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A161681 Primes that are the difference between a cube and a square (conjectured values). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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

LINKS

Table of n, a(n) for n=1..55.

R. Zumkeller, Some Examples [From Reinhard Zumkeller, Oct 31 2009]

FORMULA

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

EXAMPLE

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

PROG

(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);

    )

  );

}

CROSSREFS

Cf. A000040.

Sequence in context: A138889 A097143 A038897 * A020583 A140557 A027697

Adjacent sequences:  A161678 A161679 A161680 * A161682 A161683 A161684

KEYWORD

nonn

AUTHOR

Cino Hilliard, Jun 16 2009

EXTENSIONS

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

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

STATUS

approved

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Last modified July 3 20:23 EDT 2020. Contains 335418 sequences. (Running on oeis4.)