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number theory, postulates a very precise answer to the question of how the prime numbers are distributed. This chapter lays the foundations for our study of the theory of numbers by weaving together the themes of prime numbers, integer factorization, and the distribution of primes. In Section 1.1, we rigorously prove that the

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that equation (1.3) has at most ﬁnitely many solutions, moreover, the number of solutions can be bounded by an absolute large number (see the excellent survey of Gy˝ory [Gy3]). However, apparently a kind of such estimates is far from the actual number, indeed, many existing works in the literature suggest that equation (1.3) has few solutions

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any challenging, interesting, beautiful or historical problems in elementary number theory (by email or via the website) that you think might belong in the book. On the website you can also help me collecting solutions for the problems in the book (all available solutions will be on the website only). You can send all comments to both authors at

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For example, here are some problems in number theory that remain unsolved. (Recall that a prime number is an integer greater than 1 whose only positive factors are 1 and the number itself.) Note that these problems are simple to state — just because a topic is accessibile does not mean that it is easy. 1.

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This is a number with 79911 decimal digits! But compared with infinity, any number, no matter how large, is less than a drop in the bucket. In 1844, the Belgian mathematician Eugène Catalan conjectured that the only solutions to the equation xn − ym = 1, where x, y, n, and m are all integers greater than 1, is 3 2 − 2 3 = 1.

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An elementary number is one formalization of the concept of a closed-form number. The elementary numbers form an algebraically closed field containing the roots of arbitrary equations using field operations, exponentiation, and logarithms. References. Richardson, Daniel (1997). "How to recognize zero". Journal of Symbolic Computation.

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