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%I #18 Nov 16 2019 15:40:46
%S 2,7,11,13,19,23,47,53,61,67,71,79,83,89,107,109,127,139,151,167,191,
%T 193,199,223,233,239,251,271,277,293,307,359,431,433,439,463,487,499,
%U 503,547,557,587,593,599,631,647,673,683,719,727,769,797,859,887,919
%N Primes that are the difference between a cube and a square (conjectured values).
%C 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:
%C duplicate,count
%C 7,2
%C 11,2
%C 47,3
%C 431,7
%C 503,7
%C 1999,5
%C 28279,11
%C Yet for 3,5,17,29,... I did not find even one.
%C [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.]
%C Conjecture: The number of primes in x^3-y*2 is infinite.
%C 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?
%C Range of A167224. - _Reinhard Zumkeller_, Oct 31 2009
%H R. Zumkeller, <a href="/a161681.txt">Some Examples</a> [From _Reinhard Zumkeller_, Oct 31 2009]
%F Integers x,y such that x^3-y^2 = p where p is prime. The generation bound is 10000.
%e 3^3 - 4^2 = 15^3 - 58^2 = 11.
%o (PARI) diffcubesq(n) =
%o {
%o local(a,c=0,c2=0,j,k,y);
%o a=vector(floor(n^2/log(n^2)));
%o for(j=1,n,
%o for(k=1,n,
%o y=j^3-k^2;
%o if(ispseudoprime(y),
%o c++;
%o a[c]=y;
%o )
%o )
%o );
%o a=vecsort(a);
%o for(j=2,c/2,
%o if(a[j]!=a[j-1],
%o c2++;
%o print1(a[j]",");
%o if(c2>100,break);
%o )
%o );
%o }
%Y Cf. A000040.
%K nonn
%O 1,1
%A _Cino Hilliard_, Jun 16 2009
%E Extended and edited by _Charles R Greathouse IV_, Nov 03 2009
%E Further edits by _N. J. A. Sloane_, Nov 09 2009
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