By Leo Moser

ISBN-10: 1931705011

ISBN-13: 9781931705011

**Read Online or Download An Introduction to the Theory of Numbers PDF**

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We now return to the question raised by Gauss. What can be said about the least nonresidue np of a prime? ”. These questions were attacked in the 1920s by a number of mathematicians including Nagel, Schur, Polya, Zeitz, Landau, Vandiver, Brauer, and Vinogradov. Nagel, √ for example, proved that for p = 7, 23, np < p. Polya and Schur proved that b n=a n p < √ p log p. 51 Chapter 5. Congruences √ This implies that there are never more than p log p consecutive residues or √ nonresidues and that ranges much larger than p log p have about as many residues as nonresidues.

Theorem 1. p≤n The proof is by induction on n. We assume the theorem true for integers < n and consider the cases n = 2m and n = 2m + 1. If n = 2m then p < 42m−1 p= p≤2m p≤2m−1 23 Chapter 3. Distribution of Primes by the induction hypothesis. If n = 2m + 1 then p= p≤2m+1 p p m+1

p> p≤n √ n≤p≤n p> √ n √ π(n)−π( n) Taking logarithms we obtain √ 1 n log 4 > (π(n) − π( n)) log n 2 or √ n · 4 log 2 π(n) − π( n) < log n or π(n) < (4 log 2) √ n cn + n< .

But now if a point has a double representation, then P eiθ = R(eiθ ) and we would obtain a polynomial in eiθ which would negate the transcendental character of eiθ . Let T˜ denote the subset of S˜ which consists of those points of S˜ for which ˜ denote the subset the last operation needed to reach them is a T , and let R ˜ which consist of those points of S for which the last operation needed to reach ˜ and T˜ ∩ R ˜ = ∅. A translation of S˜ of one them is an R. Clearly S˜ = T˜ ∪ R ˜ ˜ ˜ vanish!