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'''Integers''' are the [[natural numbers]] and their negatives.<ref>Negative numbers ha These are some of the integers: ...

...two [[integer]]s is the greatest (largest) number that divides both of the integers evenly. [[Euclid]] came up with the idea of GCDs. The GCD of any two positive integers can be defined as a [[recursive]] [[function (mathematics)|function]]: ...

...satisfies the [[congruence]] <math>b^{n-1}\equiv 1\pmod{n}</math> for all integers <math>b</math> which are [[coprime|relatively prime]] to <math>n</math>. Be ...ber]]s <math>p</math> satisfy <math>b^{p-1}\equiv 1\pmod{p}</math> for all integers <math>b</math> which are relatively prime to <math>p</math>. This has been ...

...n as <math>(\mathbb{Z}, \cdot)</math>. The name of the magma would be "The integers under multiplication". ...

...s, we can say that there are "more" real numbers than integers because the integers are ''countable'' and the real numbers are ''uncountable''. ...ems are inside the real numbers. For example, the [[rational number]]s and integers are all in the real numbers. There are also more complicated number systems ...

...lso called '''clock arithmetic''', is a way of doing [[arithmetic]] with [[integers]]. Much like hours on a [[clock]], which repeat every twelve hours, once th Modular arithmetic can be used to show the idea of '''congruence'''. Two integers, ''a'' and ''b'', are '''congruent modulo n''' if they have the same [[rema ...

...'totient''' of a [[positive number|positive]] [[integer]] is the number of integers smaller than ''n'' which are [[coprime]] to ''n'' (they share no [[factor]] ...because it gives the size of the multiplicative [[group (math)|group]] of integers [[modular arithmetic|modulo]] ''n''. More precisely, <math>\phi(n)</math> i ...

...lly denoted by '''LCM(''a'', ''b'')'''. Likewise, the LCM of more than two integers is the smallest positive integer that is divisible by each of them. ...

...exists a '''nonempty''' subsequence (not necessarily consecutive) of these integers, whose sum is equal to <math> 0 </math>. (Hint: Consider <math> b_i=a_i-i < ...isets''' <math> A </math> and <math> B </math>, both with <math> n </math> integers from <math> 1 </math> to <math> n </math>. Show that there exists two '''no ...

===Integers, addition, zero=== ...of ''infinite order'') because it has an infinite number of elements, the integers. ...

...n the [[natural numbers]], subtraction does not have closure, but in the [[integers]] subtraction does have closure. Subtraction of two numbers can produce a n ...times make closure of a mathematical object by including new elements. The integers are a closure of the natural numbers by including negative numbers. The [[r ...

The number '''0''' is the "additive identity" for integers, real numbers, and complex numbers. For the real numbers, for all <math>a\i Similarly, The number '''1''' is the "multiplicative identity" for integers, real numbers, and complex numbers. For the real numbers, for all <math>a\i ...

...example, for an equation over the rationals, one can find solutions in the integers. Then, the equation is a [[diophantine equation]]. One may also look for s ...

...agorean triple''' is a set of three [[Positive number|positive]] [[integer|integers]] which satisfy the [[equation]] (make the equation work): ...

Ramanujan primes are the integers ''R_n'' that are the '''smallest''' to satisfy the condition ...

...lest Gödel numbering schemes is used every day: The correspondence between integers and their representations as strings of symbols. For example, the sequence ...ngs as well. Given a sequence <math>x_1 x_2 x_3 ... x_n</math> of positive integers, the Gödel encoding is the product of the first n primes raised to their co ...

...ven if we worked forever. If a set has the same cardinality as the set of integers, it is called a [[countable set]]. But if a set has the same cardinality a ...d <math>\begin{matrix}\frac{11}{6} \end{matrix} \in \mathbb{Q}</math>. All integers are in this set since every integer ''a'' can be expressed as the fraction ...

