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{{DISPLAYTITLE:Primes of k!+59}}<br />
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''This conversation was exported from ChatGPT''<ref>{{cite web|title=Primes of k!+59|url=https://chatgpt.com/share/69523a9e-0d00-800b-b446-9253ad0ab5a2|publisher=ChatGPT|access-date=2025-12-29}}</ref><br />
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== gpt-5-2-thinking ==<br />
=== User: Numbers k such that k! ===<br />
Numbers k such that k! + 59 is prime.<br />
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=== Assistant: Let kkk be a nonnegative integer. ===<br />
Let kkk be a nonnegative integer.<br />
* 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.<br />
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A direct primality check for 0≤k≤580\le k\le 580≤k≤58 shows that k!+59k!+59k!+59 is prime exactly for<br />
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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}.<br />
Corresponding primes:<br />
* 2!+59=612!+59=612!+59=61<br />
* 4!+59=834!+59=834!+59=83<br />
* 5!+59=1795!+59=1795!+59=179<br />
* 7!+59=50997!+59=50997!+59=5099<br />
* 11!+59=3991685911!+59=3991685911!+59=39916859<br />
* 12!+59=47900165912!+59=47900165912!+59=479001659<br />
* 17!+59=35568742809605917!+59=35568742809605917!+59=355687428096059<br />
* 21!+59=5109094217170944005921!+59=5109094217170944005921!+59=51090942171709440059<br />
* 26!+59=40329146112660563558400005926!+59=40329146112660563558400005926!+59=403291461126605635584000059<br />
* 27!+59=1088886945041835216076800005927!+59=1088886945041835216076800005927!+59=10888869450418352160768000059<br />
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== References ==<br />
<references /><br />
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[[Category:ChatGPT Shares]]<br />
[[Category:openai]]</div>Lukegao