A002645 Quartan primes: primes of the form x^4 + y^4, x > 0, y > 0. (Formerly M5042 N2178)

2, 17, 97, 257, 337, 641, 881, 1297, 2417, 2657, 3697, 4177, 4721, 6577, 10657, 12401, 14657, 14897, 15937, 16561, 28817, 38561, 39041, 49297, 54721, 65537, 65617, 66161, 66977, 80177, 83537, 83777, 89041, 105601, 107377, 119617, 121937

Offset

1,1

Comments

The largest known quartan prime is currently the largest known generalized Fermat prime: The 1353265-digit 145310^262144 + 1 = (145310^65536)^4 + 1^4, found by Ricky L Hubbard. - Jens Kruse Andersen, Mar 20 2011

Primes of the form (a^2 + b^2)/2 such that |a^2 - b^2| is a square. - Thomas Ordowski, Feb 22 2017

References

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.

N. D. Elkies, Primes of the form a^4 + b^4, Mathematical Buds, Ed. H. D. Ruderman Vol. 3 Chap. 3 pp. 22-8 Mu Alpha Theta 1984.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)

A. J. C. Cunningham, High quartan factorisations and primes, Messenger of Mathematics 36 (1907), pp. 145-174.

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]

Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

Formula

A000040 INTERSECTION A003336. - Jonathan Vos Post, Sep 23 2006

A256852(A049084(a(n))) > 1 for n > 1. - Reinhard Zumkeller, Apr 11 2015

Example

a(1) =   2 = 1^4 + 1^4.
a(2) =  17 = 1^4 + 2^4.
a(3) =  97 = 2^4 + 3^4.
a(4) = 257 = 1^4 + 4^4.

Mathematica

nn = 100000; Sort[Reap[Do[n = a^4 + b^4; If[n <= nn && PrimeQ[n], Sow[n]], {a, nn^(1/4)}, {b, a}]][[2, 1]]]
With[{nn=20}, Select[Union[Flatten[Table[x^4+y^4, {x, nn}, {y, nn}]]], PrimeQ[ #] && #<=nn^4+1&]] (* Harvey P. Dale, Aug 10 2021 *)

Prog

(PARI) upto(lim)=my(v=List(2), t); forstep(x=1, lim^.25, 2, forstep(y=2, (lim-x^4)^.25, 2, if(isprime(t=x^4+y^4), listput(v, t)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 05 2011
(PARI) list(lim)=my(v=List([2]), x4, t); for(x=1, sqrtnint(lim\=1, 4), x4=x^4; forstep(y=1+x%2, min(sqrtnint(lim-x4, 4), x-1), 2, if(isprime(t=x4+y^4), listput(v, t)))); Set(v) \\ Charles R Greathouse IV, Aug 20 2017
(Haskell)
a002645 n = a002645_list !! (n-1)
a002645_list = 2 : (map a000040 $ filter ((> 1) . a256852) [1..])
-- Reinhard Zumkeller, Apr 11 2015

Crossrefs

Subsequence of A002313 and of A028916.

Intersection of A004831 and A000040.

Cf. A002646, A000583, A003336, A000290, A256852.

Sequence in context: A219757 A297727 A358591 * A100268 A163790 A129123

Adjacent sequences: A002642 A002643 A002644 * A002646 A002647 A002648

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Nov 07 2002

Status

approved