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...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 ...