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: 126

Answer

Consider another formula to find the prime numbers, \[f\left( n \right)={{n}^{2}}-79+1601\] Now, verify the formula for prime numbers by substituting\[\left( n=0,1,2,3,4,5.....79 \right)\].Then, the prime numbers are shown below in the table,
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Work Step by Step

Now, check the above formula for\[n=80\]. \[\begin{align} & f\left( 80 \right)={{80}^{2}}-79\times 80+1601 \\ & =1681 \\ & ={{41}^{2}} \end{align}\] The prime number is the number that divisible by itself or 1. From the above table, the above formula shows the prime numbers between \[\left( n=0 \right)\]to\[\left( n=79 \right)\]. But at \[\left( n=80 \right)\] the formula fails. Hence, from the above observation, no unique formula is true for all prime numbers.
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