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By Peter Barlow

Barlow P. An ordinary research of the idea of numbers (Cornell collage Library, 1811)(ISBN 1429700467)

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1 of [1], we derive that there are primes q1 , . . 8) provided that d is not divisible by q1 , . . , qk . Let Q be the set of the least (1 − ε)u ln z ln ln z R= primes distinct from q1 , . . , qk . Denote by N their product N= q. 9) provided that z is large enough. Let D be the set of all products of r= (1 − ε) ln z ln ln z elements of Q, then for any d ∈ D, we have z 1−2ε < d < z. 10) 6 Bounds of Gaussian Sums 45 Denote by M = |D| the cardinality of D. 11. 8), we establish the existence of l with x/2z < l ≤ x/z and such that the numbers ld + 1 are prime for at least M/(4 ln x) values of d.

N, we denote by N j,t (h) the number of u ∈ V j with 1 ≤ |u| ≤ h, that is N j,t (h) = u ∈ V j : 1 ≤ |u| ≤ h , and also put S j (t) = e(v/ p) v∈V j and extend this definition for all j ∈ Z periodically with period n. Relations between H p (t) and N j,t (h), S j (t), are given by the following statement. 1. 5t j=1 49 50 Multiplicative Translations of Sets is satisfied for all k = 1, . . , n, then for any ε > 0, the bound p 1+ε h −1 H p (t) holds. Proof Let us fix some ε > 0. 5(1 + ε−1 ) , L = p 1+ε h −1 .

K. 6) where s, 1 ≤ s ≤ t − 1, is defined from the congruence sr ≡ 1 (mod t) and u i ≡ r xi (mod l), vi ≡ r yi (mod l), i = 1, . . , k. Let t (X ) be the tth cyclotomic polynomial. Since gt,l is of multiplicative s ) ≡ 0 (mod l). Therefore l divides the resultant order t modulo l then t (gt,l Rt (F) of F and t . 6) (and therefore for all ϕ(t) such polynomials). 5) has Wk (t) = O(t k ) trivial solutions. 6) such that 0 ≤ u 1 , v1 , . . , u k , vk ≤ t − 1, and Rt (F) = 0. Let us fix W such that 1 ≤ W ≤ t 2k .