Download PDF by Waclaw Sierpinski: 250 problems in elementary number theory

By Waclaw Sierpinski

ISBN-10: 0444000712

ISBN-13: 9780444000712

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But because of (118), (119) (120) 21 it also hold'sfor abs p ~ ~, i. e. for all p. From (114), (115), (121) we deduce (113). Theorem 12 is proved. THEOREM (122) 13: Let ~1 X1(q,l,xI,Y,s) be a primitive even character mod q and = (Det Y) (s2-sl) ~I ~ e(q,l,xI,Y,t)ts2-sl dt T I The integral of the right-hand side of (122) converges absolutely and is a holomorphic function for all s E ~2. Let R 2 be a nonsingular rational 2x2 matrix, j(Y) a positive number with Y ~ j(Y)E and Ac C 2 a compact domain.

From (253), (279) it follows I. We make the induction assumption that for each (1;n,m-1 B E ~1(1;n,m-1) there exists a V E Wl(1) with VB o j1 From (279) follows B = (B I *) with B 1 E ~1(1;n,m-1). Hence there exists a V I E W1(1) with )- 45 \ * \ I (283) VIB = ' am-1 ,0 \\. Here 1 (284) bI E %1(1,n+1-m,1) Hence there is a I V 2 E TI(1) I V2b I E J 1(l,n+1-m,1) with IEim-1) (285) .

By theorem 19 our problem is reduced to the case that q is a power of a prime-number. THEOREM 20: Let p (171) For be a prime-number, q = p p = 2, k = I p = 2, k ~ 2 there exists no p r i m i t i v e c respectively X X mod 2 k d x = x 4 ×2k X4, X2 k is a p r i m i t i v e For k character. For may be u n i q u e l y w r i t t e n as (c mod 2, d mod 2 k-2) are "basischaracters" 2k and k each character (172) Here k E ~ w i t h the conductors 4 2 the factor d does not appear. = X2k character mod 2 k, if and only if 31 (173) c @ 0 mod 2 (k = 2), (174) d ~ 0 rood 2 (k > 2 ) .

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250 problems in elementary number theory by Waclaw Sierpinski

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