The value of searching for the next highest prime number lies mainly in the search. First of all, the mathematics regarding prime numbers is very powerful. Much of the world's encryption technology relies on the fact that breaking a large number into prime factors is very very difficult. Nobody knows for sure, though, that there isn't a quick, easy way of factoring numbers... we just know we haven't found it yet. I would guess (and this is pure speculation) that the study of prime numbers in general would help us to figure out whether the encryption systems we use today are as secure as we think they are.
Second, even if there weren't any specific purpose right now, the study of seemingly absurd / pointless concepts in math has a very strong tendency to find applications. I would argue that the first person to suggest the imaginary number 'i' (which is the square root of negative one) was probably asked a similar if not much more derisive question. Yet the absurd idea of taking the square root of a negative number has led to a branch of mathematics that is insanely useful in physics and electrical engineering, and directly applicable to real world problems.
Answered by: Gregory Ogin, Physics Undergraduate Student, UST, St. Paul, MN
Prime numbers are essential in producing keys for cryptography. Essentially crypting a message using popular crypting algorithms require you to know 2 large primes. If p and q are such primes then the product p*q is a number which is uniquely decomposed of p and q, so if you know p*q you also know p*q - in theory. However, if the number p*q is large, it is very hard to find the primes p and q if you are given the number p*q and this is what these procedures rely upon, in effect the number p*q is known but very few people know which primes p and q make up the number - indeed, knowledge of such large value primes is classified information because common knowledge would break the cryptography algorithms.
As you probably know, computers has been increasing in speed and computing power, so a crypting algorithms that worked fine in the past can today often be cracked by a person on his home computer simply by brute force. To make stronger crypting algorithms, we need bigger numbers that are the basis for making keys. However, those numbers must have the property that they are composed of exactly 2 primes so to make a big number we need big primes.
Any book on cryptography can provide you with more details.
Answered by: Alf Salte, M.S., Oslo
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