...e say whole numbers can be negative. "Positive integers" and "non-negative integers" are another way to include zero or exclude zero, but only if people know t === Integers === ...

Fix positive integers <math>n</math> and <math>k</math>. Let <math>S</math> be a set with <math>| ...

...y zeros have been found. The "obvious" ones to find are the negative even integers. This follows from Riemann's functional equation. More have been computed a ...

...<math>[2n]</math> consisting of <math>r</math> odd and <math>s</math> even integers, with no two elements of <math>S</math> differing by <math>1</math>. Give a ...

| [[Integers]] | ℤ denotes the set of integers (-3,-2,-1,0,1,2,3...) ...

...on]] of the [[factorial]] function to all complex numbers except negative, integers. The argument of the function is shifted down by one. This means that if n ...is defined for all [[complex numbers]]. But it is not defined for negative integers and zero. For a complex number whose real part is not a negative integer, t ...

You can count the [[integers]] with <math>({0, 1, -1, 2, -2, 3, -3...})\,\!</math> ...

[[Category:Integers]] ...

:Let <math>k,\ell</math> be positive integers. Then there exists an integer <math>R(k,\ell)</math> satisfying: :Let <math>r, k_1,k_2,\ldots,k_r</math> be positive integers. Then there exists an integer <math>R(r;k_1,k_2,\ldots,k_r)</math> satisfyi ...

<li>[<strong>Surjection</strong>] For positive integers <math>m\ge n</math>, prove that the probability of a uniform random functio <li>[<strong>Coprime integers</strong>] Given positive integers <math>n \ge 2</math>, calculate the number of integer pairs <math>(x,y)</ma ...

...nal number]], which means it is impossible to write as a fraction with two integers; but some numbers, like 2.71828182845904523536, come close to the true valu ...

...e use in everyday life are rational. These include fractions and [[integer|integers]]. And also a number that can be written as a fraction while it is in its o ...

...mber system) which uses decimal [[integer]]s, [[negative number|negative]] integers, and [[0 (number)|zero]] ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

:Let <math>k,\ell</math> be positive integers. Then there exists an integer <math>R(k,\ell)</math> satisfying: :Let <math>r, k_1,k_2,\ldots,k_r</math> be positive integers. Then there exists an integer <math>R(r;k_1,k_2,\ldots,k_r)</math> satisfyi ...

:Let <math>k,\ell</math> be positive integers. Then there exists an integer <math>R(k,\ell)</math> satisfying: :Let <math>r, k_1,k_2,\ldots,k_r</math> be positive integers. Then there exists an integer <math>R(r;k_1,k_2,\ldots,k_r)</math> satisfyi ...

:Let <math>k,\ell</math> be positive integers. Then there exists an integer <math>R(k,\ell)</math> satisfying: :Let <math>r, k_1,k_2,\ldots,k_r</math> be positive integers. Then there exists an integer <math>R(r;k_1,k_2,\ldots,k_r)</math> satisfyi ...

:Let <math>k,\ell</math> be positive integers. Then there exists an integer <math>R(k,\ell)</math> satisfying: :Let <math>r, k_1,k_2,\ldots,k_r</math> be positive integers. Then there exists an integer <math>R(r;k_1,k_2,\ldots,k_r)</math> satisfyi ...

:Let <math>k,\ell</math> be positive integers. Then there exists an integer <math>R(k,\ell)</math> satisfying: :Let <math>r, k_1,k_2,\ldots,k_r</math> be positive integers. Then there exists an integer <math>R(r;k_1,k_2,\ldots,k_r)</math> satisfyi ...

:Let <math>k,\ell</math> be positive integers. Then there exists an integer <math>R(k,\ell)</math> satisfying: :Let <math>r, k_1,k_2,\ldots,k_r</math> be positive integers. Then there exists an integer <math>R(r;k_1,k_2,\ldots,k_r)</math> satisfyi ...

[[Category:Integers]] ...

