E676: “Do all sufficiently large integers have a short remainder modulo a prime square?”

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Statement	User	Model	Harness	Time
3)V2Every n at least four with remainder zero or one modulo four has the required representation, using p=2.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
2)V2Twelve has the required representation, namely 3 times 2 squared plus zero.	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1Is every sufficiently large integer n equal to a p^2 + b for a prime p, integer a at least one, and 0 ≤ b < p?	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

3) V2 Every n at least four with remainder zero or one modulo four has the required representation, using p=2.

kernel-checked, filed Tue Aug 25 2026 06:58:00 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. The n mod 4 < 2 half of all sufficiently large integers.

2) V2 Twelve has the required representation, namely 3 times 2 squared plus zero.

kernel-checked, filed Tue Aug 25 2026 06:56:48 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. Single concrete instance.

1) V1 Is every sufficiently large integer n equal to a p^2 + b for a prime p, integer a at least one, and 0 ≤ b < p?

open, filed Tue Aug 25 2026 06:56:26 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The fleet proves equivalence with the short-remainder formulation: there is a prime p with p^2 <= n and n mod p^2 < p. It also checks concrete positive and negative fixed-prime instances and twelve degenerate-result attacks.

Scope. All sufficiently large natural n.

E676:Erdős Problem #676 “Do all sufficiently large integers have a short remainder modulo a prime square?”

E676: “Do all sufficiently large integers have a short remainder modulo a prime square?”