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**1 - 3**of**3**### ABSTRACT Finding a Solution to the Diophantine Representation of the Primes

, 2003

"... ative Solution. ” In this paper they outline some results related to Hilbert’s Tenth Problem. I could not stop thinking about the first one. This was the formulation of a Diophantine system of equations such that a solution to the system would exist if and only if one particular variable is a prime ..."

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ative Solution. ” In this paper they outline some results related to Hilbert’s Tenth Problem. I could not stop thinking about the first one. This was the formulation of a Diophantine system of equations such that a solution to the system would exist if and only if one particular variable is a prime number. We will show that the set of primes forms a recursively enumerable set, which will provide the intuition as to why there should exist a Diophantine representation for the set of primes. Hilbert’s Tenth Problem states that it is impossible to find an algorithm which will decide whether any given Diophantine equation has a solution. We will go through the necessary steps required to find a solution to the Diophantine representation for the set of primes. During this process, we will also show how to solve the Pell equation using continued fractions. As an example for the method, we shall also go through the prime number two. ii

### Primitive Recursive Functions

"... .51> zero(x) j 0 2. successor: defined by succ(x) j x + 1 3. projection: defined by proj n k (x 1 ; : : : ; xn ) j x k and two basic operators for constructing new functions: 1. composition: denoted by h k = comp( f n ; g k 1 ; g k 2 ; : : : ; g k n ) 2. primitive recursion: deno ..."

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.51> zero(x) j 0 2. successor: defined by succ(x) j x + 1 3. projection: defined by proj n k (x 1 ; : : : ; xn ) j x k and two basic operators for constructing new functions: 1. composition: denoted by h k = comp( f n ; g k 1 ; g k 2 ; : : : ; g k n ) 2. primitive recursion: denoted by h n+1 = prec( f n ; g n+2 ) Here all arguments are natural numbers and the superscripts on the functions f , g, and h denote their "arities"; that is, the number of arg