Number Theory

New PDF release: An Introduction to the Theory of Numbers

By Leo Moser

ISBN-10: 1931705011

ISBN-13: 9781931705011

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Extra info for An Introduction to the Theory of Numbers

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1, −1, 1, . . 1, −1, −1, . . −1, 1, 1, . . −1, 1, −1, . . −1, −1, 1, . . It is easily seen that the general solution of p(x) (with integer coefficients) of m equations p(ak ) = ak+1 , k = 0, 1, 2, . . , m − 1, is obtained from a particular solution p1 (x) as follows. p(x) = p1 (x) + (x − a1 )(x − a2 ) · · · (x − ak−1 ) · Q(x), where Q(x) is any polynomial with integer coefficients. (3) 48 Chapter 5. Congruences Theorem 2. {pn (c)}, n ≥ 0, is relatively prime for all c if and only if p(x) belongs to one of the following six classes of polynomials.

Ak be a set of distinct residue classes (mod n). If there exists a polynomial with integer coefficients such that f (x) ≡ 0 (mod m) has roots a1 , a2 , . . , ak and no others, we call this set compatible (mod n). Let the number of compatible sets (mod n) be denoted by C(n). Since the number of subsets of the set consisting of 0, 1, 2, . . , n − 1 is 2n , we call c(n) = C(n) 2n the coefficient of compatibility of n. If n = p is a prime then the congruence (x − a1 )(x − a2 ) · · · (x − ak ) ≡ 0 (mod n) has precisely the roots a1 , a2 , .

But then {pk (0)} must be one of the following six sequences: 1, 1, 1, . . 1, −1, 1, . . 1, −1, −1, . . −1, 1, 1, . . −1, 1, −1, . . −1, −1, 1, . . It is easily seen that the general solution of p(x) (with integer coefficients) of m equations p(ak ) = ak+1 , k = 0, 1, 2, . . , m − 1, is obtained from a particular solution p1 (x) as follows. p(x) = p1 (x) + (x − a1 )(x − a2 ) · · · (x − ak−1 ) · Q(x), where Q(x) is any polynomial with integer coefficients. (3) 48 Chapter 5. Congruences Theorem 2.

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An Introduction to the Theory of Numbers by Leo Moser


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