Uses for Prime Numbers

Aside from keeping Mathematicians guessing the only significant application of prime numers and prime number theory is encryption.


Prime numbers are an integral part of modern internet security. The most common form of internet security is RSA security which has at its heart 2 very large prime numbers.

RSA security revolves around finding two large (2048 bit or 617 digit) prime numbers and multiplying them together to get your own, personal "modulus," which will be another very large number. You post your modulus for the world to see and if anybody wants to encrypt a secret number to send to you, all he has to do is cube it (multiply it by itself twice), divide by your modulus, and send you the remainder from the division.

Knowing the prime factors of the modulus, you can recompute the original secret number from the remainder that your friend sends you (the details are slightly complicated); but an enemy who intercepts the remainder -- even an enemy who knows your modulus -- in general cannot recompute the original secret number from it. The "secret number" can, of course, be some kind of letters-to-numbers translation of a textual message. (Warning: for this encryption to be strong, the secret number must be big enough that its cube greatly exceeds the modulus. In practical applications, a large number is added to the secret number before cubing.)

As far as anybody knows, there is in general no way to break this cipher faster than by finding the two prime numbers that were multiplied together to make your modulus. Consequently, cryptologists are very interested in how big a number of this sort can be factored, since that tells them how big they have to make their moduli.


Cicadas live underground for many years. They emerge every 7, 13 or 17 years, depending on species, to mate. These are prime numbers. Why do cicadas stay underground for a prime number of years? How can a cicada understand prime numbers?

It was a American palaeontologist and evolutionary biologist, Stephen Jay Gould, who posed the answer. Cicadas are not great mathematicians, the fact that they stay underground for a prime number of years is due to the process of natural selection.

Black Prince Cicada

Psaltoda plaga, commonly known as the "Black Prince", an Australian species of cicada which emerges once every 7 years.

By emerging from the ground every 7, 13 and 17 years cicadas minimize their chance that their infrequent emergences will match up with the life cycles of birds and other creatures that eat them. Many of the cicadas predators have 2-5 years cycles. For example, imagine that a bird species whose numbers wax and wane on a four year cycle. If cicadas emerged every 12 years, their arrival might coincide with the peak of its avian predator, setting up a pattern that could drive the cicadas to extinction. By cycling at a large prime number, cicadas minimize the chance that some bird or other predator can make a living off them. The emergence of a 17-year cicada species, for example, would match up with its four year predator only every (4 x 17 =68) 68 years. If, on the other hand, the cicadas emerged every 4 years it would match up with its predator every 12 years (4 x 3 =12).

Possibly there once was cicada species who emerged from the ground every 3, 4, 5,and 6 years but over many years these species have been wiped out by predators and only those cicadas who stayed underground for a prime number of years have survived until today.