Ever since Euclid proved that there is no largest prime number, mathematicians h

Question

Ever since Euclid proved that there is no largest prime number, mathematicians have spent a lot of time and energy searching for massive prime numbers. Awards are even given for this, but it’s not really about the money to them. They’ve received grants for doing this work, they take pride in doing so, and there seems to be some academic competition involved as well. What is a prime number? A prime number is only divisible by 1 and itself. The first few are 2, 3, 5, 7, 11, 13, 17, and the list goes on infinitely. All primes can be written in the form 2n -1, where n is an integer, but not all numbers calculated with that formula are prime. For example, for n = 3, 23 - 1 = 7, and 7 is prime. However, for n = 4, 24 - 1 = 15, and 15 is not prime. In January of 2013, one of the leaders in the field used a single powerful computer, along with thousands of PC's owned by people like you and me (the same people who donate their PC's computing power to search for SETI signals), and determined a world record prime number containing 17,425,170 digits. The actual number is calculated 257,885,161 - 1. Try putting that one in your calculator. My questions for you are: •Why would they do this? Don't these brilliant mathematicians have better things to do, like solving world hunger? •What’s to gain from this endless search?

Explanation / Answer

Prime numbers were not of much use prior to 19th century. But in the 19th century, people found the use for prime numbers in the form of encryption. Signals and messages of war were encrypted with the help of prime numbers. Mostly the product of two prime numbers were used for encryption as the resultant number had four factors in total which made the encryption very efficient. This usage of prime numbers have been passed on the recent era and at present it is used in public key cryptography, which has numerous and very important security applications. So the encryption is based on integer factorisation, the efficiency of which depends on the time and the number of steps spent to factorize integers into their prime factors. So if the integer is a combination of huge prime factors, it would make the encryption several times more efficient than a smaller prime number. Thus, the quest for huge prime numbers would continue to exist in the run as long as it would remain of such importance to mankind.

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