The smallest congruent number curve of rank seven
Rogers (2004) found a rank-7 curve at this \(k\). We show that no smaller \(k\) gives rank 7 — the 22-year-old upper bound is the exact value.
- \(\operatorname{rank} E_k \le 6\ \text{for all}\ k < 797507543735,\quad E_k: y^2 = x^3 - k^2x\)
- \(9\,053\,344\) candidates after the Monsky 2-Selmer sieve; sieve count \(143\,452\,705\,272\) confirmed by prime counting
- 163 residual cases closed by isogeny; mwrank cross-check \(866/866\) and \(1788/1788\)
Abstract
We prove that a(7) = 797507543735 in OEIS A194687: for every positive integer k < 797507543735 the curve y² = x³ − k²x has rank at most 6. Until now only the upper bound was known. The proof is exhaustive and unconditional (no GRH, BSD or parity conjecture) and comes with a certificate that anyone can re-check in about a minute.
Method
(1) A Monsky 2-Selmer sieve over all squarefree \(k < 797507543735\): a curve of rank \(\ge 7\) needs 2-Selmer rank \(\ge 7\) or \(\ge 8\) depending on \(k \bmod 8\). (2) PARI/GP ellrank upper bounds (2-descent and the Cassels pairing) for every candidate. (3) For the few curves where this bound is 7, the bound over the \(\mathbb{Q}\)-isogeny class — rank is an isogeny invariant. (4) An independent recomputation with Cremona's mwrank.
Check it yourself
The repository contains the candidate lists and the bound for every \(k\). The verifier verify_yakun.py re-checks completeness, coverage and all bounds in about a minute and prints 4/4. Commands are in the README.
Try it
The Telegram bot @Selloriyuzb_bot is a congruent number calculator built on the same tools: send it any \(n \le 10^{15}\) and it returns a right triangle with rational sides and area \(n\), or a proof that none exists. Send it 797507543735 to see rank 7.
Cite
Otakhon U. Kenjaev (2026). The smallest congruent number curve of rank seven. Zenodo. https://doi.org/10.5281/zenodo.23028371