Non-Wieferich Primes And Related Topics Of Erd(?)s-Tur(?)n Conjecture

In this paper, we research two distinct issues, one of which is Non-Wieferich primes under the abc conjecture. The other one is Erd(?)s-Tur(?)n Conjecture in the positive rational numbers.Recently, Graves and Murty ( [11]) proved that for any integer a ≥ 2 and any fixed integer k ≥ 2, there are>>log x/log log x primes p≤x such that ap-1 (?) 1 (mod p2) and p ≡ 1 (mod k), under the assumption of the abc conjecture. In this chapter, We improve the bound log x/ log log x to (log x/ log log x) (log log log x)M for any fixed M. This result has been published in Proceeding of American Math-ematical Society, 145(2017) , 1833-1836.The well known Erd(?)s-Tur(?)n conjecture says that if A is a subset of the natural numbers such that every sufficiently large integer can be represented as a sum of two integers of A, then the number of representation ways cannot be bounded. In Chapter 2, we prove that this is false in the positive rational numbers. Let Q+ be the set of all positive rational numbers. For any A(?) Q+ and any α∈ Q+, let R(A, +,α), R(A,-,α),R(A,·,α ) and R(A,÷, α) denote the numbers of solutions of a = a + b (a≤ b)，α=a- b (a > b), α = ab (a≤b),α = a/b with a,b ∈ A, respectively. We prove that there exists a subset A of Q+ such that, for any α∈Q+ \ {1}, R(A,+,α) = 1,R(A,-,α) = 1, R(A,·,α) = 1, R(A,÷,α) = 1 hold simultaneously and R(A, +, 1) = 1, R(A,-, 1) = 1 and R(A, ·, 1) = 1. We also prove the similar results in Q ∩ (1,+∞) and Q ∩ (0,1).