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

1,2

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

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

Example

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

Crossrefs

Sequence in context: A088769 A057986 A274204 * A105140 A107018 A247868

Adjacent sequences: A306926 A306927 A306928 * A306930 A306931 A306932

Keyword

nonn

Author

N. J. A. Sloane, Mar 16 2019

Extensions

More terms from Seiichi Manyama, Jul 15 2019

Status

approved