A002646 Half-quartan primes: primes of the form p = (x^4 + y^4)/2.

41, 313, 353, 1201, 3593, 4481, 7321, 8521, 10601, 14281, 14321, 14593, 21601, 26513, 32633, 41761, 41801, 42073, 42961, 49081, 56041, 66361, 67073, 72481, 90473, 97241, 97553, 104561, 106921, 111521, 139921, 141121, 165233, 195353, 198593

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

The 1001-digit number ((10^250 + 5659)^4 + (10^250 + 5661)^4)/2 is currently the largest known half-quartan prime. - Paul Muljadi, Mar 03 2011

The largest known is now ((2*3960926^2048 + 1)^4 + 1^4)/2 with 54051 digits. - Jens Kruse Andersen, Mar 20 2011

Primes of the form p = a^2 + b^2 with a > b > 0 such that a + b and a - b are squares. - Thomas Ordowski, Jul 07 2016

Primes p = a^2 + b^2 with a > b > 0 such that a^2 - b^2 is a square. - Thomas Ordowski, Feb 14 2017

Primes p > 5 such that the Diophantine equation X^4 + Y^2 = p^2 has a solution X,Y with nonzero X. Then X must be odd. - Thomas Ordowski and Robert G. Wilson v, Nov 29 2017

Examples

			41 is in the sequence since it is prime and 41 = (3^4 + 1^4)/2. - _Michael B. Porter_, Jul 07 2016
		

References

• A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
• J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 41, p. 16, Ellipses, Paris 2008.
• 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, Table of n, a(n) for n = 1..10000
• 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]
• Tim Evink, Jaap Top, and Jakob Dirk Top, A remark on prime (non)congruent numbers, arXiv:2105.01450 [math.NT], 2021. See p. 12.

Crossrefs

Cf. A000583, A002645, A005408, A010051.

Programs

Haskell

a002646 n = a002646_list !! (n-1)
a002646_list = [hqp | x <- [1, 3 ..], y <- [1, 3 .. x - 1],
                      let hqp = div (x ^ 4 + y ^ 4) 2, a010051' hqp == 1]
-- Reinhard Zumkeller, Jul 15 2013

Maple

N:= 10^6: # to get all terms <= N
sort(select(isprime, convert({seq(seq((x^4+y^4)/2, y=x..floor((2*N-x^4)^(1/4)),2),x=1..floor((2*N-1)^(1/4)),2)},list))); # Robert Israel, Jul 11 2016

Mathematica

nmax = 200000; jmax = Floor[(nmax/8)^(1/4)]; s = {}; Do[n = ((2 j + 1)^4 + (2 k + 1)^4)/2; If[n <= nmax && PrimeQ[n], AppendTo[s, n]], {j, 0, jmax}, {k, j,  jmax}]; Union[s] (* Jean-François Alcover, Mar 23 2011 *)
Sort[Select[Total/@(Union[Sort/@Tuples[Range[0,50],2]]^4)/2,PrimeQ]] (* Harvey P. Dale, Feb 12 2012 *)

Extensions

More terms from Len Smiley