Proven Primes
Largest prime in this catalogue
p₅# · 13⁴⁴⁷⁶³ + 1
Order k = 6, exponent n = 44,763 on the plus face: 49,867 decimal digits, proven by Pocklington's N−1 test. Compute it in your browser →
Every entry is a proven prime on one of the grid's two faces; the proof-method column identifies its criterion, whose hypotheses are set out in the paper. The underlying certificate is held with the project's records. Row 1 is the Mersenne row; the ones above the 5,000-digit census are known Mersenne primes without a certificate here (Mersenne prime): their primality is established in the literature (Lucas–Lehmer; GIMPS, OEIS A000043) and is credited there, not re-proved here. This site publishes the catalogue, while the A400659 verification page supplies standalone programs for the first prime in each of its first 100 rows.
5,781 proven primes — 2,905 on the minus face, 2,876 on the plus face. 31 orders and exponents are prime on both faces, giving a twin pair. Index v2026-11-01.
5,781 of 5,781 proven primes shown
| number | k | n | face | digits ▼ | proof method |
|---|---|---|---|---|---|
| p₅#·13⁴⁴⁷⁶³ + 1 | 6 | 44763 | +1 | 49,867 | Pocklington N−1 |
| p₂#·5⁶⁸³²⁰ − 1 | 3 | 68320 | −1 | 47,755 | Lucas N+1 |
| p₁₂#·41²⁶⁸⁰⁹ + 1 | 13 | 26809 | +1 | 43,250 | Pocklington N−1 |
| p₁#·3⁹⁰⁰¹² − 1 | 2 | 90012 | −1 | 42,947 | Lucas N+1 |
| p₁₀#·31²⁸⁰⁶⁴ + 1 | 11 | 28064 | +1 | 41,864 | Pocklington N−1 |
| p₅#·13³⁶⁸⁶⁶ − 1 | 6 | 36866 | −1 | 41,070 | Lucas N+1 |
| p₃#·7⁴⁷⁷⁴² + 1 | 4 | 47742 | +1 | 40,349 | Pocklington N−1 |
| 2¹³²⁰⁴⁹ − 1 | 1 | 132049 | −1 | 39,751 | Mersenne prime |
| p₁₀#·31²⁶⁶²³ + 1 | 11 | 26623 | +1 | 39,715 | Pocklington N−1 |
| p₄#·11³⁸¹²⁴ + 1 | 5 | 38124 | +1 | 39,705 | Pocklington N−1 |
| p₁#·3⁸²⁷⁸⁰ + 1 | 2 | 82780 | +1 | 39,497 | Pocklington N−1 |
| p₃#·7⁴⁵⁹⁷⁵ − 1 | 4 | 45975 | −1 | 38,855 | Lucas N+1 |
| p₁₁#·37²⁴³⁰⁵ − 1 | 12 | 24305 | −1 | 38,127 | Lucas N+1 |
| p₅#·13³³⁸⁶⁹ + 1 | 6 | 33869 | +1 | 37,732 | Pocklington N−1 |
| p₈#·23²⁷⁶⁸⁷ − 1 | 9 | 27687 | −1 | 37,710 | Lucas N+1 |
| p₁₂#·41²²⁹⁷⁶ + 1 | 13 | 22976 | +1 | 37,069 | Pocklington N−1 |
