A216870 A maximal length five arithmetic progression of squares in a quadratic number field.

49, 169, 289, 409, 529

Offset

1,1

Comments

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."

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.

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

Links

Table of n, a(n) for n=1..5.

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

X. Xarles, Squares in arithmetic progression over number fields, arXiv:0909.1642 [math.AG], 2009.

X. Xarles, Squares in arithmetic progression over number fields, J. Number Theory, 132 (2012), 379-389.

Formula

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

Example

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

Mathematica

NestList[120+#&, 49, 4] (* Harvey P. Dale, Apr 20 2013 *)

Crossrefs

Cf. A216869, A221671, A221672.

Sequence in context: A009431 A226353 A074216 * A254624 A256074 A369565

Adjacent sequences: A216867 A216868 A216869 * A216871 A216872 A216873

Keyword

nonn,fini,full

Author

Jonathan Sondow, Nov 20 2012

Status

approved