# Elements of number theory by I M Vinogradov By I M Vinogradov

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An) E 9Jt if and only if the system DI(XI, ... , Xm) = ai has a solution. 9. Show that every Diophantine function has a representation of the form a= F(b1. , bn) {=:::::> 3xo ... Xm [a= E(bi, ... , bn, xo, ... , Xm)] where E is a polynomial with integer coefficients. 10. Show that the following sets are Diophantine: (a) the set of all composite numbers; (b) the set of all numbers that are not powers of 2; (c) the set of all numbers that are not perfect squares. 11. For a less trivial example, show that the relation a is not a power of b is Diophantine.

2). In analogy with Diophantine equations, one can consider parametric exponential Diophantine equations and introduce exponential Diophantine representations of sets, properties, relations, and functions. 4 that exponentiation is Diophantine, we have a method for transforming any exponential Diophantine equation into an equivalent Diophantine equation with the same parameters, at the cost of an increase in the number of unknowns. Thus the class of sets (properties, relations, functions) having exponential Diophantine representations coincides with the class of Diophantine sets (properties, relations, functions, respectively).

5) This function is also Diophantine: a= arem(b, c) {:==> 2a :;:; c & [c I (b- a) V c I (b +a)]. 6) We shall also need the fact that the relation of non-divisibility is Diophantine: afb {:==> rem(b, a) > 0. 7) both relations are false when a = 0 and b > 0. This feature could easily be avoided, but in fact we shall use the relation f only in cases when it is certain that the first argument is not zero. 8) and a generalized Diophantine representation of congruence with respect to a positive modulus: a= b (mod c) {=:::::> rem(a,c) = rem(b,c).