A342390 Numbers k such that the ring of integers of Q(2^(1/k)) is not Z[2^(1/k)].

1093, 2186, 3279, 3511, 4372, 5465, 6558, 7022, 7651, 8744, 9837, 10533, 10930, 12023, 13116, 14044, 14209, 15302, 16395, 17488, 17555, 18581, 19674, 20767, 21066, 21860, 22953, 24046, 24577, 25139, 26232, 27325, 28088, 28418, 29511, 30604, 31599, 31697, 32790

Offset: 1

Author

Jianing Song, Mar 10 2021

Keywords

• nonn

Comments

For k > 1, a != 1 being a squarefree number (a != -1 unless k is a power of 2), then the ring of integers of Q(a^(1/k)) is Z[a^(1/k)] if and only if: for every p dividing k, we have a^(p-1) !== 1 (mod p^2). In other words, O_Q(a^(1/k)) = Z[a^(1/k)] if and only if none of the prime factors of k is a Wieferich prime of base a. See Theorem 5.3 of the paper of Keith Conrad.

In general, if a^d == 1 (mod p^2) for some d|(p-1), then it is easy to show that x = (1 + a^(d/p) + a^(2*d/p) + ... + a^((p-1)*d/p))/p is an algebraic integer not in Z[a^(1/p)].

Here a = 2, and the only known Wieferich primes of base 2 (A001220) are 1093, 3511 are no more below 4.97*10^17. So all known terms are multiples of either 1093 or 3511 (or both).

Examples

			2^364 == 1 (mod 1093^2), so x = (1 + 2^(364/1093) + 2^(2*364/1093) + ... + 2^(1092*364/1093))/1093 is an algebraic integer not in Z[2^(1/1093)].
		

Links

• Jianing Song, Table of n, a(n) for n = 1..11995 (all terms up to 10^7).
• Keith Conrad, The ring of integers in a radical extension

Crossrefs

Cf. A001220, A342391.

Programs

PARI

Wieferich_up_to_n(lim) = my(v=[]); forprime(p=2, lim, if(Mod(2,p^2)^(p-1)==1, v=concat(v,p))); v
Up_to_n(lim) = my(pv=Wieferich_up_to_n(lim), list=[]); for(i=1, #pv, my(p=pv[i]); for(k=1, lim\p, list=concat(list, k*p))); list=Set(list); list \\ corrected by Jianing Song, Mar 19 2022

PARI

Up_to_n(lim) = my(list=[]); for(k=1, lim\1093, list=concat(list, 1093*k)); for(k=1, lim\3511, list=concat(list, 3511*k)); list=Set(list); list \\ valid up to 4.97*10^17, corrected by Jianing Song, Mar 19 2022