Statistics Definitions > Prime Numbers in Probability and Statistics

## What is a Prime Number?

Prime numbers are whole numbers (numbers that aren’t fractions) greater than 1 that are **divisible only by itself and one.** For example, 13 is a prime number because it cannot be divided by anything but 13 and 1.

## What are primes used for in probability and statistics?

Prime numbers aren’t generally used *in* statistics (other than those number appearing in data), but statistics and probabilities are used to work with prime numbers in **number theory**. For example, you might want to find the probability of choosing a prime number from a series of numbers. The odds depend on what interval you choose:

- The probability of finding a prime in the set {0,1,2} is .333, because one out of the three numbers is a prime (1/3 = .333).
- The probability for the set of numbers from 1 to 100 is .25, because 25/100 numbers in that set are primes (which are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97).

Prime numbers *look* random, but there’s some research using statistical mechanics that suggests a chaos pattern (statistical mechanics is a branch of mathematical physics that studies the behavior of systems). “It is evident that the primes are randomly distributed but, unfortunately, we don’t know what ‘random’ means” (Vaughan, 1990).

Why do we even care about prime numbers in real life or the probability of finding them? You may not realize it, but prime numbers play an important role in many areas of science, including the math behind internet shopping. Prime numbers are the nuts and bolts behind the cryptography that keeps your personal information secure when you shop online.

## Is 2 a prime number?

Yes. It’s the first (and only) even prime number.

## How many prime numbers are there?

There are an infinite amount of prime numbers. Although new prime numbers are being discovered every day, there’s no end to the amount of primes to be discovered. The proof of this fact goes back to 300 B.C.E. when Euclid outlined it in book IX of *The Elements*. Proposition 20 states every list of primes (no matter how large) is missing at least one prime number.

## What is the Largest Prime Number?

As of the beginning of 2016, the largest prime number was 2^{74,207,281}-1 (record stands as of time of writing—September 17, 2017). It is calculated by multiplying 2 by itself 74,207,281 times, then subtracting one. New primes are being found all the time. For the most up-to-date list of the largest prime number, see: GIMPS.

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## References

Gonick, L. (1993). The Cartoon Guide to Statistics. HarperPerennial.

Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences, Wiley.

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