (/wiki/317(number)), (/wiki/373(number)), (/wiki/797(number)), (/wiki/3137(number)), (/wiki/3797(number)), 739397 (OEIS: A020994)

Twin primes

((/wiki/3), (/wiki/5)), (5, (/wiki/7)), ((/wiki/11(number)), (/wiki/13(number))), ((/wiki/17(number)), (/wiki/19(number))), ((/wiki/29(number)), (/wiki/31(number))), ((/wiki/41(number)), (/wiki/43(number))), ((/wiki/59(number)), (/wiki/61(number))), ((/wiki/71(number)), (/wiki/73(number))), ((/wiki/101(number)), (/wiki/103(number))), ((/wiki/107(number)), (/wiki/109(number))), ((/wiki/137(number)), (/wiki/139(number))), ((/wiki/149(number)), (/wiki/151(number))), ((/wiki/179(number)), (/wiki/181(number))), ((/wiki/191(number)), (/wiki/193(number))), ((/wiki/197(number)), (/wiki/199(number))), ((/wiki/227(number)), (/wiki/229(number))), ((/wiki/239(number)), (/wiki/241(number))), ((/wiki/269(number)), (/wiki/271(number))), ((/wiki/281(number)), (/wiki/283(number))), ((/wiki/311(number)), (/wiki/313(number))), ((/wiki/347(number)), (/wiki/349(number))), ((/wiki/419(number)), (/wiki/421(number))), ((/wiki/431(number)), (/wiki/433(number))), ((/wiki/461(number)), (/wiki/463(number))) (OEIS: A001359, OEIS: A006512)

Unique primes

The list of primes p for which the period length of the decimal expansion of 1/p is unique (no other prime gives the same period).

(/wiki/3), (/wiki/11(number)), (/wiki/37(number)), (/wiki/101(number)), 9091, 9901, 333667, 909091, 99990001, 999999000001, 9999999900000001, 909090909090909091, 1111111111111111111, 11111111111111111111111, 900900900900990990990991 (OEIS: A040017)

Wagstaff primes

(/wiki/3), (/wiki/11(number)), (/wiki/43(number)), (/wiki/683(number)), (/wiki/2731(number)), 43691, 174763, 2796203, 715827883, 2932031007403, 768614336404564651, 201487636602438195784363, 845100400152152934331135470251, 56713727820156410577229101238628035243 (OEIS: A000979)

3, (/wiki/5), (/wiki/7), 11, (/wiki/13(number)), (/wiki/17(number)), (/wiki/19(number)), (/wiki/23(number)), (/wiki/31(number)), 43, (/wiki/61(number)), (/wiki/79(number)), (/wiki/101(number)), (/wiki/127(number)), (/wiki/167(number)), (/wiki/191(number)), (/wiki/199(number)), (/wiki/313(number)), (/wiki/347(number)), (/wiki/701(number)), (/wiki/1709(number)), (/wiki/2617(number)), (/wiki/3539(number)), (/wiki/5807(number)), (/wiki/10501(number)), 10691, 11279, 12391, 14479, 42737, 83339, 95369, 117239, 127031, 138937, 141079, 267017, 269987, 374321 (OEIS: A000978)

Wall–Sun–Sun primes

A prime p > 5, if p2 divides the Fibonacci number ${\displaystyle F{p-\left({\frac {p}{5}}\right)}}$, where the Legendre symbol ${\displaystyle \left({\frac {p}{5}}\right)}$ is defined as

${\displaystyle \left({\frac {p}{5}}\right)={\begin{cases}1&{\textrm {if}}\;p\equiv \pm 1{\pmod {5}}\\-1&{\textrm {if}}\;p\equiv \pm 2{\pmod {5}}.\end{cases}}}$

As of 2022, no Wall-Sun-Sun primes have been found below ${\displaystyle 2^{64}}$ (about ${\displaystyle 18\cdot 10^{18}}$).

Wieferich primes

Primes p such that ap − 1 ≡ 1 (mod p2) for fixed integer a > 1.

2p − 1 ≡ 1 (mod p2): (/wiki/1093(number)), (/wiki/3511(number)) (OEIS: A001220) 3p − 1 ≡ 1 (mod p2): (/wiki/11(number)), 1006003 (OEIS: A014127) 4p − 1 ≡ 1 (mod p2): (/wiki/1093(number)), (/wiki/3511(number)) 5p − 1 ≡ 1 (mod p2): (/wiki/2), 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801 (OEIS: A123692) 6p − 1 ≡ 1 (mod p2): 66161, 534851, 3152573 (OEIS: A212583) 7p − 1 ≡ 1 (mod p2): (/wiki/5), 491531 (OEIS: A123693) 8p − 1 ≡ 1 (mod p2): (/wiki/3), (/wiki/1093(number)), (/wiki/3511(number)) 9p − 1 ≡ 1 (mod p2): (/wiki/2), (/wiki/11(number)), 1006003 10p − 1 ≡ 1 (mod p2): (/wiki/3), (/wiki/487(number)), 56598313 (OEIS: A045616) 11p − 1 ≡ 1 (mod p2): (/wiki/71(number)) 12p − 1 ≡ 1 (mod p2): (/wiki/2693(number)), 123653 (OEIS: A111027) 13p − 1 ≡ 1 (mod p2): (/wiki/2), (/wiki/863(number)), 1747591 (OEIS: A128667) 14p − 1 ≡ 1 (mod p2): (/wiki/29(number)), (/wiki/353(number)), 7596952219 (OEIS: A234810) 15p − 1 ≡ 1 (mod p2): 29131, 119327070011 (OEIS: A242741) 16p − 1 ≡ 1 (mod p2): (/wiki/1093(number)), (/wiki/3511(number)) 17p − 1 ≡ 1 (mod p2): (/wiki/2), (/wiki/3), 46021, 48947 (OEIS: A128668) 18p − 1 ≡ 1 (mod p2): (/wiki/5), (/wiki/7), (/wiki/37(number)), (/wiki/331(number)), 33923, 1284043 (OEIS: A244260) 19p − 1 ≡ 1 (mod p2): (/wiki/3), (/wiki/7), (/wiki/13(number)), (/wiki/43(number)), (/wiki/137(number)), 63061489 (OEIS: A090968) 20p − 1 ≡ 1 (mod p2): (/wiki/281(number)), 46457, 9377747, 122959073 (OEIS: A242982) 21p − 1 ≡ 1 (mod p2): (/wiki/2) 22p − 1 ≡ 1 (mod p2): (/wiki/13(number)), (/wiki/673(number)), 1595813, 492366587, 9809862296159 (OEIS: A298951) 23p − 1 ≡ 1 (mod p2): (/wiki/13(number)), 2481757, 13703077, 15546404183, 2549536629329 (OEIS: A128669) 24p − 1 ≡ 1 (mod p2): (/wiki/5), 25633 25p − 1 ≡ 1 (mod p2): (/wiki/2), 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801

As of 2018, these are all known Wieferich primes with a ≤ 25.

Wilson primes

(/wiki/5), (/wiki/13(number)), (/wiki/563(number)) (OEIS: A007540)

As of 2018, these are the only known Wilson primes.

Wolstenholme primes

Primes p for which the binomial coefficient ${\displaystyle {{2p-1} \choose {p-1}}\equiv 1{\pmod {p^{4}}}.}$

16843, 2124679 (OEIS: A088164)

As of 2018, these are the only known Wolstenholme primes.

Woodall primes

(/wiki/7), (/wiki/23(number)), 383, 32212254719, 2833419889721787128217599, 195845982777569926302400511, 4776913109852041418248056622882488319 (OEIS: A050918)