OFFSET
1,1
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
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.
EXAMPLE
41 is in the sequence since it is prime and 41 = (3^4 + 1^4)/2. - Michael B. Porter, Jul 07 2016
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
EXTENSIONS
More terms from Len Smiley
STATUS
approved