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# Qwiz5 Quizbowl Essentials: Prime Numbers

A prime number is one that has no factors other than one and itself. Prime numbers are central to the fundamental theorem of arithmetic which states that all numbers greater than one can be expressed as the product of primes or is a prime itself. By analyzing questions, you can see patterns emerge - patterns that will help you answer more questions. Qwiz5 is all about those patterns. In each installment of Qwiz5, we take an answer line and look at its five most common clues. Here we explore five clues that will help you answer a tossup on prime numbers.

MERSENNE

Mersenne primes are a special type of prime number that is equal to one less than a power of two (aka 2 to the N minus 1). To date, fifty-one Mersenne primes have been discovered, but a distributed computing project known as GIMPS is actively searching for more, using the Lucas-Lehmer test for primality.

GOLDBACH’S CONJECTURE

Goldbach’s conjecture states that every even integer greater than two can be written as the sum of two primes. Goldbach also wrote a weak conjecture which states that every odd number greater than five can be written as the sum of three primes.

SIEVE OF ERATOSTHENES

The Sieve of Eratosthenes is an ancient algorithm for finding prime numbers wherein you cross out all numbers that are multiples of other numbers in sequence. The Miller-Rabin test is another common algorithm for finding prime numbers.

RIEMANN HYPOTHESIS

The Riemann Hypothesis is an unsolved problem in mathematics that, if solved, would provide information about the distribution of primes. Collectively, the Riemann Hypothesis, Goldbach’s conjecture, and the twin prime conjecture comprise the eighth of twenty-three Hilbert’s problems.

PIERRE DE FERMAT

French mathematician Pierre de Fermat is the namesake of a special case of prime numbers. He also names Fermat’s Little Theorem which can be used to test if a number is prime. However, In certain situations, this returns a false-positive. Those numbers that falsely satisfy the Fermat test, are known as Carmichael numbers or pseudoprimes.

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Quizbowl is about learning, not rote memorization, so we encourage you to use this as a springboard for further reading rather than as an endpoint. Here are a few things to check out:

* This site explains Goldbach’s conjecture well and even has an interactive Goldbach calculator.

* Here’s an interesting article about the role of prime numbers in RSA encryption.

* In December 2018, GIMPS (Great Internet Mersenne Prime Search) was used to discover the 51st Mersenne Prime. Read more about GIMPS here. You can even download the software and start your own search for the 52nd Mersenne Prime.

* This short video explains how to use the sieve of Eratosthenes to find prime numbers:

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