%I #68 Oct 07 2024 01:26:29
%S 1009,1129,1201,1801,2521,2689,3049,3361,3529,3889,4201,4561,4729,
%T 5209,5569,5881,6841,7561,7681,8089,8521,8689,8761,8929,9241,9601,
%U 9769,10369,12049,12289,12601,12721,12889,13441,13729,14281,14401,14449,15121,15241
%N Primes of the form x^2 + 840*y^2.
%C Discriminant = -3360. See A139643 for more information.
%C The primes are congruent to {1, 121, 169, 289, 361, 529} (mod 840).
%C Also, primes that in 1969 were unverified values for n for the Erdos-Straus conjecture (that 4/n = 1/x + 1/y + 1/z is always solvable in natural numbers), see Mordell 1969. - _Ron Knott_, Dec 11 2013
%C There are 273 terms < 100000 in this sequence. Of these, 59 are of the form 11n-3 or 11n-4. Since 4/(11*n-3)= 1/(3*n) + 1/(3*(11*n-3)) + 1/(n*(11*n-3)) and 4/(11*n-4)= 1/(3*n-1) + 1/(3*(11*n-4)) + 1/(3*(3*n-1)*(11*n-4)), these terms can be removed from the set of primes not proved for the Erdős-Straus conjecture. For example: 1009 = 11*92-3 so 4/1009 = 1/(3*92)+1/(3*1009)+1/(92*1009). These formulas were taken from the tables in Chapter 1 of the Mishima link. - _Gary Detlefs_, Jan 27 2014
%D R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section D11.
%D L. J. Mordell, Diophantine Equations, Academic press, 1969, pages 287-290.
%H Vincenzo Librandi and Ray Chandler, <a href="/A139665/b139665.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Vincenzo Librandi).
%H Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/egyptian.html#fxdlen">Egyptian Fractions</a>.
%H H. H. Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/overview.htm">Overview of "Mathematician's Secret Room"</a>.
%H N. J. A. Sloane et al., <a href="https://oeis.org/wiki/Binary_Quadratic_Forms_and_OEIS">Binary Quadratic Forms and OEIS</a> (Index to related sequences, programs, references).
%F From _Gary Detlefs_, Jan 22 2014: (Start)
%F a(n) == {1,25,49,73} (mod 96);
%F a(n)^2 == {1,49} (mod 96);
%F a(n)^4 == 1 (mod 96). (End)
%F a(n) == {1,9,25} (mod 56). - _Gary Detlefs_, Jan 27 2014
%t QuadPrimes2[1, 0, 840, 10000] (* see A106856 *)
%t Select[Table[Prime[n], {n, 1, 5000}], MemberQ[{1, 11^2, 13^2, 17^2, 19^2, 23^2}, Mod[#, 840]] &] (* _Ron Knott_, Dec 11 2013 *)
%o (Magma) [ p: p in PrimesUpTo(15000) | p mod 840 in {1, 121, 169, 289, 361, 529}]; // _Vincenzo Librandi_, Jul 29 2012
%o (Magma) k:=840; [p: p in PrimesUpTo(16000) | NormEquation(k, p) eq true]; // _Bruno Berselli_, Jun 01 2016
%Y Cf. A106856, A139643.
%K nonn,easy
%O 1,1
%A _T. D. Noe_, Apr 29 2008