A216869 - OEIS

%I #20 Nov 03 2013 08:27:54

%S 1,25,49

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

%C 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.

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

%H A. Alvarado and E. H. Goins, <a href="http://arxiv.org/abs/1210.6612">Arithmetic progressions on conic sections</a>, arXiv 2012.

%H A. Bremner, <a href="http://jointmathematicsmeetings.org/amsmtgs/2141_abstracts/1086-11-296.pdf">Arithmetic progressions of squares in cubic fields</a>, Abstract 2012.

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

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

%Y Cf. A216870, A221671, A221672.

%K nonn,fini,full,bref

%O 1,2

%A _Jonathan Sondow_, Nov 20 2012