A356467 Smallest prime congruent to 1 (mod prime(n)) which is the norm of some principal ideal in the ring of prime(n)-th cyclotomic integers.

7, 11, 29, 23, 53, 103, 191, 599, 4931, 5953, 32783, 101107, 178021, 549149

Offset: 2

Author

Paul Vanderveen, Aug 08 2022

Keywords

• nonn
• more

Comments

For the p-th cyclotomic ring of integers, primes which are norms of principal ideals should have asymptotic density 1/(h(p)*(p-1)) where h(p) is the class number of the p-th cyclotomic field.

Examples

			a(3) = 11 since 11 is the smallest prime congruent to 1 mod 5 (prime(3) = 5), which is the norm of some element in the 5th cyclotomic ring of integers. The algebraic integer x^2-x-1 has norm 11 where x is a primitive 5th root of unity.
a(2) - a(8) are the smallest primes congruent to 1 mod prime(n) as those corresponding cyclotomic fields have class number 1.
a(9) = 599. The 23rd cyclotomic ring of integers does not have class number 1. The smallest prime congruent to 1 (mod 23) is 47, and there is no cyclotomic integer with norm 47. The algebraic integer x^3-x-1 has norm 599 where x is a primitive 23rd root of unity.
		

Crossrefs

Cf. A035095.

Programs

PARI

a(n)={ p=prime(n); t=0; K=bnfinit(polcyclo(p)); q=1; while(t==0, q=nextprime(q+1); if(q%p==1 && #bnfisintnorm(K,q)>0, t=1); ); return(q); }