ELEMENTS OF THE THEORY OF INTEGERS - Tapa blanda

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. 1903 Excerpt: ... 765. Theorem. If fj. is any given modulus all the integers, when written in the order of increasing magnitude', may be divided into successiv groups ofi each, so that the integers in each of these groups, taken in order, ar congruent with o, I, 2, 3, , i--1. The first integers in the successiv groups will be the various multiples of fi-' ,-3ft 2ft--ft o, ft, 2fx, 3//, .-, if fjt is positiv, , 3ft 2ft ft --ft 2ft 3ft if // is negativ. If p/jt is either of these multiples, the group of which it is the first integer will be W PV + l W + 2 W + 3, ", Pf + ( f-1). These integers ar in each case evidently congruent with o, I, 2, 3,..., //--1. 766. Definition. This principle is calld the periodicity of the integers with respect to a given modulus. 767. Definition. A cyclical arrangement of any set of things, given in a certain order, as, for example, the letters a, b, c, d, e, f is an arrangement obtaind by dividing the set into two parts at any place and then making the first part the second and vice versa, the order of the symbols in each part being unchanged. Thus b, c, d, e, f a and e, f a, b, c, d ar cyclical arrangements of the above set of letters. The original set unchanged is also said to be a cyclical arrangement of itself. 768. Theorem. If p is any given modulus and any group of p successiv integers ar taken,.they ar congruent, in order, with some cyclical arrangement of the integers o, 1, 2, 3,..., p -2, p -1. Let (pfj. + p be the first of the given group of integers, wher p is positiv and numerically less than p. § 522, Then the group of integers will be W + P flx + P + l VV-+ P + 2 ' w + P + (I t1 I-2) w + P + (I P I-0 These differ each by a multiple of p from the cor responding integers of the group P, P + I, / + 2, ., p + ( I /...