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%I #17 Mar 21 2022 17:09:17
%S 20771,40487,62313,103855,121461,145397,186939,202435,228481,270023,
%T 283409,311565,353107,364383,394649,436191,445357,477733,519275,
%U 526331,560817,602359,607305,643901,685443,688279,726985,768527,769253,810069,850227,851611,893153,931201,934695,976237
%N Odd numbers k such that the ring of integers of Q(5^(1/k)) is not Z[5^(1/k)].
%C 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.
%C 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)].
%C Here a = 5. Since 2 is Wieferich prime of base 5, for all even k we have O_Q(a^(1/k)) != Z[a^(1/k)]. There are only 6 other known Wieferich primes of base 5 (A123692) up to 9.7*10^14.
%H Jianing Song, <a href="/A342391/b342391.txt">Table of n, a(n) for n = 1..7286</a> (all terms up to 2*10^8).
%H Keith Conrad, <a href="https://kconrad.math.uconn.edu/blurbs/gradnumthy/integersradical.pdf">The ring of integers in a radical extension</a>
%e 5^10385 == 1 (mod 20771^2), so x = (1 + 5^(10385/20771) + 5^(2*10385/20771) + ... + 5^(20770*10385/20771))/20771 is an algebraic integer not in Z[5^(1/20771)].
%o (PARI) Wieferich_up_to_n(lim) = my(v=[]); forprime(p=2, lim, if(Mod(5,p^2)^(p-1)==1, v=concat(v,p))); v
%o Up_to_n(lim) = my(pv=Wieferich_up_to_n(lim), list=[]); for(i=2, #pv, my(p=pv[i]); forstep(k=1, lim\p, 2, list=concat(list, k*p))); list=Set(list); list \\ corrected by _Jianing Song_, Mar 21 2022
%Y Cf. A123692, A342390.
%K nonn
%O 1,1
%A _Jianing Song_, Mar 10 2021
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