Congruent numbers in residue classes 5, 6, and 7 modulo 8

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A positive rational number $q$ is congruent if there are positive rational numbers $a,b,c$ with $a^2+b^2=c^2$ and $ab/2=q$; thus $q$ is the area of a right triangle with rational side lengths.

Statement recorded in the supplied source.

Is every positive integer $n$ with $n\equiv5,6,$ or $7\pmod8$ congruent? [S2]

Short English statement

Does every positive integer in residue class five, six or seven modulo eight occur as the area of a rational right triangle?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 8900009 (AMR-088-0009). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Christian Wuthrich, Open problems in number theory, lecture slides (December 2011) slides headed Congruent number problem and Conjecture (following the triangle examples). https://www.maths.nottingham.ac.uk/plp/pmzcw/download/open.pdf.