In mathematics, 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, … (sequence A003273 in 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 is not a congruent number.
If q is a congruent number then s2q is also a congruent number for any rational number s (just by multiplying each side of the triangle by s). This leads to the observation that whether a nonzero rational number q is a congruent number depends only on its residue in the group
- .
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.
Read more about Congruent Number: Congruent Number Problem, Relation To Elliptic Curves, Current Progress
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