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A080169 Numbers that are cubes of primes of the form 4k+1 (A002144).

%I A080169 #17 Dec 02 2022 07:04:51
%S A080169 125,2197,4913,24389,50653,68921,148877,226981,389017,704969,912673,
%T A080169 1030301,1295029,1442897,2571353,3307949,3869893,5177717,5929741,
%U A080169 7189057,7645373,12008989,12649337,13997521,16974593,19465109,21253933
%N A080169 Numbers that are cubes of primes of the form 4k+1 (A002144).
%C A080169 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 A080169 a(n) is the hypotenuse of three and only three right triangles with integral arms.
%C A080169 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 A080169   See the Dickson reference, (A) on p. 227.
%D A080169 L. E. Dickson, History of the Theory of Numbers, Carnegie Institution, Publ. No. 256, Vol. II, Washington D.C., 1920, p. 227.
%D A080169 Morris Kline, Mathematical Thought from Ancient to Modern Times, 1972, pp. 275-276.
%H A080169 Amiram Eldar, <a href="/A080169/b080169.txt">Table of n, a(n) for n = 1..10000</a>
%F A080169 a(n) = A002144(n)^3, n >= 1.
%F A080169 Product_{n>=1} (1 - 1/a(n)) = A334425. - _Amiram Eldar_, Dec 02 2022
%e A080169 a(2) = 2197 is the hypotenuse of the three triangles 825, 2035, 2197; 845, 2028, 2197; 1547, 1560, 2197.
%e A080169 a(2) = 9^2 + 46^2  = 39^2 + 26^2, and these are the only decompositions. - _Wolfdieter Lang_, Jan 15 2015
%t A080169 Select[Prime[Range[60]], Mod[#, 4] == 1 &]^3 (* _Amiram Eldar_, Dec 02 2022 *)
%o A080169 (PARI) fermat(n) = { for(x=1,n, y=4*x+1; if(isprime(y),print1(y^3" ")) ) }
%Y A080169 Cf. A002144, A080109, A080175, A334425.
%K A080169 easy,nonn
%O A080169 1,1
%A A080169 _Cino Hilliard_, Mar 16 2003
%E A080169 Edited: New name, part of old one now as a comment. Dickson reference, formula and cross references added. - _Wolfdieter Lang_, Jan 15 2015