| p₇#·19²⁸²⁷¹ − 1 | 8 | 28271 | −1 | 36,158 | Lucas N+1 |
| p₁#·3⁷⁵⁵⁵¹ − 1 | 2 | 75551 | −1 | 36,048 | Lucas N+1 |
| p₄#·11³³⁶⁹⁰ + 1 | 5 | 33690 | +1 | 35,087 | Pocklington N−1 |
| p₈#·23²⁵⁷³⁷ − 1 | 9 | 25737 | −1 | 35,054 | Lucas N+1 |
| p₄#·11³³⁵¹⁷ − 1 | 5 | 33517 | −1 | 34,907 | Lucas N+1 |
| p₆#·17²⁷⁶²³ − 1 | 7 | 27623 | −1 | 33,994 | Lucas N+1 |
| 2¹¹⁰⁵⁰³ − 1 | 1 | 110503 | −1 | 33,265 | Mersenne prime |
| p₁#·3⁶⁴⁸²² + 1 | 2 | 64822 | +1 | 30,929 | Pocklington N−1 |
| p₄#·11²⁸⁹⁷⁶ − 1 | 5 | 28976 | −1 | 30,178 | Lucas N+1 |
| p₂#·5⁴²⁵⁰⁰ − 1 | 3 | 42500 | −1 | 29,708 | Lucas N+1 |
| p₆#·17²³⁹¹⁷ + 1 | 7 | 23917 | +1 | 29,434 | Pocklington N−1 |
| p₉#·29¹⁸⁴³⁴ − 1 | 10 | 18434 | −1 | 26,967 | Lucas N+1 |
| p₁₂#·41¹⁶²¹⁹ + 1 | 13 | 16219 | +1 | 26,171 | Pocklington N−1 |
| 2⁸⁶²⁴³ − 1 | 1 | 86243 | −1 | 25,962 | Mersenne prime |
| p₅#·13²²⁸⁷² + 1 | 6 | 22872 | +1 | 25,482 | Pocklington N−1 |
| p₁₁#·37¹⁶¹³⁸ + 1 | 12 | 16138 | +1 | 25,319 | Pocklington N−1 |
| p₈#·23¹⁷⁹³⁹ − 1 | 9 | 17939 | −1 | 24,436 | Lucas N+1 |
| p₁₁#·37¹⁵⁴⁴⁹ + 1 | 12 | 15449 | +1 | 24,239 | Pocklington N−1 |
| p₁₂#·41¹⁴⁷⁸⁷ − 1 | 13 | 14787 | −1 | 23,862 | Lucas N+1 |
| p₁₁#·37¹⁵⁰⁴⁷ − 1 | 12 | 15047 | −1 | 23,609 | Lucas N+1 |
| p₁#·3⁴⁹³⁴⁰ − 1 | 2 | 49340 | −1 | 23,542 | Lucas N+1 |
| p₆#·17¹⁸⁶⁷⁰ + 1 | 7 | 18670 | +1 | 22,977 | Pocklington N−1 |
| p₅#·13²⁰⁵⁸⁷ + 1 | 6 | 20587 | +1 | 22,937 | Pocklington N−1 |
| p₁₃#·43¹³²⁸⁸ − 1 | 14 | 13288 | −1 | 21,721 | Lucas N+1 |
| p₁₁#·37¹³⁶¹⁶ − 1 | 12 | 13616 | −1 | 21,364 | Lucas N+1 |
| p₅#·13¹⁹⁰⁰² + 1 | 6 | 19002 | +1 | 21,171 | Pocklington N−1 |
| p₁#·3⁴³⁹⁵⁶ + 1 | 2 | 43956 | +1 | 20,973 | Pocklington N−1 |
| p₃#·7²⁴²⁹¹ + 1 | 4 | 24291 | +1 | 20,530 | Pocklington N−1 |
| p₁₁#·37¹²⁹⁴² + 1 | 12 | 12942 | +1 | 20,307 | Pocklington N−1 |
| p₈#·23¹⁴⁷⁸¹ + 1 | 9 | 14781 | +1 | 20,135 | Pocklington N−1 |
| p₂#·5²⁸⁷⁹³ + 1 | 3 | 28793 | +1 | 20,127 | Pocklington N−1 |
| p₄#·11¹⁹⁰²⁰ + 1 | 5 | 19020 | +1 | 19,810 | Pocklington N−1 |
| p₁₀#·31¹²⁹²⁵ − 1 | 11 | 12925 | −1 | 19,286 | Lucas N+1 |
| p₉₉₉#·7919³⁹¹⁸ + 1 | 1000 | 3918 | +1 | 18,664 | Pocklington N−1 |
