

A002645


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


13



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
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OFFSET

1,1


COMMENTS

Primes in the set {A000583 + A000583}.  Jonathan Vos Post, Sep 23 2006
The largest known quartan prime is currently the largest known generalized Fermat prime: The 1353265digit 145310^262144+1 = (145310^65536)^4+1^4, found by Ricky L Hubbard.  Jens Kruse Andersen, Mar 20 2011


REFERENCES

A. J. C. Cunningham, Binomial Factorisations, Vols. 19, Hodgson, London, 19231929; see Vol. 1, pp. 245259.
N. D. Elkies, Primes of the form a^4 + b^4, Mathematical Buds, Ed. H. D. Ruderman Vol. 3 Chap. 3 pp. 228 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

T. D. Noe and Moshe Levin, 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. 145174.


FORMULA

A000040 INTERSECTION A003336.  Jonathan Vos Post, Sep 23 2006


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]]]


PROG

(PARI) upto(lim)=my(v=List(2), t); forstep(x=1, lim^.25, 2, forstep(y=2, (limx^4)^.25, 2, if(isprime(t=x^4+y^4), listput(v, t)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 05 2011


CROSSREFS

Subsequence of A002313.
Cf. A002646, A000040, A000583, A003336.
Sequence in context: A181546 A081744 A219757 * 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



