Balancing non-Wieferich primes in arithmetic progression and $abc$ conjecture

Lucas non-Wieferich primes in arithmetic progressions and the $abc$ conjecture

“…The following result found in [6]. The following result available in [8] Lemma 3.3. For any real number α with |α| > 1, there exists C > 0 such that…”

A note on balancing non-Wieferich primes in arithmetic progressions

“…If we replace integer a with real number α in Lemma 2.2, a similar type of inequality holds for real α, which is available in [9]. It is stated as follows.…”

“…Moreover, for any prime p > 2, B p−( 8p ) ≡ 0 (mod p), where 8 p denotes the Jacobi symbol [6]. Then S. S. Rout [9] called the prime p as a balancing Wieferich prime if it satisfies the congruence B p−( 8 p ) ≡ 0 (mod p 2 ). Otherwise it is called balancing non-Wieferich prime and he proved that for an integer r ≥ 2, under the assumption of the abc conjecture for the number field Q[ √ 2], there are at least O(log x/ log log x) balancing non-Wieferich primes p with p ≡ 1 (mod r) (see [9]).…”

“…Then S. S. Rout [9] called the prime p as a balancing Wieferich prime if it satisfies the congruence B p−( 8 p ) ≡ 0 (mod p 2 ). Otherwise it is called balancing non-Wieferich prime and he proved that for an integer r ≥ 2, under the assumption of the abc conjecture for the number field Q[ √ 2], there are at least O(log x/ log log x) balancing non-Wieferich primes p with p ≡ 1 (mod r) (see [9]). Then U. K. Dutta et al improved the lower bound from log x/ log log x to (log x/ log log x)(log log log x) M , where M is any fixed positive integer.…”