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A002645
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Quartan primes: primes of the form x^4 + y^4, x>0, y>0.
(Formerly M5042 N2178)
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5
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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 1353265-digit 145310^262144+1 = (145310^65536)^4+1^4, found by Ricky L Hubbard. - Jens Kruse Andersen, Mar 20 2011
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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).
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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. 145-174.
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FORMULA
| A000040 INTERSECTION A003336. - Jonathan Vos Post, Sep 23 2006
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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.
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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]]]
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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
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CROSSREFS
| Subsequence of A002313.
Cf. A002646, A000040, A000583, A003336.
Sequence in context: A053786 A181546 A081744 * A100268 A163790 A129123
Adjacent sequences: A002642 A002643 A002644 * A002646 A002647 A002648
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Nov 07 2002
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