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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 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.)