A161681 Primes that are the difference between a cube and a square (conjectured values).

2, 7, 11, 13, 19, 23, 47, 53, 61, 67, 71, 79, 83, 89, 107, 109, 127, 139, 151, 167, 191, 193, 199, 223, 233, 239, 251, 271, 277, 293, 307, 359, 431, 433, 439, 463, 487, 499, 503, 547, 557, 587, 593, 599, 631, 647, 673, 683, 719, 727, 769, 797, 859, 887, 919

Offset: 1

Author

Cino Hilliard, Jun 16 2009

Keywords

• nonn

Comments

The primes found among the differences are sorted in ascending order and unique primes are then extracted. I call this a "conjectured" sequence since I cannot prove that somewhere on the road to infinity there will never exist an integer pair x,y such that x^3-y^2 = 3,5,17,..., missing prime. For example, testing x^3-y^2 for x,y up to 10000, the count of some duplicates are:

duplicate,count

7,2

11,2

47,3

431,7

503,7

1999,5

28279,11

Yet for 3,5,17,29,... I did not find even one.

[Comment from Charles R Greathouse IV, Nov 03 2009: 587 = 783^3 - 21910^2, 769 = 1025^3 - 32816^2, and 971 = 1295^3 - 46602^2 were skipped in the original.]

Conjecture: The number of primes in x^3-y*2 is infinite.

Conjecture: The number of duplicates for a given prime is finite. Then there is the other side - the primes that are not in the sequence 3, 5, 17, 29, 31, 37, 41, 43, 59, 73, 97, 101, 103, ... Looks like a lot of twin components here. Do these have an analytical form? Is there such a thing as a undecidable sequence?

Range of A167224. - Reinhard Zumkeller, Oct 31 2009

Examples

			3^3 - 4^2 = 15^3 - 58^2 = 11.
		

Links

• R. Zumkeller, Some Examples [From _Reinhard Zumkeller_, Oct 31 2009]

Crossrefs

Cf. A000040.

Programs

PARI

diffcubesq(n) =
{
  local(a,c=0,c2=0,j,k,y);
  a=vector(floor(n^2/log(n^2)));
  for(j=1,n,
    for(k=1,n,
      y=j^3-k^2;
      if(ispseudoprime(y),
        c++;
        a[c]=y;
      )
    )
  );
  a=vecsort(a);
  for(j=2,c/2,
    if(a[j]!=a[j-1],
      c2++;
      print1(a[j]",");
      if(c2>100,break);
    )
  );
}

Formula

Integers x,y such that x^3-y^2 = p where p is prime. The generation bound is 10000.

Extensions

Extended and edited by Charles R Greathouse IV, Nov 03 2009

Further edits by N. J. A. Sloane, Nov 09 2009