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
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
KEYWORD
nonn,fini,full
AUTHOR
Jonathan Sondow, Nov 20 2012
STATUS
approved