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

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

Offset

1,1

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

Links

Table of n, a(n) for n=1..7.

Example

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.

Prog

(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(#t<p, my(V=vector(#v*#t), idx); for(i=1, #v, my(vim=Mod(v[i], m)); for(j=1, #t, V[idx++]=lift(chinese(vim, Mod(t[j], p))))); v=V; m*=p)); v=Set(v); diff(concat(v, m+v[1]))
a(n)=forstep(b=0, 9e99, makeU(31, n), if(ok(b), return(b)))
\\ Charles R Greathouse IV, Apr 24 2014

Crossrefs

Cf. A096594, A172994.

Sequence in context: A049300 A339017 A084957 * A235812 A343471 A035307

Adjacent sequences: A239941 A239942 A239943 * A239945 A239946 A239947

Keyword

nonn

Author

James G. Merickel, Mar 31 2014

Status

approved