Congruent number

In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property.

The sequence of (integer) congruent numbers starts with

5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37, 38, 39, 41, 45, 46, 47, 52, 53, 54, 55, 56, 60, 61, 62, 63, 65, 69, 70, 71, 77, 78, 79, 80, 84, 85, 86, 87, 88, 92, 93, 94, 95, 96, 101, 102, 103, 109, 110, 111, 112, 116, 117, 118, 119, 120, ... (sequence A003273 in the OEIS)

For example, 5 is a congruent number because it is the area of a (20/3, 3/2, 41/6) triangle. Similarly, 6 is a congruent number because it is the area of a (3,4,5) triangle. 3 and 4 are not congruent numbers. The triangle sides demonstrating a number is congruent can have very large numerators and denominators, for example 263 is the area of a triangle whose two shortest sides are 16277526249841969031325182370950195/2303229894605810399672144140263708 and 4606459789211620799344288280527416/61891734790273646506939856923765.

If q is a congruent number then s²q is also a congruent number for any natural number s (just by multiplying each side of the triangle by s), and vice versa. This leads to the observation that whether a nonzero rational number q is a congruent number depends only on its residue in the group

${\displaystyle \mathbb {Q} ^{*}/\mathbb {Q} ^{*2},}$

where ${\displaystyle \mathbb {Q} ^{*}}$ is the set of nonzero rational numbers.

Every residue class in this group contains exactly one square-free integer, and it is common, therefore, only to consider square-free positive integers when speaking about congruent numbers.