A306929 Numbers k (>0) such that x^2+y^2 and x^2+k*y^2 can be simultaneously squares.

1, 7, 10, 11, 17, 20, 22, 23, 24, 27, 30, 31, 34, 41, 42, 45, 47, 49, 50, 52, 53, 57, 58, 59, 60, 61, 68, 71, 72, 74, 76, 77, 79, 82, 83, 85, 86, 90, 92, 93, 94, 97, 99, 100, 101, 102, 104, 105, 107, 110, 111, 112, 113, 114, 115, 119, 120, 121, 122, 124, 126, 127, 130, 133, 134, 137

Offset: 1

Author

N. J. A. Sloane, Mar 16 2019

Keywords

• nonn

Comments

Note that Dickson refers to C. H. Brooks and S. Watson, 1857 and lists "the following 41 positive integers A<=100." but 47, 53 and 83 are missing. - Michael Somos, Feb 09 2020

Examples

			From _Seiichi Manyama_, Jul 15 2019: (Start)
14663^2 + 111384^2 = 112345^2 and 14663^2 + 47*111384^2 = 763751^2. So 47 is a term.
2873161^2 + 2401080^2 = 3744361^2 and 2873161^2 + 83*2401080^2 = 22062761^2. So 83 is a term. (End)
From the K. S. Brown link, 1141^2 + 13260^2 = 13309^2, 1141^2 + 53*13260^2 = 96541^2, so 53 is a term. - _Michael Somos_, Feb 10 2020
		

References

• The Lady's and Gentleman's Diary, London, 1857, pp. 61-63. See question 1911.

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..1000
• L. E. Dickson, History of the Theory of Numbers, vol. 2. Carnegie Institute Public. 256, Washington, DC, 1920, see p. 475.
• H. Mishima, Tables of Representation of concordant number n
• K. S. Brown, Concordant Forms

Extensions

More terms from Seiichi Manyama, Jul 15 2019