Paul Vanderveen has authored 3 sequences.
A356531
Primes p == 1 (mod 23) which are norms of elements in the 23rd cyclotomic field.
Original entry on oeis.org
599, 691, 829, 1151, 2347, 2393, 3037, 3313, 3359, 4463, 4831, 5107, 5521, 5659, 6763, 8741, 9109, 9661, 10627, 10949, 11593, 12743, 13249, 14537, 14767, 14951, 15319, 15733, 16883, 17573
Offset: 1
2347 is in this sequence since it is the norm of the element x^7-x^3-x-1 where x is a 23rd primitive root of unity.
- Reimer Bruchmann, Quadratic and cyclotomic rings of integers, March 26th, 2022, 487-534.
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.
Original entry on oeis.org
7, 11, 29, 23, 53, 103, 191, 599, 4931, 5953, 32783, 101107, 178021, 549149
Offset: 2
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.
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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); }
A350121
Increasing sequence of primes p == 3 (mod 4) such that all of 2,3,5,...,prime(n) are primitive roots mod p.
Original entry on oeis.org
3, 19, 907, 1747, 2083, 101467, 350443, 916507, 1014787, 6603283, 27068563, 45287587, 226432243, 243060283, 3946895803, 5571195667, 9259384843, 19633449763, 229012273627, 965558895907, 2793054173947, 5142304754563
Offset: 1
a(2) = 19 since 19 is the smallest prime (congruent to 3 (mod 4)) such that the first two primes (2 and 3) are primitive roots.
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max=0;Do[n=Prime@i;If[Mod[n,4]==3,k=1;While[MultiplicativeOrder[Prime@k,n]==n-1,k++];If[k-1>max,Print@n;max++]],{i,10^6}] (* Giorgos Kalogeropoulos, Dec 17 2021 *)
-
N=10^10;
default(primelimit, N);
A=2;
{ forprime (p=3, N,
if (p%4==3,
q = 1;
forprime (a=2, A,
if ( znorder(Mod(a, p)) != p-1, q=0; break() );
);
if ( q, A=nextprime(A+1); print1(p, ", ") );
);
); }
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