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...cture''' is a [[conjecture]] about the distribution of prime divisors of [[Mersenne numbers]] and was made by [[Donald B. Gillies]] in a 1964 paper<ref>{{cite | title = Three new Mersenne primes and a statistical theory ...

...s.txt Williams primes]</ref> The Williams numbers base 2 are exactly the [[Mersenne number]]s. ...s conjectured that for every ''b'' ≥ 2, there are infinitely many Williams primes for base ''b''. ...

...hapter = Generalized Mersenne Prime|isbn = 978-1-4419-5905-8}}</ref> These primes allow fast modular reduction algorithms and are widely used in [[cryptograp * [[Mersenne primes]], which have the form <math>2^k-1</math>, ...

{{About|the theorem on perfect numbers and Mersenne primes|the theorem on the infinitude of prime numbers|Euclid's theorem}} ...s a [[theorem]] in [[number theory]] that relates [[perfect number]]s to [[Mersenne prime]]s. It states that an even number is perfect [[if and only if]] it ha ...

...n number theory was sparked by her sister's use of computers to discover [[Mersenne prime]]s. She published an article on a closely related topic, [[perfect nu ...ber|large prime numbers]], and Reid's relatives' discovery of new Mersenne primes.{{r|singmaster|belle}} ...

...]] glossary</ref> obtained the following factorization into [[prime number|primes]]: ...3681 is an example of a [[Mersenne prime#Gaussian Mersenne primes|Gaussian Mersenne]] norm.<ref name="primepages" /> ...

...</math> for every prime less than 100. The theorem and its application to primes <math>p</math> less than 100 were attributed to Germain by [[Adrien-Marie L ...th> the auxiliary prime may be arbitrarily large similarly to the Mersenne primes she most likely proved the theorem in the general case by her consideration ...

| title = Patterns in primes are a clue to the strong law of small numbers ...Guy gives is the conjecture that <math>2^p-1</math> is prime—in fact, a [[Mersenne prime]]—when <math>p</math> is prime; but this conjecture, while true for < ...

...the reciprocals of the primes#Partial sums|reciprocals of the first ''n'' primes]] is not an integer for any ''n''. ...orbert |title=Tight upper and lower bounds for the reciprocal sum of Proth primes |journal=The Ramanujan Journal |date=1 September 2022 |volume=59 |issue=1 | ...

...Norbert|title=Tight upper and lower bounds for the reciprocal sum of Proth primes|date=2022|journal=Ramanujan Journal|volume=59 |pages=181–198 |publisher=Spr |OEIS_name=Proth primes: primes of the form ''k''*2^''m'' + 1 with odd ''k'' < 2^''m'', ''m'' ≥ 1 ...

==Largest known primes== ...st]] for their form, for example the [[Lucas–Lehmer primality test]] for [[Mersenne number]]s. “!” is the [[factorial]], “#” is the [[primorial]], and <math>\P ...

...greater than 1 can be written uniquely (up to reordering) as a product of primes.}} ...hat states that every even natural number greater than 2 is the sum of two primes.}} ...

...ty of ''N'' depends on the primality of ''q'' and indeed smaller 'probable primes' until some threshold is reached where ''q'' is considered small enough to ...n find a suitable ''E'' in polynomially many attempts. The distribution of primes on the Hasse interval ...

| Products of two primes, not necessarily distinct. | {{math|''p''_''n''#}}, the product of the first {{math|''n''}} primes. ...

...ntil 1884 (whereas the ''Harmonie Universelle'' written in 1636 by [[Marin Mersenne]] is considered the first publication in Europe outlining equal temperament ...a larger even integer as the sum of a prime and the product of at most two primes |journal=Sci. Sinica |volume=16 |year=1973 |pages=157–176}}</ref> ...

...this and proved the theorem for all [[regular prime]]s, leaving irregular primes to be analyzed individually. Building on Kummer's work and using sophistica ...'' = 3}} as challenges to his mathematical correspondents, such as [[Marin Mersenne]], [[Blaise Pascal]], and [[John Wallis]],<ref>Ribenboim, pp. 13, 24</ref> ...

== Base 2 repunit primes == {{main|Mersenne prime}} ...