Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 5 - Number Theory and the Real Number System - 5.1 Number Theory: Prime and Composite Numbers - Exercise Set 5.1 - Page 258: 122


Euclid theory: Euclid theory is the part of the number theory and this theory represents the infinite set of the prime numbers. The prime number of the set do not divisible by other prime numbers of the set. If all prime numbers are multiplied to each other, then add\[1\]. Such as the product of all finite prime numbers is \[p\]of the set of finite prime numbers. Then, \[P=1+p\]

Work Step by Step

For example: Consider the set of finite prime number is \[\{1,2,3,5,7,11\}\]. Then, the product of the all prime numbers is, \[\begin{align} & p=1\times 2\times 3\times 5\times 7\times 11 \\ & =2310 \end{align}\] Then, the value of \[P\]is, \[\begin{align} & P=1+2310 \\ & =2311 \end{align}\] Now, check the divisibility of this number with the prime numbers, \[\begin{align} & \frac{2311}{2}=1155.5, \\ & \frac{2311}{3}=770.33, \\ & \frac{2311}{5}=462.2 \\ \end{align}\] Hence, from the above example the statement of Euclid theory is true. Number theory: The number theory is directly interlinked with the arithmetic of the mathematics to find the integers such as rational numbers, algebraic integers etc. The number theory is described through analytical objects of Riemann zeta function.
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