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A maximal length five arithmetic progression of squares in a quadratic number field.
3

%I #14 Dec 04 2018 11:03:22

%S 49,169,289,409,529

%N A maximal length five arithmetic progression of squares in a quadratic number field.

%C Bremner (2102): "Xarles (2011) investigated arithmetic progressions (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."

%C Euler showed that K(1) = 3. See A216869 for the smallest non-constant example. Another example is a(1), a(2), a(3) = 49, 169, 289 = 7^2, 13^2, 17^2.

%C It is known that K(3) >= 4.

%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.

%H X. Xarles, <a href="http://arxiv.org/abs/0909.1642">Squares in arithmetic progression over number fields</a>, arXiv:0909.1642 [math.AG], 2009.

%H X. Xarles, <a href="https://doi.org/10.1016/j.jnt.2011.07.010">Squares in arithmetic progression over number fields</a>, J. Number Theory, 132 (2012), 379-389.

%F a(n+1) - a(n) = 120 for n = 1, 2, 3, 4.

%e a(n) = 7^2, 13^2, 17^2, sqrt(409)^2, 23^2 for n = 1, 2, 3, 4, 5.

%t NestList[120+#&,49,4] (* _Harvey P. Dale_, Apr 20 2013 *)

%Y Cf. A216869, A221671, A221672.

%K nonn,fini,full

%O 1,1

%A _Jonathan Sondow_, Nov 20 2012