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A139665 Primes of the form x^2 + 840*y^2. 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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 Erdos Strauss 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

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)

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

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 *)

PROG

(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

CROSSREFS

Sequence in context: A210759 A171119 A153640 * A261405 A024974 A025400

Adjacent sequences:  A139662 A139663 A139664 * A139666 A139667 A139668

KEYWORD

nonn,easy

AUTHOR

T. D. Noe, Apr 29 2008

STATUS

approved

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Last modified November 17 20:46 EST 2018. Contains 317278 sequences. (Running on oeis4.)