User:Kenneth J Scheller

Retired therapist and pattern hobbyist exploring number theory. Currently studying prime number behavior through Excel-based simulations and analytical reasoning, especially in the context of Greatest Prime Divisor (GPD) patterns.

Developed the GPD Modality Theorem as a structural criterion for prime exclusion from the list of popular primes (A385503).

Submissions are offered with assistancefrom AI (ChatGPT-4) as a thought partner, not as a source of original content.

GPD Modality Theorem (informal observation):

Let p<q<r be primes. Suppose that at some integer N, both p and r are tied as the most frequent greatest prime divisors of numbers <=N, and q is not. Then it appears that q can never become modal (even in a tie) at any later value n>N .

The reasoning is structural: – GPD counts increase at most by 1 at each step (so overtaking requires a tie first). – Smaller primes like p tend to lead early due to higher density. – Larger primes like r gain by priority when they appear in factorizations. A middle prime like q, caught between these advantages and behind both modal counts, seems to have no remaining mechanism to ascend.

Example: At N=1456, both 7 and 13 are modal; 11 is not. After this point, 11 never has a remaining mechanism for becoming modal.

This may be related to the work of McNew on popular primes and prime gaps, but I haven’t found this particular structural criterion stated in this form.

—Submitted Kenneth J Scheller, July 2025