OFFSET
1,1
COMMENTS
a(n) is the sum of two squares in exactly two ways (Fermat). See the Dickson reference, (B) on p. 277. - Wolfdieter Lang, Jan 15 2015
a(n) is the hypotenuse of three and only three right triangles with integral arms.
In 1640 Fermat generalized the 3,4,5 triangle with the theorem: A prime of the form 4n+1 is the hypotenuse of one and only one right triangle with integral arms. The square of a prime of the form 4n+1 is the hypotenuse of two and only two... The cube of three and only three... .
See the Dickson reference, (A) on p. 227.
REFERENCES
L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.
Morris Kline, Mathematical Thought from Ancient to Modern Times, 1972, pp. 275-276.
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = A002144(n)^3, n >= 1.
Product_{n>=1} (1 - 1/a(n)) = A334425. - Amiram Eldar, Dec 02 2022
EXAMPLE
a(2) = 2197 is the hypotenuse of the three triangles 825, 2035, 2197; 845, 2028, 2197; 1547, 1560, 2197.
a(2) = 9^2 + 46^2 = 39^2 + 26^2, and these are the only decompositions. - Wolfdieter Lang, Jan 15 2015
MATHEMATICA
Select[Prime[Range[60]], Mod[#, 4] == 1 &]^3 (* Amiram Eldar, Dec 02 2022 *)
PROG
(PARI) fermat(n) = { for(x=1, n, y=4*x+1; if(isprime(y), print1(y^3" ")) ) }
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Cino Hilliard, Mar 16 2003
EXTENSIONS
Edited: New name, part of old one now as a comment. Dickson reference, formula and cross references added. - Wolfdieter Lang, Jan 15 2015
STATUS
approved