[[Category:Integers]] ...

:Let <math>k,\ell</math> be positive integers. Then there exists an integer <math>R(k,\ell)</math> satisfying: :Let <math>r, k_1,k_2,\ldots,k_r</math> be positive integers. Then there exists an integer <math>R(r;k_1,k_2,\ldots,k_r)</math> satisfyi ...

...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math Let <math>U=\{1,2,\ldots,n\}</math> be the universe. The number of positive integers from <math>U</math> which is divisible by some <math>p_{i_1},p_{i_2},\ldots ...

...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math Let <math>U=\{1,2,\ldots,n\}</math> be the universe. The number of positive integers from <math>U</math> which is divisible by some <math>p_{i_1},p_{i_2},\ldots ...

...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math Let <math>U=\{1,2,\ldots,n\}</math> be the universe. The number of positive integers from <math>U</math> which is divisible by some <math>p_{i_1},p_{i_2},\ldots ...

for non-negative integers ''n'' and ''k'' where ''n'' ≥ ''k'' and with the initial condition ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

...olor="red">''ordered''</font> sum of <font color="red">''positive''</font> integers. A '''<math>k</math>-composition''' of <math>n</math> is a composition of < ...umber of solutions to <math>x_1+x_2+\cdots+x_k=n</math> in ''nonnegative'' integers. We call such a solution a '''weak <math>k</math>-composition''' of <math>n ...

:'''Instance''': <math>n</math> positive integers <math>x_1<x_2<\cdots <x_n</math>. ...

where <math>\mathbb{N}</math> is the set of all nonnegative integers. Compared to the LP for the max-flow problem, we just replace the last line Due to the Flow Integrality Theorem, when capacities are integers, there must be an integral flow whose value is maximum among all flows (int ...

...s: it guarantees that the median is always a list element (e.g., a list of integers will never have a fractional median), and it guarantees that the median exi ...

[[Category:Integers]] ...

...math>k</math>-compositions (the ''ordered'' sum of <math>k</math> positive integers). There are <math>{n-1\choose k-1}</math> many <math>k</math>-compositions ...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math ...

...and <math>N</math> be a random variable taking values in the non-negative integers and independent of the <math>X_n</math> for all <math>n \ge 1</math>. Prove ...th> uniformly at random, where <math>n</math> and <math>q</math> are given integers with <math>q \ge 1</math>. Find the expected number of inversions in <math> ...

:'''Instance''': <math>n</math> positive integers <math>x_1<x_2<\cdots <x_n</math>. ...

:Let <math>k,\ell</math> be positive integers. Then there exists an integer <math>R(k,\ell)</math> satisfying: :Let <math>r, k_1,k_2,\ldots,k_r</math> be positive integers. Then there exists an integer <math>R(r;k_1,k_2,\ldots,k_r)</math> satisfyi ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

For any tuple <math>\mathbf{v}=(n_1,n_2,\ldots,n_m)</math> of nonnegative integers satisfying that <math>n_1+n_2+\cdots +n_m=n</math>, let <math>a_{\mathbf{v} Let <math>\mathbf{v}=(n_1,\ldots,n_m)</math> be a vector of nonnegative integers satisfying that <math>n_1+\cdots+n_m=n</math>, and let <math>a_{\mathbf{v}} ...

...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math Let <math>U=\{1,2,\ldots,n\}</math> be the universe. The number of positive integers from <math>U</math> which is divisible by some <math>p_{i_1},p_{i_2},\ldots ...

...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math Let <math>U=\{1,2,\ldots,n\}</math> be the universe. The number of positive integers from <math>U</math> which is divisible by some <math>p_{i_1},p_{i_2},\ldots ...

...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math Let <math>U=\{1,2,\ldots,n\}</math> be the universe. The number of positive integers from <math>U</math> which is divisible by some <math>p_{i_1},p_{i_2},\ldots ...

