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

49, 169, 289, 409, 529

Offset: 1

Author

Jonathan Sondow, Nov 20 2012

Keywords

• nonn
• fini
• full

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.

Examples

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

Links

• 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.

Crossrefs

Cf. A216869, A221671, A221672.

Programs

Mathematica

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

Formula

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