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...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. ...

...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}}: ...

..., 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 ...

...refer the definition of a Mersenne number where exponent ''n'' has 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 ...

...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 ...

...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''. ...

| factorization = [[prime number|prime]] | prime = 1st ...

...'' is a special kind of [[prime number]]. A prime number ''p'' is a Wilson prime if (and only if [ [[iff]] ]) ...jCNGuZ93z06kDmxXutGU8S_ADA6FgZw&cad=rja On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson], Ann. of. Math. '''39'''(1938), 350 ...

...ecause of its relation to the [[prime number theorem|distribution of prime numbers]]. It also has applications in other areas such as [[physics]], [[probabili ...ww.claymath.org/sites/default/files/ezeta.pdf|title=On the Number of Prime Numbers less than a Given Quantity}}</ref> ...

| factorization = [[prime number|prime]] ...it is '''III'''. It is the only prime that p and <math>p-1</math> are both prime. ...

...(number)|5]], and [[7 (number)|7]]. The next following eleven odd abundant numbers are *Odd abundant numbers below 500000 are in On-Line Encyclopedia of Integer Sequences [http://oeis. ...

|<center>'''Natural numbers example''' Numbers less than [[0]] (such as [[−1]]) are not natural numbers. ...

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

...he number itself: 6) are 1, 2, and 3 and 1 + 2 + 3 equals 6. Other perfect numbers include 28, 496 and 8128. ==Perfect numbers that are even== ...

...ian]]s to better describe how the [[prime number]]s are placed among whole numbers. The Riemann hypothesis is so important, and so difficult to prove, that th ...

...point out the dot that is sometimes used to separate the positions of the numbers in this [[system]]. Almost everyone uses this nowadays and prefers the conv ...han ten distinct values (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent any numbers, no matter how large. These digits are often used with a [[decimal separato ...

...d also the range of the fingerprints is <math>[p]</math>, thus we want the prime <math>p</math> to be reasonably small, but still has a good chance to disti :* choose uniformly at random a prime <math>p\in[k]</math>; ...

For example, <math>\phi(8) = 4</math>, because the four numbers: 1, 3, 5 and 7 don't share any factors with 8. For any prime number, p, <math>\phi(p) = p-1</math>. ...

...an then be represented by a stream of Gödel numbers (also called effective numbers). [[Rogers' equivalence theorem]] states criteria for which those numberi ...of arithmetic]], any code obtained this way can be uniquely factored into prime factors, so it is possible to recover the original sequence. ...

...th>k</math>. Let <math>S</math> be a set with <math>|S|=n</math>. Find the numbers of <math>k</math>-tuples <math>(T_1,T_2,\dots,T_k)</math> of subsets <math> Let <math>p</math> be a prime integer and <math>a</math> be a positive integer. Show '''combinatorially'' ...

...ods, if the public key is large enough, only someone with knowledge of the prime factors can feasibly decode the message.<ref name="rsa">{{cite journal # Choose two different large random [[prime number]]s <math>p \,</math> and <math>q \,</math> ...

...utation <math>\pi</math> of <math>n</math> numbers, we want to reorder the numbers in <math>\pi</math> in increasing order. This is the problem of sorting, on ...l numbers in <math>\pi'</math> are smaller than <math>\pi_i</math> and all numbers in <math>\pi''</math> are larger than <math>\pi_i</math>; ...

...ndom variables ranging over <math>[p]=\{0,1,2,\ldots,p-1\}</math> for some prime <math>p</math>. Unlike the above construction, now we only need two indepen For prime <math>p</math>, for any <math>i,j,a\in[p]</math>, there is exact one value ...

...groups <math> \mathbf Z/p^n \mathbf Z </math>, where ''p'' is the fixed [[prime number]] and <math> n = 1,2, \dots </math>. We can express this by [[Pontry ...able to move ahead finally from Kummer's century-old results on [[regular prime]]s. ...

...ndom variables ranging over <math>[p]=\{0,1,2,\ldots,p-1\}</math> for some prime <math>p</math>. Unlike the above construction, now we only need two indepen For prime <math>p</math>, for any <math>i,j,a\in[p]</math>, there is exact one value ...

...ndom variables ranging over <math>[p]=\{0,1,2,\ldots,p-1\}</math> for some prime <math>p</math>. Unlike the above construction, now we only need two indepen For prime <math>p</math>, for any <math>i,j,a\in[p]</math>, there is exact one value ...

Consider a [http://en.wikipedia.org/wiki/Sequence ''sequence''] of numbers with the following properties: the sum of every <math>n</math> consecutive ...ove <math>L(n,m)\geq m+n-2</math>, for any <math>n,m</math> are relatively prime(the only positive integer factor that divides both of them is 1). ...

...ed a [[countable set]]. But if a set has the same cardinality as the real numbers, it is called an [[uncountable set]]. * <math>\mathbb{P}</math>, denoting the set of all [[prime number|primes]]. ...

The construction of pairwise independent random variables via modulo a prime introduced in Section 1 already provides a way of constructing a strongly 2 Let <math>p</math> be a prime. The function <math>h_{a,b}:[p]\rightarrow [p]</math> is defined by ...

== Construction via modulo a prime == ...ndom variables ranging over <math>[p]=\{0,1,2,\ldots,p-1\}</math> for some prime <math>p</math>. Unlike the above construction, now we only need two indepen ...

