Concordant Form

A concordant form is an integer Triple $(a,b,N)$ where

$$\cases{ a^2 + b^2 = c^2\cr a^2 + Nb^2 = d^2,\cr}$$

with $c$ and $d$ integers. Examples include

$\cases{ 14663^2 + 111384^2 = 112345^2\cr 14663^2 + 47\cdot 111384^2 = 763751^2\cr}$
$\cases{ 1141^2 + 13260^2 = 13309^2\cr 1141^2 + 53\cdot 13260^2 = 96541^2\cr}$
$\cases{ 2873161^2 + 2401080^2 = 3744361^2\cr 2873161^2 + 83\cdot 2401080^2 = 22062761^2.\cr}$

Dickson (1962) states that C. H. Brooks and S. Watson found in The Ladies' and Gentlemen's Diary (1857) that $x^2 + y^2$ and $x^2 + Ny^2$ can be simultaneously squares for $N<100$ only for 1, 7, 10, 11, 17, 20, 22, 23, 24, 27, 30, 31, 34, 41, 42, 45, 49, 50, 52, 57, 58, 59, 60, 61, 68, 71, 72, 74, 76, 77, 79, 82, 85, 86, 90, 92, 93, 94, 97, 99, and 100 (which evidently omits 47, 53, and 83 from above). The list of concordant primes less than 1000 is now complete with the possible exception of the 16 primes 103, 131, 191, 223, 271, 311, 431, 439, 443, 593, 607, 641, 743, 821, 929, and 971 (Brown).

See also Congruum

References

Brown, K. S. ``Concordant Forms.'' http://www.seanet.com/~ksbrown/kmath286.htm.

Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Chelsea, p. 475, 1952.