Unsolved problems in number theory | Arithmetic problems of plane geometry | Triangle geometry | Elliptic curves
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 in the OEIS)Congruent number table: n ≤ 120 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. 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 , where 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. (Wikipedia).