...inite field <math>\mathbb{Z}_p=\{0,1,\ldots,p-1\}</math> for some suitable prime <math>p</math> (to be specified later), which means the additions and multi ...://en.wikipedia.org/wiki/Bertrand%27s_postulate Chebyshev's theorem], such prime <math>p</math> always exists), the above randomized communication protocol ...

...inite field <math>\mathbb{Z}_p=\{0,1,\ldots,p-1\}</math> for some suitable prime <math>p</math> (to be specified later), which means the additions and multi ...://en.wikipedia.org/wiki/Bertrand%27s_postulate Chebyshev's theorem], such prime <math>p</math> always exists), the above randomized communication protocol ...

...inite field <math>\mathbb{Z}_p=\{0,1,\ldots,p-1\}</math> for some suitable prime <math>p</math> (to be specified later), which means the additions and multi ...://en.wikipedia.org/wiki/Bertrand%27s_postulate Chebyshev's theorem], such prime <math>p</math> always exists), the above randomized communication protocol ...

...inite field <math>\mathbb{Z}_p=\{0,1,\ldots,p-1\}</math> for some suitable prime <math>p</math> (to be specified later), which means the additions and multi ...://en.wikipedia.org/wiki/Bertrand%27s_postulate Chebyshev's theorem], such prime <math>p</math> always exists), the above randomized communication protocol ...

...inite field <math>\mathbb{Z}_p=\{0,1,\ldots,p-1\}</math> for some suitable prime <math>p</math> (to be specified later), which means the additions and multi ...://en.wikipedia.org/wiki/Bertrand%27s_postulate Chebyshev's theorem], such prime <math>p</math> always exists), the above randomized communication protocol ...

...inite field <math>\mathbb{Z}_p=\{0,1,\ldots,p-1\}</math> for some suitable prime <math>p</math> (to be specified later), which means the additions and multi ...://en.wikipedia.org/wiki/Bertrand%27s_postulate Chebyshev's theorem], such prime <math>p</math> always exists), the above randomized communication protocol ...

...inite field <math>\mathbb{Z}_p=\{0,1,\ldots,p-1\}</math> for some suitable prime <math>p</math> (to be specified later), which means the additions and multi ...://en.wikipedia.org/wiki/Bertrand%27s_postulate Chebyshev's theorem], such prime <math>p</math> always exists), the above randomized communication protocol ...

* Numbers (example 3+6=9) ...theorem]] – [[Fermat's last theorem]] – [[Goldbach's conjecture]] – [[Twin Prime Conjecture]] – [[Gödel's incompleteness theorems]] – [[Poincaré conjecture] ...

...ndom variables ranging over <math>[p]=\{0,1,2,\ldots,p-1\}</math> for some prime <math>p</math>. We assume <math>X_0,X_1</math> to be two independent random For prime <math>p</math>, for any <math>i,j,a\in[p]</math>, there is exact one value ...

...numbers, or the finite field <math>\mathbb{Z}_p</math> of integers modulo prime <math>p</math>). We suppose that each field operation (addition, subtractio ...inite field <math>\mathbb{Z}_p=\{0,1,\ldots,p-1\}</math> for some suitable prime <math>p</math> (to be specified later), which means the additions and multi ...

...numbers, or the finite field <math>\mathbb{Z}_p</math> of integers modulo prime <math>p</math>). We suppose that each field operation (addition, subtractio ...> and <math>p\in[2^k,2^{k+1}]</math> be an arbitrary prime number. (Such a prime <math>p</math> always exists.) The input strings <math>a,b</math> can be re ...

Many amounts can be variables, which can change their value unlike numbers such as 5 or 200. Some examples of variables are [[distance]] and [[time]]. ...''f''<small>&nbsp;</small>&prime;(''x'') (marked with the [[Prime (symbol)|prime symbol]]) will be 2''x'' at any point. This process of working out a slope ...

...an [[wikt:extend|extension]] of the [[factorial]] function to all complex numbers except negative, integers. The argument of the function is shifted down by The gamma function is defined for all [[complex numbers]]. But it is not defined for negative integers and zero. For a complex numb ...

...orking with a linear function, we replace ''m'' and ''b'' with actual real numbers. ...' and 3 constants ''A'', ''B'', and ''C'' that are replaced by actual real numbers before working. This form is often used in [[geometry]] and in [[system of ...

=== Construction via modulo a prime === ...ndom variables ranging over <math>[p]=\{0,1,2,\ldots,p-1\}</math> for some prime <math>p</math>. Unlike the above construction, now we only need two indepen ...

...ndom variables ranging over <math>[p]=\{0,1,2,\ldots,p-1\}</math> for some prime <math>p</math>. Unlike the above construction, now we only need two indepen For prime <math>p</math>, for any <math>i,j,a\in[p]</math>, there is exact one value ...

...ndom variables ranging over <math>[p]=\{0,1,2,\ldots,p-1\}</math> for some prime <math>p</math>. Unlike the above construction, now we only need two indepen For prime <math>p</math>, for any <math>i,j,a\in[p]</math>, there is exact one value ...

==== Construction via modulo a prime ==== ...ndom variables ranging over <math>[p]=\{0,1,2,\ldots,p-1\}</math> for some prime <math>p</math>. Unlike the above construction, now we only need two indepen ...

...ndom variables ranging over <math>[p]=\{0,1,2,\ldots,p-1\}</math> for some prime <math>p</math>. Unlike the above construction, now we only need two indepen For prime <math>p</math>, for any <math>i,j,a\in[p]</math>, there is exact one value ...