Proof of D. J. Newman's Coprime Mapping Conjecture

We call the function / described in the theorem a coprime mapping. Theorem 1 settles in the affirmative a conjecture of D. J. Newman. The special case when / = {JV + l.JV + 2, ...,2Af} was proved by D. E. Daykin and M. J. Baines [2]. V. Chvatal [1] established Newman's conjecture for each JV ^ 1002. We prove Theorem 1 constructively by giving an algorithm for the construction of a coprime mapping / . This algorithm will be discussed in §2. If u is a real number and n is a natural number, let D{u, ri) denote the number of odd integers /, 1 < / < 2n — 1, with denotes Euler's function. If also k is a natural number, let E(k, ri) denote the maximal number of integers coprime to k that can be found in every set of n consecutive integers. Thus, for example, £(3,4) = 2, since in every set of 4 consecutive integers there are at least 2 integers coprime to 3 and the set {0,1,2, 3} has exactly 2 integers coprime to 3. If n > 1, let p^ri) denote the largest prime not exceeding 2n — \. We shall prove