In  and , Chahal showed that there exist infinitely many congruent numbers contained in every residue class modulo 8.
Chahal, Congruent numbers and elliptic curves, Amer.
A positive integer n is called a congruent number if it is equal to the area of a right triangle with rational sides.
Finding all possible congruent numbers is a problem that mathematicians have tackled for centuries.
That's why mathematicians from around the globe recently worked together, relying heavily on computer code and hard-drive space, to find congruent numbers up to one trillion.
The first few congruent numbers are 5, 6, 7, 13, 14, 15, 20, and 21.
Many congruent numbers were known prior to the new calculation.
The calculation found 3,148,379,694 of these more mysterious congruent numbers up to a trillion.
These results have motivated us to search for other representations of congruent numbers and relations among these numbers.
We present below a few interesting relations among these congruent numbers by means of theorems in which n is denoted by the notation n(r,s).
Chahal, [2,3] studied the connection between elliptic curves and congruent numbers, and employed an identity of Desboves to show that there are infinitely many congruent numbers in each residue class modulo 8.
The idea of a congruent number can be generalized by requiring only that n be equal to the area of a triangle with rational sides.