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
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Product_{n>=1} (1 - 1/a(n)) = A334446. - Amiram Eldar, Dec 02 2022
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[100] + 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
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