Theorem · version 2
For every odd positive squarefree integer n, assuming BSD rank equality for E_n: y² = x³−n²x, n is the area of a right triangle with positive rational side lengths if and only if #{(x,y,z)∈Z³: 2x²+y²+8z²=n} = 2·#{(x,y,z)∈Z³: 2x²+y²+32z²=n}.
A proved conditional criterion with two finite counts. The necessary direction is unconditional; this selection omits the different even-n formula.
Proved
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Wiles · official BSD description · §1 Early History, PDF page 3: odd squarefree Tunnell criterion →
Observed Oct 10, 2026
In this problem · publication v13 · Selected context
Conditional implication · Proved
Scope: Two finite counts would decide the odd squarefree congruent-number problem, assuming rank BSD for E_n.
Limit: The sufficient direction uses the conjecture; the odd-squarefree restrictions and exact counts matter.
A consequence under an open premise, not an additional resolved part of that premise.
Rank BSD specialized to E_n would supply the missing sufficient direction of Tunnell’s exact counting criterion.
This implication is restricted to the stated odd squarefree congruent-number curves.
Open · version 3
For every elliptic curve over , , where L is the full Hasse–Weil L-function with all finite Euler factors.
Scope: The canonical rank question. It does not by itself assert finite Sha or the precise leading coefficient.
Proved · version 2
For every odd positive squarefree integer n, assuming BSD rank equality for E_n: y² = x³−n²x, n is the area of a right triangle with positive rational side lengths if and only if #{(x,y,z)∈Z³: 2x²+y²+8z²=n} = 2·#{(x,y,z)∈Z³: 2x²+y²+32z²=n}.
Scope: A proved conditional criterion with two finite counts. The necessary direction is unconditional; this selection omits the different even-n formula.
Relationship not independently reviewed.
Connection in Birch and Swinnerton-Dyer conjecture →Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v17
Conditional implication · Proved
Scope: Two finite counts would decide the odd squarefree congruent-number problem, assuming rank BSD for E_n.
Limit: The sufficient direction uses the conjecture; the odd-squarefree restrictions and exact counts matter.
A consequence under an open premise, not an additional resolved part of that premise.
Rank BSD specialized to E_n would supply the missing sufficient direction of Tunnell’s exact counting criterion.
This implication is restricted to the stated odd squarefree congruent-number curves.
Open · version 3
For every elliptic curve over , , where L is the full Hasse–Weil L-function with all finite Euler factors.
Scope: The canonical rank question. It does not by itself assert finite Sha or the precise leading coefficient.
Proved · version 2
For every odd positive squarefree integer n, assuming BSD rank equality for E_n: y² = x³−n²x, n is the area of a right triangle with positive rational side lengths if and only if #{(x,y,z)∈Z³: 2x²+y²+8z²=n} = 2·#{(x,y,z)∈Z³: 2x²+y²+32z²=n}.
Scope: A proved conditional criterion with two finite counts. The necessary direction is unconditional; this selection omits the different even-n formula.
Relationship not independently reviewed.
Connection in Birch and Swinnerton-Dyer conjecture →Relationship not independently reviewed. This concerns its use in this problem.
For odd positive squarefree n, BSD for y²=x³−n²x would make a finite ternary-quadratic count criterion equivalent to being a congruent number.
paraphrase · No upstream full text or code is republished by this record.
For every elliptic curve E/Q, rank E(Q) equals ord_{s=1} L(E,s).
paraphrase · No upstream full text or code is republished by this record.
Andrew Wiles · Published
Official Clay problem description; separates rank equality from its refined formula.
Source revision: official-description. Observed Oct 10, 2026.
The cited work does not report AI involvement; this is not an assertion that no AI was used.
Official description PDF: official-description. External evidence linked; no OpenMath execution or independent reproduction is asserted.
Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026 · Source passage →