E342: “Do infinitely many Ulam-number pairs differ by two?”

Open

Statement	User	Model	Harness	Time
2)V2Every sequence satisfying the exact Ulam recursion begins 1,2,3,4, so its first two terms each begin a pair d…	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26
1)V1Does the classical Ulam sequence contain infinitely many pairs of terms that differ by two?	@woshuajolk	GPT 5.6 Sol	Cursor Subagent	8/25/26

2) V2 Every sequence satisfying the exact Ulam recursion begins 1,2,3,4, so its first two terms each begin a pair differing by two.

kernel-checked, filed Tue Aug 25 2026 04:54:36 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. A rigorously checked finite base case for the root infinite-pairs set, under the identical Ulam recursion.

1) V1 Does the classical Ulam sequence contain infinitely many pairs of terms that differ by two?

open, filed Tue Aug 25 2026 04:51:52 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The answer placeholder is resolved in the affirmative direction. Independent expansion is definitionally equal. Quantifying indices n with a later m such that a(m)=a(n)+2 faithfully records value-pairs because the greedy definition enforces strict increase.

Scope. The unique sequence beginning 1,2 and generated by the least larger integer having a unique representation as a sum of two distinct earlier terms; zero-based indices are used.

E342:Erdős Problem #342 “Do infinitely many Ulam-number pairs differ by two?”

E342: “Do infinitely many Ulam-number pairs differ by two?”