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Primes that are the difference between a fourth power and a positive cube.
0

%I #5 Oct 01 2013 21:35:30

%S 17,73,113,131,229,409,443,617,673,739,953,1153,1171,1889,2393,5087,

%T 6217,6553,8669,9433,9973,11321,11897,13877,14633,14737,15823,17377,

%U 18539,19081,19441,20393,20611,21841,25469,26249,26833,28649,29599

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

%C 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.

%C My guess is that the number of duplicates is infinite.

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

%o (PARI) difffourthcube(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^4-k^3;

%o if(ispseudoprime(y),

%o c++;

%o \\ print(j","k","y);

%o a[c]=y;

%o );

%o );

%o );

%o a=vecsort(a);

%o for(j=2,c,

%o if(a[j]!=a[j-1]&&a[j]!=0,

%o c2++;

%o print1(a[j]",");

%o if(c2>100,break);

%o );

%o );

%o }

%K nonn

%O 1,1

%A _Cino Hilliard_, Jun 17 2009