| p₄#·11¹⁷⁸⁴⁰ + 1 | 5 | 17840 | +1 | 18,581 | Pocklington N−1 |
| p₈#·23¹³²³⁸ − 1 | 9 | 13238 | −1 | 18,034 | Lucas N+1 |
| p₂#·5²⁵¹⁷⁶ − 1 | 3 | 25176 | −1 | 17,599 | Lucas N+1 |
| p₅#·13¹⁵⁴³⁷ − 1 | 6 | 15437 | −1 | 17,200 | Lucas N+1 |
| p₉#·29¹⁰⁸⁸⁹ − 1 | 10 | 10889 | −1 | 15,933 | Lucas N+1 |
| p₉#·29¹⁰⁸⁶⁴ − 1 | 10 | 10864 | −1 | 15,896 | Lucas N+1 |
| p₁₀#·31¹⁰⁴⁸⁹ + 1 | 11 | 10489 | +1 | 15,653 | Pocklington N−1 |
| p₁₀#·31¹⁰²⁷² − 1 | 11 | 10272 | −1 | 15,330 | Lucas N+1 |
| p₃#·7¹⁷⁶⁵⁸ − 1 | 4 | 17658 | −1 | 14,925 | Lucas N+1 |
| p₈#·23¹⁰⁶⁹⁰ − 1 | 9 | 10690 | −1 | 14,564 | Lucas N+1 |
| p₅#·13¹³⁰³⁸ − 1 | 6 | 13038 | −1 | 14,527 | Lucas N+1 |
| p₄#·11¹³⁶¹⁸ − 1 | 5 | 13618 | −1 | 14,185 | Lucas N+1 |
| p₃#·7¹⁶⁶⁵⁰ − 1 | 4 | 16650 | −1 | 14,073 | Lucas N+1 |
| p₁₁#·37⁸⁷⁴⁰ + 1 | 12 | 8740 | +1 | 13,718 | Pocklington N−1 |
| p₅#·13¹²²¹³ − 1 | 6 | 12213 | −1 | 13,608 | Lucas N+1 |
| p₅#·13¹²¹⁶⁰ + 1 | 6 | 12160 | +1 | 13,549 | Pocklington N−1 |
| p₁₂#·41⁸³⁵² + 1 | 13 | 8352 | +1 | 13,483 | Pocklington N−1 |
| 2⁴⁴⁴⁹⁷ − 1 | 1 | 44497 | −1 | 13,395 | Mersenne prime |
| p₁₀#·31⁸⁹⁶³ − 1 | 11 | 8963 | −1 | 13,377 | Lucas N+1 |
| p₈#·23⁹⁷⁸⁹ + 1 | 9 | 9789 | +1 | 13,337 | Pocklington N−1 |
| p₃#·7¹⁴⁶⁰⁰ − 1 | 4 | 14600 | −1 | 12,340 | Lucas N+1 |
| p₄#·11¹¹⁸²⁷ − 1 | 5 | 11827 | −1 | 12,319 | Lucas N+1 |
| p₅#·13¹⁰⁸¹⁷ + 1 | 6 | 10817 | +1 | 12,053 | Pocklington N−1 |
| p₅#·13¹⁰⁷⁶² − 1 | 6 | 10762 | −1 | 11,992 | Lucas N+1 |
| p₄₉#·229⁵⁰¹⁰ − 1 | 50 | 5010 | −1 | 11,912 | Lucas N+1 |
| p₇#·19⁹¹⁹⁷ − 1 | 8 | 9197 | −1 | 11,767 | Lucas N+1 |
| p₂#·5¹⁶²⁹⁴ − 1 | 3 | 16294 | −1 | 11,390 | Lucas N+1 |
| p₄₇#·223⁴⁶⁴³ − 1 | 48 | 4643 | −1 | 10,988 | Lucas N+1 |
| p₇#·19⁸¹²⁵ + 1 | 8 | 8125 | +1 | 10,396 | Pocklington N−1 |
| p₄₈#·227⁴³¹⁹ − 1 | 49 | 4319 | −1 | 10,263 | Lucas N+1 |
Click a number to compute its exact value with Explore the Grid; click the k, n or digits headings to sort. A cell whose two faces are both prime gives a twin pair — two primes differing by 2 — and is marked accordingly. The catalogue is also available as primes.json, and the per-row primality index as index.json.