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 A080175 Fourth power of primes of the form 4k+1 (A002144). 7
 625, 28561, 83521, 707281, 1874161, 2825761, 7890481, 13845841, 28398241, 62742241, 88529281, 104060401, 141158161, 163047361, 352275361, 492884401, 607573201, 895745041, 1073283121, 1387488001, 1506138481, 2750058481 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS a(n) is the hypotenuse of four and only four right triangles with integral legs (Fermat). See the Dickson reference, (A) on p. 227. 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... REFERENCES 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. Morris Kline, Mathematical Thought from Ancient to Modern Times, 1972, pp. 275-276. LINKS FORMULA a(n) = A002144(n)^4 = A080109(n)^2, n >= 1. EXAMPLE 625 is the hypotenuse of triangles 175, 600, 625; 220, 585, 625; 336, 527, 625; 375, 500, 625. MAPLE seq(p^4, p = select(isprime, [seq(4*k+1, k=1..100)])); # Robert Israel, Jan 14 2015 MATHEMATICA Select[4 Range + 1, PrimeQ[#] &]^4 (* Vincenzo Librandi, Jun 24 2015 *) PROG (PARI) fermat(n) = { for(x=1, n, y=4*x+1; if(isprime(y), print1(y^4, " ")) ) } (MAGMA) [a^4: n in [0..40] | IsPrime(a) where a is 4*n + 1 ]; // Vincenzo Librandi, Jun 24 2015 CROSSREFS Cf. A002144, A080109. Sequence in context: A016972 A156162 A017044 * A017128 A180259 A017224 Adjacent sequences:  A080172 A080173 A080174 * A080176 A080177 A080178 KEYWORD easy,nonn AUTHOR Cino Hilliard, Mar 16 2003 EXTENSIONS Edited: name shortened, part of old name as a comment, comment changed, Dickson reference, formula and cross references added. - Wolfdieter Lang, Jan 14 2015 STATUS approved

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Last modified March 30 10:09 EDT 2020. Contains 333125 sequences. (Running on oeis4.)