# An Introduction to the Theory of Numbers by G. H. & Wright, E. M. Hardy PDF

By G. H. & Wright, E. M. Hardy

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We introduce some more notation. Let W be a finite word. For a positive integer `, we write W ` for the word W : : : W (` times repeated concatenation of the word W ) and W 1 for the infinite word constructed by concatenation of infinitely many copies of W . x bxc/jW je. Here, d e denotes the upper integer part function. a` /` 1 be a sequence of elements from A. Let w > 1 be a real number. We say that a satisfies Condition . Wn /n 1 such that: (i) For every n 1, the word Wn Znw is a prefix of the word a.

Zwegers, Mock #-functions and real analytic modular forms. In q-Ser. Appl. Comb. Number Theory Phys. (ed. B. Berndt and K. Ono), Contemp. Math. 291, AMS (2001), 269–277. Chapter 2 Expansions of algebraic numbers Yann Bugeaud Contents 1 Representation of real numbers . . . . . . . . . . 2 Combinatorics on words and complexity . . . . . . . . 3 Complexity of algebraic numbers . . . . . . . . . . 4 Combinatorial transcendence criteria . . . . . . . . . 5 Continued fractions .

Second proof. Here, we follow an alternative approach presented in [2]. 31) that pn pn0 is divisible by an integer multiple of b un . 33) `jb 1Äj Ä4 Since a satisfies Condition . xn /j` Ä b un 1Äj Ä4 `jb 1Äj Ä4 Ä maxfb wn Cvn C2un ; b un Cwn ; pn ; pn0 g " ; for every n 1. 7 that all the points xn lie in a finite number of proper subspaces of Q4 . z1 ; z2 ; z3 ; z4 / and an infinite set of distinct positive integers N1 such that z1 b wn Cvn C2un C z2 b un Cwn C z3 pn C z4 pn0 D 0; for any n in N1 .