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Under ABC, Silverman showed that there are infinitely many non-Wieferichprimes with respect to any (non-trivial) base $a$. Recently Srinivas andSubramani proved an analogous result over number fields with trivial classgroup. In the first part of this article, we extend their result to anyarbitrary number fields. Secondly, we give an asymptotic lower bound for thenumber of non-Wieferich prime ideals. Furthermore, we show a lower bound ofsame order is achievable for non-Wieferich prime ideals having norm congruentto $1 \pmod k.$ Lastly, we generalize Silverman's work for elliptic curves overarbitrary number fields following the treatment by K\"uhn and M\"uller.

Distribution of Non-Wieferich primes in certain algebraic groups under ABC