History of the Theory of Numbers - Hardcover

Dickson, Leonard E.

Hardcover

ISBN 10: 0828400865 ISBN 13: 9780828400862

Publisher: American Mathematical Society, 1971

Synopsis

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1919 Excerpt: ...di Mat., 25, 1887, 161-173. 1"Casopis, Prag, 18, 1889, 97; cf. Fortschritte Math., 1889, 30. F. J. Studnicka191 treated at length the solution in integers x, y (yV) of 6x+l=?/2, discussed by Leibniz in 1716. L. Gegenbauer192 gave a new derivation of the equations of Berger188 leading to asymptotic expressions for the number of solutions of x2=D. A. Tonelli193 gave a method of solving x2=c (mod p), when p is a prime 4/i+l and some quadratic non-residue g of p is known. Set p = 2'y + l, where y is odd. By Euler's criterion, the power y2" l of c and g are congruent to +1,--1. Set eo = 0 or 1, according as the power y2'2 of c is congruent to +1 or--1. Then 2-12-2= +i(mod p). For s3, set ii = 0 or 1 according as the square root of the left member is = +lor-1. Then G. B. Mathews77 (p. 53) treated the cases in which x2=a (mod p) is solvable by formulas. Cf. Legendre.166 S. Dickstein194 noted that H. Wronski169 gave the solution-d +MJ of zn--ay"=0 (mod M) with (1/1)2 in place of K, and gave, as the condition for solvability, o(1/1)'-1«0mod.M). But there may be solutions when the last condition is satisfied by no integer k. This is due to the fact that the value assigned to y imposes a limitation, which may be avoided by using the same expressions for y, z in a parameter K, subject to the condition aKn--1=0 (mod M). M. F. J. Mann194" proved that, if n=2V-., where X, M,... are distinct odd primes, the number of solutions of xp=l(mod n) is GGiG2... 0i02 where G= 1 if n or p is odd, otherwise G is the g. c. d. of 2p and 2-1, and where Gi, G2,.., gi, g2l.. are the g. c. d.'s of p with X0-1, M-1)-., X--1,M--1,..., respectively. A. Tonelli196 gave an explicit formula for the roots of x2==c (mod px), 0, 1,2,... until we reach a number of the form z...

Edition details

• Publisher: American Mathematical Society
• Publication date: 1971
• Language: English
• ISBN 10: 0828400865
• ISBN 13: 9780828400862
• Binding: Hardcover
• Number of pages: 3