A prime p is called balancing non-Wieferich prime if B p−( 8 p ) ≡ 0 (mod p 2 ), where B n be the n-th balancing number and 8 p denotes the Jacobi symbol. Under the assumption of the abc conjecture for the number field Q[ √ 2], S. S. Rout proved that there are at least O(log x/ log log x) said primes p ≡ 1 (mod r), where r > 2 be any fixed integer. In this paper, we improve the lower bound such that for any given integer r > 2 there are ≫ log x primes p ≤ x satisfies B p−( 8p ) ≡ 0 (mod p 2 ) and p ≡ 1 (mod r), under the abc conjecture for the number field. This improves the recent result of Y. Wang and Y. Ding by adding an additional condition that the primes p are in arithmetic progression.

A prime p is called Lucas non-Wieferich prime associated to the pair of nonzero fixed integers, where U n be the Lucas sequence of first kind and ∆ p denotes the Jacobi symbol. Assuming the abc conjecture for the number field Q( √ ∆), S. S. Rout proved that there are at least O(log x/ log log x)(log log log x) M said primes p ≡ 1 (mod r), where M be the fixed positive integer. In this paper, we improve the lower bound such that for any given integer r ≥ 2 there are ≫ log x primes p ≤ x satisfies U p−( ∆ p ) ≡ 0 (mod p 2 ) and p ≡ 1 (mod r), under the assumption of the abc conjecture for the number field Q( √ ∆).

The negative polynomial Pell's equation is Pwhere d is an integer. In this paper, we prove that it has no non-trivial integer polynomial solutions P (X) and Q(X) if and only if d = ±1, ±2. For d = ±1, ±2, we investigate the existence of polynomial solutions P (X), Q(X) with integer coefficients.

We prove the lower bound for the number of Lucas non-Wieferich primes in arithmetic progressions. More precisely, for any given integer k ≥ 2 k\ge 2 , there are ≫ log x \gg \hspace{0.25em}\log x Lucas non-Wieferich primes p ≤ x p\le x such that p ≡ ± 1 ( mod k ) p\equiv \pm 1\hspace{0.25em}\left({\rm{mod}}\hspace{0.33em}k) , assuming the a b c abc conjecture for number fields.