Problem 1214

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claims/: The 1 claim page of Problem 1214, one per claimant's result; the problem's standing derives from them.

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Statement. Let $x,y\geq 1$ be integers such that, for all $n\geq 1$, the set of primes dividing $x^{n}-1$ is equal to set of primes dividing $y^n-1$. Must $x=y$?

Status. Proved: the site credits Corrales-Rodrigáñez and Schoof [CoSc97] with the positive answer; the accepted claim page is Corrales-Rodrigáñez and Schoof 1997.

Source. erdosproblems.com/1214, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #1214, https://www.erdosproblems.com/1214.

References.

• [CoSc97] Corrales-Rodrigáñez, Capi and Schoof, René, The support problem and its elliptic analogue. J. Number Theory (1997), 276-290.

Formalization. Statement in formal-conjectures (read at the commit linked, the last to touch the file), tagged solved with a formal-proof link to a Lean file in Boris Alexeev's repository that declares itself a formalization of the paper's result; the file is linked from the claim page, and this corpus has not built it.

Current assessment

The question, in the site's formulation accessed, asks whether integers $x,y\ge1$ with the same set of primes dividing $x^n-1$ and $y^n-1$ for every $n\ge1$ must be equal. The answer is yes: Corrales-Rodrigáñez and Schoof 1997, Theorem 1, proves for any number field that if $x^n\equiv1$ modulo a prime ideal implies $y^n\equiv1$ modulo that ideal, for all $n$ and almost all prime ideals, then $y$ is a power of $x$; the two-sided hypothesis over $\mathbb{Q}$ makes each of $x,y$ a power of the other and so $x=y$. The paper also proves the elliptic-curve analogue (Theorem 2) and poses the abelian-variety case, which lies outside this question.

Acceptance rests on the refereed paper and on the site's curator labeling the problem proved with that credit; this corpus has not verified the proof. A third-party Lean file declaring itself a formalization of the result is linked from the claim page; this corpus has read it as text and has not built or audited it, so it gives no formalized evidence. Search scope: the site's problem page (no forum comments), the community database, the paper, the formal-conjectures file and the linked Lean file.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• corralesrodriganez_1997_support_problem_elliptic_analogue
• corralesrodriganez_1997_support_problem_elliptic_analogue / theorem_1
• corralesrodriganez_1997_support_problem_elliptic_analogue / theorem_2