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...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 ...

...ver half running instances return "yes", and output "no" if otherwise. The numbers of "yes"s and "no"s in the <math>t</math> trials follow the Binomial distri ...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 ...

...ver half running instances return "yes", and output "no" if otherwise. The numbers of "yes"s and "no"s in the <math>t</math> trials follow the Binomial distri ...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 ...

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 ...

Let <math>p>n</math> be the smallest prime strictly greater than <math>n</math>. The function <math>g:\mathbb{Z}_p^n\t (b) Use Jensen's inequality to prove log-sum inequality: For nonnegative numbers, ...

||<math> e^{\frac{1}{12}-\zeta^{\prime}(-1)} = e^{\frac{1}{8}-\frac{1}{2}\sum\limits_{n=0}^{\infty} \frac{1}{n+1} ||Brun constant ₄ = Σ inverse of [[twin prime]] ...

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 ...

...math>n</math>. This fact is very useful in proving theorems for partitions numbers. ...of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math>. This function, called the Euler <math>\phi</math> functi ...