...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math Let <math>U=\{1,2,\ldots,n\}</math> be the universe. The number of positive integers from <math>U</math> which is divisible by some <math>p_{i_1},p_{i_2},\ldots ...

...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math Let <math>U=\{1,2,\ldots,n\}</math> be the universe. The number of positive integers from <math>U</math> which is divisible by some <math>p_{i_1},p_{i_2},\ldots ...

...integer <math>n</math>, let <math>\phi(n)</math> be the number of positive integers from <math>\{1,2,\ldots,n\}</math> that are relative prime to <math>n</math Let <math>U=\{1,2,\ldots,n\}</math> be the universe. The number of positive integers from <math>U</math> which is divisible by some <math>p_{i_1},p_{i_2},\ldots ...

...roblem <math>f:\{0,1\}^*\to \mathbb{Z}^+</math> whose outputs are positive integers. ...roblem <math>f:\{0,1\}^*\to \mathbb{Z}^+</math> whose outputs are positive integers. ...

where <math>\mathbb{N}</math> is the set of all nonnegative integers. Compared to the LP for the max-flow problem, we just replace the last line Due to the Flow Integrality Theorem, when capacities are integers, there must be an integral flow whose value is maximum among all flows (int ...

...). For example, Briggs' first table contained the common logarithms of all integers in the range 1–1000, with a precision of 8 digits. As the function {{nowrap ...of logarithm tables: given a table listing log₁₀(''x'') for all integers ''x'' ranging from 1 to 1000, the logarithm of 3542 is approximated by ...

where <math>\mathbb{N}</math> is the set of all nonnegative integers. Compared to the LP for the max-flow problem, we just replace the last line Due to the Flow Integrality Theorem, when capacities are integers, there must be an integral flow whose value is maximum among all flows (int ...

where <math>\mathbb{N}</math> is the set of all nonnegative integers. Compared to the LP for the max-flow problem, we just replace the last line Due to the Flow Integrality Theorem, when capacities are integers, there must be an integral flow whose value is maximum among all flows (int ...

...d the smallest natural number. Natural numbers are always whole numbers ([[integers]]) and never less than zero. ...

...nstruction, a 2-universal hash function can be uniquely represented by two integers <math>a</math> and <math>b</math>, which can be stored in two entries (or j ...

:'''Input''': <math>n</math> positive integers <math>p_1,p_2,\ldots,p_n</math> and a positive integer <math>m</math>; :'''Input''': a set of <math>n</math> positive integers <math>S=\{x_1,x_2,\ldots,x_n\}</math>; ...

:'''Input''': <math>n</math> positive integers <math>p_1,p_2,\ldots,p_n</math> and a positive integer <math>m</math>; :'''Input''': a set of <math>n</math> positive integers <math>S=\{x_1,x_2,\ldots,x_n\}</math>; ...

:'''Input''': <math>n</math> positive integers <math>p_1,p_2,\ldots,p_n</math> and a positive integer <math>m</math>; :'''Input''': a set of <math>n</math> positive integers <math>S=\{x_1,x_2,\ldots,x_n\}</math>; ...

:'''Input''': <math>n</math> positive integers <math>p_1,p_2,\ldots,p_n</math> and a positive integer <math>m</math>; :'''Input''': a set of <math>n</math> positive integers <math>S=\{x_1,x_2,\ldots,x_n\}</math>; ...

:'''Input''': <math>n</math> positive integers <math>p_1,p_2,\ldots,p_n</math> and a positive integer <math>m</math>; :'''Input''': a set of <math>n</math> positive integers <math>S=\{x_1,x_2,\ldots,x_n\}</math>; ...

:'''Input''': <math>n</math> positive integers <math>p_1,p_2,\ldots,p_n</math> and a positive integer <math>m</math>; :'''Input''': a set of <math>n</math> positive integers <math>S=\{x_1,x_2,\ldots,x_n\}</math>; ...