Chapter 1.1, Problem 60E

Large prime numbers The number $2^{4497}-1$ is prime. At the time that this number was determined to be prime, it took one of the world's fastest computers about 60 days to verify that it was prime. This computer was capable of performing $2\times {10}^{11}$ calculations per second. Use scientific form to estimate the number of calculations needed to perform this computation. (More recently, in $2005,2^{\text{m},\text{ miles }}-1,$ a number containing 9,152,052 digits. was shown to be prime.)