A216869 The smallest non-constant arithmetic progression of integer squares of maximal length three.

1, 25, 49

Offset

1,2

Comments

Bremner (2012): "Euler showed that the length of the longest arithmetic progression (AP) of integer squares is equal to three. [See Dickson.] Xarles (2011) investigated APs in number fields, and proved the existence of an upper bound K(d) for the maximal length of an AP of squares in a number field of degree d. He shows that K(2) = 5." See A216870.

References

L. E. Dickson, History of the Theory of Numbers, Vol. II, Chelsea, New York, 1952, pp. 435-440.

Links

Table of n, a(n) for n=1..3.

A. Alvarado and E. H. Goins, Arithmetic progressions on conic sections, arXiv 2012.

A. Bremner, Arithmetic progressions of squares in cubic fields, Abstract 2012.

Formula

a(2) - a(1) = a(3) - a(2) = 24.

Example

a(1) = 1^2, a(2) = 5^2, a(3) = 7^2.

Crossrefs

Cf. A216870, A221671, A221672.

Sequence in context: A028505 A367496 A154082 * A143278 A106632 A284666

Adjacent sequences: A216866 A216867 A216868 * A216870 A216871 A216872

Keyword

nonn,fini,full,bref

Author

Jonathan Sondow, Nov 20 2012

Status

approved