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A216870 A maximal length five arithmetic progression of squares in a quadratic number field. 3
49, 169, 289, 409, 529 (list; graph; refs; listen; history; text; internal format)
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 A016922

Adjacent sequences:  A216867 A216868 A216869 * A216871 A216872 A216873

KEYWORD

nonn,fini,full

AUTHOR

Jonathan Sondow, Nov 20 2012

STATUS

approved

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Last modified April 7 13:36 EDT 2020. Contains 333305 sequences. (Running on oeis4.)