A080169 - OEIS

%I #17 Dec 02 2022 07:04:51

%S 125,2197,4913,24389,50653,68921,148877,226981,389017,704969,912673,

%T 1030301,1295029,1442897,2571353,3307949,3869893,5177717,5929741,

%U 7189057,7645373,12008989,12649337,13997521,16974593,19465109,21253933

%N Numbers that are cubes of primes of the form 4k+1 (A002144).

%C 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

%C a(n) is the hypotenuse of three and only three right triangles with integral arms.

%C 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... .

%C   See the Dickson reference, (A) on p. 227.

%D L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.

%D Morris Kline, Mathematical Thought from Ancient to Modern Times, 1972, pp. 275-276.

%H Amiram Eldar, <a href="/A080169/b080169.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A002144(n)^3, n >= 1.

%F Product_{n>=1} (1 - 1/a(n)) = A334425. - _Amiram Eldar_, Dec 02 2022

%e a(2) = 2197 is the hypotenuse of the three triangles 825, 2035, 2197; 845, 2028, 2197; 1547, 1560, 2197.

%e a(2) = 9^2 + 46^2  = 39^2 + 26^2, and these are the only decompositions. - _Wolfdieter Lang_, Jan 15 2015

%t Select[Prime[Range[60]], Mod[#, 4] == 1 &]^3 (* _Amiram Eldar_, Dec 02 2022 *)

%o (PARI) fermat(n) = { for(x=1,n, y=4*x+1; if(isprime(y),print1(y^3" ")) ) }

%Y Cf. A002144, A080109, A080175, A334425.

%K easy,nonn

%O 1,1

%A _Cino Hilliard_, Mar 16 2003

%E Edited: New name, part of old one now as a comment. Dickson reference, formula and cross references added. - _Wolfdieter Lang_, Jan 15 2015