A129830 Conjectured first occurrence of numbers n with the property that there exist two consecutive primes p and q such that pq + n is a fourth power.

1, 4, 9, 10, 25, 35, 36, 46, 49, 64, 66, 75, 113, 144, 149, 179, 188, 196, 203, 221, 241, 250, 290, 302, 380, 395, 397, 400

Offset: 1

Author

Cino Hilliard, May 21 2007

Keywords

• easy
• nonn

Comments

It remains to prove that for certain n, pq+n != y^4 for all consecutive primes p and q. This list was computed for p and q with prime indices up to 10000.

Examples

			p=7,q=11,k=4. 7*11+4 = 81 = 3^4.
		

Crossrefs

Cf. A129783.

Programs

PARI

primefourth(n,m) = { local(c,k,x,p1,p2,j); c=0; for(k=1,m, for(x=1,n, p1=prime(x); p2=(prime(x+1)); y=p1*p2+k; if(isfourth(y), c++; print1(k","); break; ) ) ); c; } isfourth(n) = \Return 1 if n is a fourth power { local(r); r = sqrt(sqrt(n)); if(floor(r+.5)^4== n,1,0) }