关于平衡非Wieferich素数的一个注记

A Note On Balancing Non-Wieferich Primes

令Bn是第n个平衡数。最近,Routh证明了在数域Q[√2]的abc猜想成立的前提下,对于任意正整数k>2和n>1,有>>logx/loglogx的素数p满足p≡1(mod k)和Bp-(8/p)■0(mod p^2),其中(·/p)表示Jacobi符号。对于任意给定的正数M,Dutta,Pate和Ray将下界改进到(logx/loglogx)(logloglogx)^M。但是,我们发现他们的证明中有一些漏洞。尽管我们现在没能弥补这些漏洞,但在去掉mod k的限制条件下,我们将下界进一步改进到logx,即在数域Q[√2]的abc猜想成立的前提下有>>logx的素数p满足Bp-(8/p)■0(mod p^2)。

Let Bn be the n-th balancing number.Recently,Routh proved that for any positive integers k≥2 and n>1,there are>>logx/loglogx primes p≤x such that p≡1(mod k)and B p-(8/p)■0(mod p^2)under the assumption of the abc conjecture for the number field Q[√2],where(·/p)is the Jacobi symbol.Dutta,Patel and Ray improved this lower bound to(logx/loglogx)(logloglogx)M for any fixed M.However,we found that their proofs contain some gaps.Although we failed to fill the gaps right now,the lower bound is improved to logx but without the restriction of mod k in this paper,i.e.,we prove that there are>>logx primes p≤x such that B p-(8/p)■0(mod p^2)assuming the abc conjecture for the number field Q[√2].