A261117 Smallest positive integer b such that b^(2^n)+1 is divisible by the square of A035089(n+1).

8, 7, 110, 40, 1497, 894, 315, 48, 166107, 95853, 63609, 71589, 492348, 209628, 388440, 48853, 6118793, 2684186, 25787045, 49643800, 54302036, 3969770538, 17592956651, 7347360617, 991255542, 8249087392, 11518171450, 51385581002, 2268777293, 21252616802, 2822082710511

Offset: 0

Author

Jeppe Stig Nielsen, Aug 08 2015

Keywords

• nonn

Comments

For given n, if A035089(n+1) exists (which is true by Dirichlet's theorem on arithmetic progressions), then a(n) exists. Proof: p := A035089(n+1) is a prime of the form p=k*2^(n+1)+1, then the group (Z/(p^2)Z)* is cyclic of order p*(p-1) = p*k*2^(n+1). It therefore has an element b of order exactly 2^(n+1). For that b we have then b^(2^n) == -1 (mod p^2).

For given n, a(n) is not necessarily the smallest b such that b^(2^n)+1 is nonsquarefree; see A260824.

Examples

			Consider n=4, hence generalized Fermat numbers b^16+1. The first prime (A035089(4+1)) of the form 32*k+1 is 97. It follows that 97 is the smallest prime whose square divides a number of the form b^16+1. The first time 97^2 divides b^16+1 is for b=1497. Hence a(4)=1497. However, A260824(4) is smaller, A260824(4)=392. This is because already 392^16+1 is nonsquarefree (but the prime with a square dividing it, 769, exceeds 97).
		

Crossrefs

Cf. A260824, A248214, A035089.

Programs

PARI

a(n)=for(k=1,10^10,p=(k<<(n+1))+1;if(isprime(p),break()));for(b=1,p^2,b%p!=0&Mod(b,p^2)^(1<
				

PARI

a(n)=for(k=1, 10^10, p=(k<<(n+1))+1; if(isprime(p), break())); e=p*(p-1)/(1<<(n+1)); h=znprimroot(p^2)^e; g=h^2; m=p^2; for(i=1,1<