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a=b mod Alexander Macfarlane arithmetic progression belongs modulo chapter common factor congruence ax=c mod congruent modulo COROLLARY Definition denote different from zero different primes Diophantine equations divisible by p2 ELEMENTARY PROPERTIES Euler's criterion evidently EXERCISES exist integers exist primitive roots exists an integer exponent dt Fermat's general theorem finite number following theorem follows at once fundamental theorem greatest common divisor gruence Hence the number highest power hypothesis integers less integers not greater irreducible fractions least common multiple Mansfield Merriman necessary and sufficient notation number of integers number of primes numerical right triangle obtained obviously odd prime number pip2 primitive root modulo proof PROPERTIES OF CONGRUENCES prove the following quadratic character quadratic non-residue modulo quadratic residue modulo ratic relatively prime integers root modulo 2pa second member Sieve of Eratosthenes simple Fermat theorem solution square number theorem follows theory of numbers tion whence Wilson's Theorem X-root
Page 3 - In these reissues it will generally be found that the monographs are enlarged by additional articles or appendices which either amplify the former presentation or record recent advances. This plan of publication has been arranged in order to meet the demand of teachers and the convenience of classes, but it is also thought that it may prove advantageous to readers in special lines of mathematical literature. It is the intention of the publishers and editors to add other monographs to the series...
Page 3 - Among the topics which are under consideration are those of elliptic functions, the theory of quantics, the group theory, the calculus of variations, and non-Euclidean geometry; possibly also monographs on branches of astronomy, mechanics, and mathematical physics may be included. It is the hope of the editors that this...
Page 20 - If the sum or the difference of two irreducible fractions is an integer, the denominators of the fractions are equal. 5. The algebraic sum of any number of irreducible fractions, whose denominators are prime each to each, cannot be an integer. 6*. The number of divisions to be effected in finding the greatest common divisor of two numbers by the Euclidian algorithm does not exceed five times the number of digits in the smaller number (when this number is written in the usual scale of 10). § 10....
Page 3 - This plan of publication has been arranged in order to meet the demand of teachers and the convenience of classes, but it is also thought that it may prove advantageous to readers in special lines of mathematical literature. It is the intention of the publishers and editors to add other monographs to the series from time to time, if the call for the same seems to warrant it. Among the topics which are under consideration are those of elliptic functions, the theory of numbers, the group theory, the...
Page 46 - Ifa + 6 + c+d + ... isa multiple of 9, then N is a multiple of 9. This is the well-known criterion: a number is a multiple of 9 if and only if the sum of its digits is a multiple of 9.
Page 59 - We shall now prove the following more complete theorem, without the use of I or II. III. // p is an odd prime number and a is an integer not divisible by p, then a is a quadratic residue or a quadratic nonresidue modulo p according as a^(pi} = +1 or a^(p~l^-imodp.
Page 8 - It is clear from this definition that a is also divisible by c. The integers b and c are said to be divisors or factors of a; and a is said to be a multiple of b or of c. The...