A161735 Primes that are the difference between a fourth power and a positive cube.

17, 73, 113, 131, 229, 409, 443, 617, 673, 739, 953, 1153, 1171, 1889, 2393, 5087, 6217, 6553, 8669, 9433, 9973, 11321, 11897, 13877, 14633, 14737, 15823, 17377, 18539, 19081, 19441, 20393, 20611, 21841, 25469, 26249, 26833, 28649, 29599

Offset: 1

Author

Cino Hilliard, Jun 17 2009

Keywords

• nonn

Comments

There are primes like p = 20393, 3905513, 5177033, 28398833, or 10877895569 which have more than one representation p=x^4-y^3, with x,y>=1.

My guess is that the number of duplicates is infinite.

Programs

PARI

difffourthcube(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^4-k^3;
if(ispseudoprime(y),
c++;
\\ print(j","k","y);
a[c]=y;
);
);
);
a=vecsort(a);
for(j=2,c,
if(a[j]!=a[j-1]&&a[j]!=0,
c2++;
print1(a[j]",");
if(c2>100,break);
);
);
}

Formula

If x^4-y^3 is prime for integers x >=1, y>=1, list it.