A161682 Primes that are not of the form x^3 - y^2.

3, 5, 17, 29, 31, 37, 41, 43, 59, 73, 97, 101, 103, 113, 131, 137, 149, 157, 163, 173, 179, 181, 197, 211, 227, 229, 241, 257, 263, 269, 281, 283, 311, 313, 317, 331, 337, 347, 349, 353, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 443, 449, 457, 461, 467, 479, 491, 509, 521, 523, 541

Offset: 1

Author

Cino Hilliard, Jun 16 2009

Keywords

• nonn

Comments

The current values are conjectural as they have been reduced from a finite list of values x^3 - y^2 within a search radius x,y < 10000.

Conjecture: The sequence is infinite.

No more solutions with x < 2.2*10^9. - Daniel Starodubtsev, Jan 22 2020

Crossrefs

Equals A000040 \ A161681.

Programs

Mathematica

(* assuming x < 10^4 *) notOfTheForm[p_] := Do[r = Reduce[ y > 0 && p == x^3 - y^2, {y}, Integers]; If[r =!= False, If[x > xmax, xmax = x; Print["xmax = ", xmax]]; Return[True]], {x, 1, 10^4}] =!= True; xmax = 1; Reap[ Do[ If[ notOfTheForm[p], Print["p = ", p]; Sow[p]], {p, Prime /@ Range[100]}]][[2, 1]] (* Jean-François Alcover, Oct 09 2012 *)

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););
);
}

Extensions

Worthless link removed by R. J. Mathar, Jul 16 2009

a(28)-a(61) from Jean-François Alcover, Oct 09 2012