Story Prime

What are some reasons prime numbers are so important to mathematicians?

I've seen a few unproved theorems (The Rieman Hypothesis was one of them) in which prime numbers played a critical role. And I also heard that prime numbers are the building blocks for other prime numbers. Can anyone explain why?

Public Comments

1. Well, you factor things into primes. Beyond that, the biggest reason is probably that all finite fields depend intimately on prime numbers.
2. Large prime numbers are heavily used in cryptography.
3. Many mathematical domains make great use of prime numbers. An example from the theory of finite groups are the Sylow theorems: if G is a finite group and pn is the highest power of the prime p which divides the order of G, then G has a subgroup of order pn. Also, any group of prime order is cyclic (Lagrange's theorem). Several public-key cryptography algorithms, such as RSA or the Diffie-Hellman key exchange are based on large prime numbers (for example with 512 bits). They rely on the fact that it is thought to be much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. Inevitably, some of the numbers that occur in nature are prime. There are, however, relatively few examples of numbers that appear in nature because they are prime. One example of the use of prime numbers in nature is as an evolutionary strategy used by cicadas of the genus Magicicada. These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences makes it very difficult for predators to evolve that could specialise as predators on Magicicadas. If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas. Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects. There is speculation that the zeros of the zeta function are connected to the energy levels of complex quantum systems.