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

1, 25, 49

Offset: 1

Author

Jonathan Sondow, Nov 20 2012

Keywords

• nonn
• fini
• full
• bref

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.

Examples

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

References

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

Links

• A. Alvarado and E. H. Goins, Arithmetic progressions on conic sections, arXiv 2012.
• A. Bremner, Arithmetic progressions of squares in cubic fields, Abstract 2012.

Crossrefs

Cf. A216870, A221671, A221672.

Formula

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