Openai/69523a9e-0d00-800b-b446-9253ad0ab5a2

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gpt-5-2-thinking[edit | edit source]

Numbers k such that k! + 59 is prime.

Let kkk be a nonnegative integer.

If k≥59k \ge 59k≥59, then 59∣k!59 \mid k!59∣k!, so k!+59≡0(mod59),k!+59 \equiv 0 \pmod{59},k!+59≡0(mod59), and since k!+59>59k!+59>59k!+59>59, it’s composite. So we only need to check 0≤k≤580\le k\le 580≤k≤58.

A direct primality check for 0≤k≤580\le k\le 580≤k≤58 shows that k!+59k!+59k!+59 is prime exactly for

k∈{2,4,5,7,11,12,17,21,26,27}.\boxed{k \in \{2,4,5,7,11,12,17,21,26,27\}.}k∈{2,4,5,7,11,12,17,21,26,27}. Corresponding primes:

2!+59=612!+59=612!+59=61
4!+59=834!+59=834!+59=83
5!+59=1795!+59=1795!+59=179
7!+59=50997!+59=50997!+59=5099
11!+59=3991685911!+59=3991685911!+59=39916859
12!+59=47900165912!+59=47900165912!+59=479001659
17!+59=35568742809605917!+59=35568742809605917!+59=355687428096059
21!+59=5109094217170944005921!+59=5109094217170944005921!+59=51090942171709440059
26!+59=40329146112660563558400005926!+59=40329146112660563558400005926!+59=403291461126605635584000059
27!+59=1088886945041835216076800005927!+59=1088886945041835216076800005927!+59=10888869450418352160768000059