For example (see Figure aside), 7 is congruent because
Equivalently, a number is congruent if there exist 3 rational squares , , and in arithmetic progression such that .
The problem of determining if a number is congruent is old and difficult. The numbers involved are often very large. For example, according to Zeigel, 157 is proved congruent by the right triangle with legs and .
A major advancement has been the characterization provided by J.Tunnell. He has proved that if the Birch and Swinnerton-Dyer conjecture is true, then an odd squarefree number is congruent if and only if the two sets
You can download a text file (primitive-congruent-mod123.txt) of 1.6 MB, containing a list of the squarefree congruent numbers below . To save further space the numbers of the form which are all congruent, are omitted from the list.
The smallest 3 × 3 magic square whose entries are consecutive congruent numbers is
Below, the spiral pattern of congruent numbers up to 2500. See the page on prime numbers for an explanation and links to similar pictures.