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%I #27 Dec 02 2022 07:05:01
%S 625,28561,83521,707281,1874161,2825761,7890481,13845841,28398241,
%T 62742241,88529281,104060401,141158161,163047361,352275361,492884401,
%U 607573201,895745041,1073283121,1387488001,1506138481,2750058481
%N Fourth power of primes of the form 4k+1 (A002144).
%C a(n) is the hypotenuse of four and only four right triangles with integral legs (Fermat). See the Dickson reference, (A) on p. 227.
%C In 1640 Fermat generalized the 3,4,5 Pythagorean triangle with the theorem: A prime of the form 4k+1 is the hypotenuse of one and only one right triangle with integral legs. The square of a prime of the form 4k+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 Publication 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="/A080175/b080175.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = A002144(n)^4 = A080109(n)^2, n >= 1.
%F Product_{n>=1} (1 - 1/a(n)) = A334446. - _Amiram Eldar_, Dec 02 2022
%e 625 is the hypotenuse of triangles 175, 600, 625; 220, 585, 625; 336, 527, 625; 375, 500, 625.
%p seq(p^4, p = select(isprime,[seq(4*k+1,k=1..100)])); # _Robert Israel_, Jan 14 2015
%t Select[4 Range[100] + 1, PrimeQ[#] &]^4 (* _Vincenzo Librandi_, Jun 24 2015 *)
%o (PARI) fermat(n) = { for(x=1,n, y=4*x+1; if(isprime(y),print1(y^4, " ")) ) }
%o (Magma) [a^4: n in [0..40] | IsPrime(a) where a is 4*n + 1 ]; // _Vincenzo Librandi_, Jun 24 2015
%Y Cf. A002144, A080109, A334446.
%K easy,nonn
%O 1,1
%A _Cino Hilliard_, Mar 16 2003
%E Edited: name shortened, part of old name as a comment, comment changed, Dickson reference, formula and cross references added. - _Wolfdieter Lang_, Jan 14 2015
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