Concordant Form

A concordant form is an integer triple  where

(1)

with  and  integers. Examples include

(2)

Dickson (2005) states that C. H. Brooks and S. Watson found in The Ladies' and Gentlemen's Diary (1857) that  and  can be simultaneously squares for  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.

REFERENCES:

Dickson, L. E. History of the Theory of Numbers, Vol. 1: Divisibility and Primality. New York: Dover, p. 475, 2005.

MathPages. "Concordant Forms." https://www.mathpages.com/home/kmath286.htm.