[29.12.2025 10:29] Ян Дененберг: Numbers k such that k! + 59 is prime. https://chatgpt.com/c/69523961-0ef8-8326-b5f4-59b9aae4a7d6 Let (k) be a nonnegative integer.
* If (k \ge 59), then (59 \mid k!), so
[
k!+59 \equiv 0 \pmod{59},
]
and since (k!+59>59), it’s composite.
So we only need to check (0\le k\le 58).
A direct primality check for (0\le k\le 58) shows that (k!+59) is prime exactly for
[
\boxed{k \in {2,4,5,7,11,12,17,21,26,27}.}
]
Corresponding primes:
* (2!+59=61)
* (4!+59=83)
* (5!+59=179)
* (7!+59=5099)
* (11!+59=39916859)
* (12!+59=479001659)
* (17!+59=355687428096059)
* (21!+59=51090942171709440059)
* (26!+59=403291461126605635584000059)
* (27!+59=10888869450418352160768000059) https://chatgpt.com/share/69523a9e-0d00-800b-b446-9253ad0ab5a2 Понедельник, 29122025, 10:26:06.
Numbers k such that k! 59 is prime
© Copyright: Ян Дененберг, 2025
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Свидетельство о публикации №225122900865
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