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..., and needs to test if a certain number is prime. The official proof of a prime is through its primality certificate. ...cient): <math>(p-1)!</math> evaluates to -1 mod p where p is the candidate prime number ...

...vely prime means that they do not have common divisors, other than 1. Such numbers are named after [[Robert Daniel Carmichael|Robert Carmichael]]. ...umbers are composite numbers that behave a little bit like they would be a prime number. ...

...smaller numbers are called factors or [[divisor]]s. 1 is a factor of all numbers. ...larger number. Note that since 1 is not prime, it is not included in the prime factorization. ...

...other than 1. In other words, each [[prime number]] that appears in its [[prime factorization]] appears exactly once. </math> in the prime factorization is to a power larger than one. ...

...satisfies a result proven by [[Srinivasa Ramanujan]]. It relates to the [[prime counting function]]. where <math>\pi</math>''(x)'' is the [[prime counting function]]. The prime counting function is the number of primes less than or equal to ''x''. ...

...ns prefer the definition of a Mersenne number where exponent ''n'' to be a prime number. ...d ''2⁴ − 1 = 15'' can be shown to be composite because 4 is not prime. ...

...wever: There are numbers that pass the test, and that are not prime. These numbers are called [[Carmichael number]]s. [[Fermat's little theorem]] states that if ''p'' is prime and <math>1 \le a < p</math>, then ...

...mat number''' is a special [[positive number|positive]] [[number]]. Fermat numbers are named after [[Pierre de Fermat]]. The formula that generates them is where ''n'' is a nonnegative [[integer]]. The first nine Fermat numbers are {{OEIS|id=A000215}}: ...

...ion''' is the [[Function (mathematics)|function]] counting the number of [[prime number]]s less than or equal to some [[real number]] ''x''. It is written a ...<math>\pi(p_n)=n</math>.<ref>{{Cite web|last=Weisstein|first=Eric W.|title=Prime Counting Function|url=https://mathworld.wolfram.com/PrimeCountingFunction.h ...

...test common divisor]] ('''GCD''', or '''highest common factor''') of these numbers is 1.<ref name=":0" /> 5 = 5*1 (Prime). The only common factor is 1, so they are coprime. ...

'''Willan's Formula''' is a formula that can find the ''nth'' '''prime number'''. ...is either a prime number or <math>1</math>, meaning when <math>j</math> is prime, <math>\frac{(j-1)!+1}{j}</math> is an integer. ...

...'' is a special kind of [[prime number]]. A prime number ''p'' is a Wilson prime if (and only if [ <math>\iff</math> ]) ...an.aebi/.ws_gen/14/Emma_Lehmer_1938.pdf On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson], Ann. of. Math. '''39'''(1938), 350 ...

...lowing problem: Given an integer <math> N </math>, find its [[prime number|prime factor]]s. [[Category:Numbers]] ...

...pt for the rectangle with sides of lengths for 11 and 1. 11 is therefore a prime number.]] ...imes the number itself, so those two numbers are two of its divisors. A '''prime number''' is a number with ''only'' two divisors: one and itself.<ref name= ...

The first few superabundant numbers are [[1 (number)|1]], [[2 (number)|2]], [[4 (number)|4]], [[6 (number)|6]], where ''p''_i is the ''i''-th [[prime number]], and ...

| prime = ...</math>.<ref>Wells, D. ''The Penguin Dictionary of Curious and Interesting Numbers'' London: Penguin Group. (1987): 41–44</ref> An [[integer]] is ''[[even]]'' ...

...=Divisor|archive-url=|archive-date=|access-date=September 20, 2020|website=Prime Glossary}}</ref> ...on by zero]]). Numbers divisible by 2 are called [[even number|even]], and numbers not divisible by 2 are called [[odd number|odd]]. ...

==Important numbers 7001–7999== * '''7043''' – [[Sophie Germain prime]] ...

| factorization = [[prime number|prime]] | prime = 4th ...

...heory]]. It is related to the [[prime number theorem|distribution of prime numbers]]. It also has uses in other areas such as [[physics]], [[probability theor ...ww.claymath.org/sites/default/files/ezeta.pdf|title=On the Number of Prime Numbers less than a Given Quantity|access-date=2016-03-09|archive-date=2016-03-04|a ...