A239944 Smallest b > 0 making b^(2k) + b^k - 1 prime for k=1 through n.

2, 2, 2, 2, 460724, 610357585, 37748311920

Offset: 1

Author

James G. Merickel, Mar 31 2014

Keywords

• nonn

Comments

This sequence is a natural companion to A172994, which holds numbers b 'outperforming' 10 in this polynomial (relative to the sequence A096594). a(8) is known to exceed 4*10^12.

Note that a(n) exists for all n under Schinzel's hypothesis H. - Charles R Greathouse IV, Apr 24 2014

Examples

			a(5)=460724 means that for no integer k from 2 through 460723 does it hold that k^2n + k^n - 1 is prime for all n from 1 through 5, but for k=460724 this prime simultaneity occurs.
		

Crossrefs

Cf. A096594, A172994.

Programs

PARI

ok(b,n)=my(bk=1);for(k=1,n,bk*=b;if(!ispseudoprime(bk^2+bk-1),return(0)));b>0
a(n)=my(b=2); while(!is(b,n), b++); b \\ Charles R Greathouse IV, Apr 24 2014

PARI

\\ Reasonably efficient code, using precomputed modulus tables to speed the searches.
diff(v)=vector(#v-1,i,v[i+1]-v[i])
ok(b,n)=my(bk=1);for(k=1,n,bk*=b;if(!ispseudoprime(bk^2+bk-1),return(0)));b>0
okMod(b,p,n)=for(k=1,n,my(m=Mod(b,p)^k);if(m^2+m==1,return(0)));1
lst(p,n)=select(b->okMod(b,p,n),[0..p-1])
makeU(lim,n)=my(v=[0],m=1,t);forprime(p=5,lim,t=lst(p);if(#tCharles R Greathouse IV, Apr 24 2014