# 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

**which states that all numbers greater than one can be expressed as the product of primes or is a prime itself.**

*fundamental theorem of arithmetic**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

**). To date, fifty-one Mersenne primes have been discovered, but a distributed computing project known as**

*2 to the N minus 1***is actively searching for more, using the**

*GIMPS***test for primality.**

*Lucas-Lehmer*__GOLDBACHâ€™S CONJECTURE__

** Goldbachâ€™s conjecture** states that every even integer greater than two can be written as the

**Goldbach also wrote a**

*sum of two primes.***which states that every odd number greater than five can be written as the**

*weak conjecture***.**

*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

**test is another common algorithm for finding prime numbers.**

*Miller-Rabin*__RIEMANN HYPOTHESIS__

The ** Riemann Hypothesis** is an unsolved problem in mathematics that, if solved, would provide information about the

**. Collectively, the Riemann Hypothesis,**

*distribution of primes***, and the**

*Goldbachâ€™s conjecture***comprise the eighth of twenty-three**

*twin prime conjecture***.**

*Hilbertâ€™s problems*__PIERRE DE FERMAT__

French mathematician ** Pierre de Fermat **is the namesake of a special case of prime numbers. He also names

**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**

*Fermatâ€™s Little Theorem***or**

*Carmichael numbers***.**

*pseudoprimes******

*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 ** Goldbachâ€™s conjecture** well and even has an interactive Goldbach calculator.

* Hereâ€™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

**. Read more about GIMPS**

*Mersenne Prime*__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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