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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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23
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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;
text;
internal format)
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OFFSET
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1,1
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COMMENTS
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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
Primes of the form (a^2 + b^2)/2 such that |a^2 - b^2| is a square. - Thomas Ordowski, Feb 22 2017
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REFERENCES
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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
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T. D. Noe and 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.
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FORMULA
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A000040 INTERSECTION A003336. - Jonathan Vos Post, Sep 23 2006
A256852(A049084(a(n))) > 1 for n > 1. - Reinhard Zumkeller, Apr 11 2015
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EXAMPLE
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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
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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 *)
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PROG
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(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
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CROSSREFS
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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
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Victoria A Sapko (vsapko(AT)canes.gsw.edu), Nov 07 2002
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STATUS
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approved
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