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

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_This conversation was exported from ChatGPT_[[1]](https://freemwiki.com/index.php?mobileaction=toggle_view_mobile&title=Openai%2F69523a9e-0d00-800b-b446-9253ad0ab5a2#cite_note-1)

## gpt-5-2-thinking[edit](https://freemwiki.com/index.php?title=Openai/69523a9e-0d00-800b-b446-9253ad0ab5a2&action=edit&section=1 "Edit section: gpt-5-2-thinking")

### User: Numbers k such that k![edit](https://freemwiki.com/index.php?title=Openai/69523a9e-0d00-800b-b446-9253ad0ab5a2&action=edit&section=2 "Edit section: User: Numbers k such that k!")

Numbers k such that k! + 59 is prime.

### Assistant: Let kkk be a nonnegative integer.[edit](https://freemwiki.com/index.php?title=Openai/69523a9e-0d00-800b-b446-9253ad0ab5a2&action=edit&section=3 "Edit section: Assistant: Let kkk be a nonnegative integer.")

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

## References[edit](https://freemwiki.com/index.php?title=Openai/69523a9e-0d00-800b-b446-9253ad0ab5a2&action=edit&section=4 "Edit section: References")

1.   [↑](https://freemwiki.com/index.php?mobileaction=toggle_view_mobile&title=Openai%2F69523a9e-0d00-800b-b446-9253ad0ab5a2#cite_ref-1 "Jump up")["Primes of k!+59"](https://chatgpt.com/share/69523a9e-0d00-800b-b446-9253ad0ab5a2). ChatGPT. Retrieved 2025-12-29.