%I #31 Dec 02 2022 07:04:56

%S 25,169,289,841,1369,1681,2809,3721,5329,7921,9409,10201,11881,12769,

%T 18769,22201,24649,29929,32761,37249,38809,52441,54289,58081,66049,

%U 72361,76729,78961,85849,97969,100489,113569,121801,124609,139129

%N Square of primes of the form 4k+1 (A002144).

%C a(n) is the sum of two positive squares in only one way. See the Dickson reference, (B) p. 227.

%C a(n) is the hypotenuse of two and only two right triangles with integral legs (modulo leg exchange). See the Dickson reference, (A) p. 227.

%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...

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

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

%H Amiram Eldar, <a href="/A080109/b080109.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A002144(n)^2 = A070079(n)^2 + (4*A070151(n))^2, for n >= 1. - _Wolfdieter Lang_, Jan 13 2015

%F From _Amiram Eldar_, Dec 02 2022: (Start)

%F Product_{n>=1} (1 + 1/a(n)) = A243380

%F Product_{n>=1} (1 - 1/a(n)) = A088539. (End)

%e a(7) = 2809 is the hypotenuse of triangles 1241, 2520, 2809 and 1484, 2385, 2809, and only of these.

%e a(7) = 53^2 = 2809 = 45^2 + (4*7)^2, and this is the only way. - _Wolfdieter Lang_, Jan 13 2015

%t Select[4 Range[96] + 1, PrimeQ]^2 (* _Michael De Vlieger_, Dec 27 2016 *)

%o (PARI) fermat(n) = { for(x=1,n, y=4*x+1; if(isprime(y),print1(y^2" ")) ) }

%Y Cf. A002144, A080175. - _Wolfdieter Lang_, Jan 13 2015

%Y Cf. A070079, A070151, A088539, A243380.

%K nonn,easy

%O 1,1

%A _Cino Hilliard_, Mar 16 2003

%E Edited: Name changed, part of old name as comment. Comments added and changed. Dickson reference added. - _Wolfdieter Lang_, Jan 13 2015