A139665 Primes of the form x^2 + 840*y^2.

1009, 1129, 1201, 1801, 2521, 2689, 3049, 3361, 3529, 3889, 4201, 4561, 4729, 5209, 5569, 5881, 6841, 7561, 7681, 8089, 8521, 8689, 8761, 8929, 9241, 9601, 9769, 10369, 12049, 12289, 12601, 12721, 12889, 13441, 13729, 14281, 14401, 14449, 15121, 15241

Offset: 1

Author

T. D. Noe, Apr 29 2008

Keywords

• nonn
• easy

Comments

Discriminant = -3360. See A139643 for more information.

The primes are congruent to {1, 121, 169, 289, 361, 529} (mod 840).

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

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

References

• R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section D11.
• L. J. Mordell, Diophantine Equations, Academic press, 1969, pages 287-290.

Links

• Vincenzo Librandi and Ray Chandler, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi).
• Ron Knott, Egyptian Fractions.
• H. H. Mishima, Overview of "Mathematician's Secret Room".
• N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).

Crossrefs

Cf. A106856, A139643.

Programs

Magma

[ p: p in PrimesUpTo(15000) | p mod 840 in {1, 121, 169, 289, 361, 529}]; // Vincenzo Librandi, Jul 29 2012

Magma

k:=840; [p: p in PrimesUpTo(16000) | NormEquation(k, p) eq true]; // Bruno Berselli, Jun 01 2016

Mathematica

QuadPrimes2[1, 0, 840, 10000] (* see A106856 *)
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 *)

Formula

From Gary Detlefs, Jan 22 2014: (Start)

a(n) == {1,25,49,73} (mod 96);

a(n)^2 == {1,49} (mod 96);

a(n)^4 == 1 (mod 96). (End)

a(n) == {1,9,25} (mod 56). - Gary Detlefs, Jan 27 2014