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Let $\mathbf{Z}$ have the integers as objects, with $\mathbf{Z}(n,p)=\mathbb{Z}$ for $p=n$ or $p=n-1$ and zero otherwise. Composition is bilinear, the generator of $\mathbf{Z}(n,n)$ is the identity, and the composite of two degree-lowering generators is zero. Additive functors $\mathbf{Z}\to \mathcal{A}$ are exactly chain complexes in the additive category $\mathcal{A}$.

If $a,b$ are nonzero integers, $d=\gcd (a,b)$, and $(x_0,y_0)$ is one solution of the linear Diophantine equation $ax+by=c$, then all solutions areSubtract two solutions and use the fact that $a/d$ and $b/d$ are coprime integers. The linear Diophantine equation solvability criterion says that a particular solution exists exactly when $d$ divides $c$; the extended Euclidean algorithm constructs one. This parametrization separates existence from the complete family of answers.

Analytic number theory studies integers, prime numbers, and arithmetic functions using tools from real analysis and complex analysis, especially Dirichlet series and their singularities.

The Archimedean property says that the natural numbers are unbounded in the real numbers. In particular, for every positive real number $a$ there is a positive integer $n$ with $1/n<a$. Otherwise a purported upper bound for all natural numbers contradicts the least-upper-bound property by translating that bound by one.

For an integer base $p\ge 2$, a nonnegative integer has a finite positional representation with digits $k_i\in \{0,\dots ,p-1\}$. Repeated Euclidean division gives the digits uniquely; appending zero high-place digits has no effect. The base need not be prime for uniqueness, but a prime base is useful in Lucas theorem and modular binomial coefficients.

For every $\epsilon >0$ there is a positive integer $K(\epsilon )$ such that every finite bipartite graph has an $\epsilon$-regular pair of partitions with at most $K(\epsilon )$ cells on each side.

To prove this, give $X\times Y$ the uniform probability measure, let $f$ be the indicator function of the edge set, and for partitions $\mathcal{P},\mathcal{Q}$ define their energy byThis conditional expectation is an orthogonal projection, so $0\le \mathcal{E}\le \Vert f{\Vert }_2^2\le 1$. If the partitions are not $\epsilon$-regular, choose witnesses $A_{ij}\subseteq X_i$ and $B_{ij}\subseteq Y_j$ in every irregular cell pair and refine each $X_i$ by all its $A_{ij}$ and each $Y_j$ by all its $B_{ij}$. The refined energy exceeds the old energy byOn $A_{ij}\times B_{ij}$ the absolute value function of the mean of the difference is greater than $\epsilon$, while this rectangle occupies at least an ${\epsilon }^2$ fraction of $X_i\times Y_j$. The Cauchy-Schwarz inequality therefore contributes more than ${\epsilon }^4|X_i||Y_j|/(|X||Y|)$ from that cell pair. Irregular pairs have total weight greater than $\epsilon$, so every refinement raises the energy by more than ${\epsilon }^5$. The process stops after at most $\lceil {\epsilon }^{-5}\rceil$ refinements. If the current cell counts are $r,s$, the construction gives at most $r{2}^s,s{2}^r$ new cells, so a finite iterated bound depending only on $\epsilon$ supplies $K(\epsilon )$.

For positive integers $k,s,\ell$, some $N$ ensures that every $k$-color finite coloring of $[N]$ contains a monochromatic setIt follows from the Van der Waerden theorem by mathematical induction on the number of colors. If $M$ is a bound for $k-1$ colors, take a long one-color arithmetic progression of length $(\ell -1)M+1$ and step $d$. Either one of $sd,2sd,\dots ,Msd$ has its color, giving the result immediately, or these multiples use only $k-1$ colors. Pull back their finite coloring to $[M]$ and use mathematical induction, then multiply the resulting configuration by $sd$. Enlarge the ambient finite integer interval to include all these multiples. The cases $k=1$ and $\ell =1$ are immediate. The parameter $s$ makes this useful for positive monochromatic solutions of a partition regular equation with arbitrary integer coefficients.

Cancellation occurs when different complex numbers in an exponential sum have directions that reduce the absolute value of their sum. For example, the orthogonality of roots of unity makes ${\sum }_{a=0}^{p-1}e(a/p)=0$ for every integer $p>1$, despite all terms having absolute value one.

The Cartan integer associated with two roots isA crystallographic root system requires every Cartan integer to be an integer.

For coefficients supported on $N$ consecutive integers,The star restricts to primitive Dirichlet characters. Their finite Fourier transform of a primitive Dirichlet character has normalization of absolute value $\sqrt{q}$. Orthogonality of Dirichlet characters therefore bounds the weighted contribution for one modulus by ${\sum }_{\begin{array}{c}a\mathrm{mod}q\\ (a,q)=1\end{array}}|{\sum }_n{a}_{n}e(an/q){|}^2$. Sum over $q$ and use the exponential-sum large sieve on the distinct reduced fractions of denominators at most $Q$.

For a complex semisimple Lie algebra and an integer $k\ge 0$, the Weyl character formula and Weyl denominator formula giveApply the dilation $e^{\mu }\mapsto e^{(k+1)\mu }$ to the denominator identity to obtain the numerator. Cancellation leaves a finite geometric series for each positive root of a root system. Evaluating the resulting polynomial at all formal exponentials equal to one gives the dimension, without taking an undefined quotient at the identity.

For a prime number $p$ and nonnegative integers $h+k=p-1$, Wilson theorem gives $(-1{)}^{k}h!k!\equiv -1(\mathrm{mod}p)$ by replacing the last $k$ factors of $(p-1)!$ with $-1,-2,\dots ,-k$. For odd $p$, $h,k$ have the same parity; for $p=2$, both signs have the same residue class. Thus $h!k!+(-1{)}^h\equiv 0(\mathrm{mod}p)$, including the endpoint cases $0!=1$.

A composite number is an integer greater than one that is not a prime number.

Two integers $a$ and $b$ are coprime when $\gcd (a,b)=1$.

A corner consists of these three points in the integer grid, all contained in the set being studied. The displacement may be positive or negative, but must be nonzero, so the points are distinct. The corners theorem guarantees a corner in every subset of positive fixed density of a finite subset of a sufficiently large square grid.

If every sequence of distinct elements of a positive-real set $A$ has arithmetic averages tending to zero, then $A$ is a countable set. For each positive integer $n$, the threshold set $\{a\in A:a\ge 1/n\}$ must be finite: an infinite sequence inside it would have averages at least $1/n$. The Archimedean property makes $A$ their union, so the countable union of countable sets theorem applies. The hypothesis concerns every distinct-element sequence, not merely one enumeration.

For $b>1$ and an integer $q\ge 2$, assign $x\ge 1$ the color $\lfloor {\log }_{b}x\rfloor \mathrm{mod}q$. The floor function partitions the domain into half-open geometric bins. If $1<{\log }_b(y/x)<q-1$, their bin indices differ by an integer in $\{1,\dots ,q-1\}$, so their colors differ. For example, $b=3/2$ and $q=3$ separates every ratio in $[1.9,2]$. Composing this finite coloring with another logarithm can obstruct monochromatic multiplicative configurations whose logarithms have such ratios.

The bracket enumerates the positive integers not divisible by $10$. Every positive integer has exactly one decomposition into a power of $10$ times such an integer, so this formula is a bijection from ${\mathbb{N}}^2$ to $\mathbb{N}$. Subtracting one in the codomain includes zero without requiring a trailing-zero convention for its decimal representation. For the inverse, write $N+1=10^{i}q$ with $10\nmid q$, then $j=q-1-\lfloor q/10\rfloor$.