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

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For the definition of convergence for p-adic numbers we thus must decide under what conditions two p-adic numbers are to be considered close to one another. In the example of the first section, we spoke of the p-nearness of two p-adic integers x and y, meaning by this that the difference of x and y should be divisible by a high power of p. It was under this definition of nearness that the analogy between the definitions of real numbers and of p-adic integers became apparent. If we use the concept of thep-value v,, then thep-nearness of x and x will be characterized by the value of v,(x - y).

31 p-ADIC N U M B E R S Sec. 31 Convergent p-adic series can clearly be termwise added and subtracted and multiplied by a constant p-adic number. The associativity property of series also holds for them. Theorem 9. If the terms of a convergent p-adic series are rearranged, its convergence is not affected and its sum does not change. The simple proof of this theorem is left to the reader. In analysis it is proved that the property described in Theorem 9, when applied to real numbers, characterizes absolutely convergent series.

1. Congruences and Equations in the Ring 0, At the beginning of Section 3 we considered the question of the solvability of the congruence x2 = 2 (mod 7") for n = I, 2, ... ,and this led us to the concept of a p-adic integer. 1). This connection is described more fully in the following theorem. Sec. 51 CONGRUENCES A N D p-ADIC INTEGERS 41 Theorem 1. Let F(xl, . , x,) be a polynomial whose coefficients are rational integers. The congruence F(x,,... 1) is solvable for all k 2 1 if and only if the equation F(x,, ...

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