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 ?? Q+, let R(A, +,?), R(A,-,?),R(A,·,? ) and R(A,÷, ?) denote the numbers of solutions of a = a + b (a? b)??=a- b (a > b), ? = ab (a?b),? = a/b with a,b ? A, respectively. We prove that there exists a subset A of Q+ such that, for any ??Q+ \ {1}, R(A,+,?) = 1,R(A,-,?) = 1, R(A,·,?) = 1, R(A,÷,?) = 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).