hcju/proofwikips huggingface dataset:
Sigma Function of Integer
Let $n$ be an integer such that $n \ge 2$. Let $\map \sigma n$ be the sigma function of $n$. That is, let $\map \sigma n$ be the sum of all positive divisors of $n$. Let the prime decomposition of $n$ be: :$\displaystyle n = \prod_{1 \mathop \le i \mathop \le r} p_i^{k_i} =  p_1^{k_1} p_2^{k_2} \cdots p_r^{k_r}$ Then: :$\displaystyle \map \sigma n = \prod_{1 \mathop \le i \mathop \le r} \frac {p_i^{k_i + 1} - 1} {p_i - 1}$
Product of Sigma and Euler Phi Functions
Let $n$ be an integer such that $n \ge 2$. Let the prime decomposition of $n$ be: :$n = p_1^{k_1} p_2^{k_2} \ldots p_r^{k_r}$ Let $\sigma \left({n}\right)$ be the sigma function of $n$. Let $\phi \left({n}\right)$ be the Euler phi function of $n$. Then: :$\displaystyle \sigma \left({n}\right) \phi \left({n}\right) = n^2 \prod_{1 \mathop \le i \mathop \le r} \left({1 - \frac 1 {p_i^{k_i + 1}}}\right)$
Order Modulo n of Power of Integer
Let $a$ have multiplicative order $c$ modulo $n$. Then for any $k \ge 1$, $a^k$ has multiplicative order $\dfrac c {\gcd \left\{{c, k}\right\}}$ modulo $n$.
Integers with Primitive Roots
Let $n \in \Z: n > 1$. Then $n$ has a primitive root  one of the following holds: : $n = 2$ : $n = 4$ : $n = p^k$ : $n = 2 p^k$ where $p > 2$ is prime and $k \ge 1$.
Tau Function of Square-Free Integer is Power of 2
Let $n$ be a square-free integer. Let $\tau: \Z \to \Z$ be the $\tau$ function. Then $\map \tau n = 2^r$ for some $r \ge 1$. The converse is not true in general. That is, if $\map \tau n = 2^r$ for some $r \ge 1$, it is not necessarily the case that $n$ is square-free.
Tau Function Odd Iff Argument is Square
Let $\tau: \Z \to \Z$ be the $\tau$ function. Then $\tau \left({n}\right)$ is odd  $n$ is square.
Sigma Function Odd iff Argument is Square or Twice Square
Let $\sigma: \Z \to \Z$ be the sigma function. Then $\sigma \left({n}\right)$ is odd  $n$ is either square or twice a square.
Cancellability of Congruences
Let $a, b, c, n \in \Z$ be integers. Then: :$c a \equiv c b \pmod n \iff a \equiv b \pmod {n / d}$ where $d = \gcd \set {c, n}$.
Solution of Linear Diophantine Equation
The linear Diophantine equation: :$a x + b y  = c$ has solutions : :$\gcd \set {a, b} \divides c$ where $\divides$ denotes divisibility. If this condition holds with $\gcd \set {a, b} > 1$ then division by $\gcd \set {a, b}$ reduces the equation to: :$a' x + b' y = c'$ where $\gcd \set {a', b'} = 1$. If $x_0, y_0$ is one solution of the latter equation, then the general solution is: :$\forall k \in \Z: x = x_0 + b' k, y = y_0 - a' k$ or: :$\forall k \in \Z: x = x_0 + \dfrac b d k, y = y_0 - \dfrac a d k$
GCD with Prime
Let $p$ be a prime number. Then: :$\forall n \in \Z: \gcd \set {n, p} = \begin{cases} p & : p \divides n \\ 1 & : p \nmid n \end{cases}$
Prime Divides Power
Let $p$ be a prime number. Let $a, n \in \Z$ be integers. Then $p$ divides $a^n$  $p^n$ divides $a^n$.
Existence of Laurent Series
Let $z_0 \in \C$ be a complex number. Let $R \in \R_{>0}$ be a real number. Let $\map {B'} {z_0, R}$ be the open punctured disk at $z_0$ of radius $R$. Let $f: \map {B'} {z_0, R} \to \C$ be holomorphic. Then there exists a sequence $\sequence {a_n}_{n \mathop \in \Z}$ such that: :$\map f z = \displaystyle \sum_{n = -\infty}^\infty a_n \paren {z - z_0}^n$ for all $z \in B'(z_0, R)$.
Congruence by Divisor of Modulus
Let $z \in \R$ be a real number. Let $a, b \in \R$ such that $a$ is congruent modulo $z$ to $b$, that is: :$a \equiv b \pmod z$ Let $m \in \R$ such that $z$ is an integer multiple of $m$: :$\exists k \in \Z: z = k m$ Then: : $a \equiv b \pmod m$
Congruence of Quotient
Let $a, b \in \Z$ and $n \in \N$. Let $a$ be congruent to $b$ modulo $n$, i.e. $a \equiv b \pmod n$. Let $d \in \Z: d > 0$ such that $d$ is a common divisor of $a, b$ and $n$. Then: : $\displaystyle \frac a d \equiv \frac b d \pmod {n / d}$
Divisibility by 11
Let $N \in \N$ be expressed as: :$N = a_0 + a_1 10 + a_2 10^2 + \cdots + a_n 10^n$ Then $N$ is divisible by $11$  $\displaystyle \sum_{r \mathop = 0}^n \paren {-1}^r a_r$ is  divisible by $11$. That is, a divisibility test for $11$ is achieved by alternately adding and subtracting the digits and taking the result modulo $11$.
Solutions of Polynomial Congruences
Let $P \left({x}\right)$ be an integral polynomial. Let $a \equiv b \pmod n$. Then $P \left({a}\right) \equiv P \left({b}\right) \pmod n$. In particular, $a$ is a solution to the polynomial congruence $P \left({x}\right) \equiv 0 \pmod n$ iff $b$ is also.
Solution of Linear Congruence
Let $a x \equiv b \pmod n$ be a linear congruence. The following results hold:
Chinese Remainder Theorem/General Result
Let $n_1, n_2, \ldots, n_r$ be positive integers such that $n_i \perp n_j$ for all $i \ne j$ (that is, $\gcd \set {n_i, n_j} = 1$). Then the system of linear congruences: :$x \equiv b_1 \pmod {n_1}$ :$x \equiv b_2 \pmod {n_2}$ :$\ldots$ :$x \equiv b_r \pmod {n_r}$ has a simultaneous solution which is unique modulo $\displaystyle \prod_{i \mathop = 1}^r n_i$.
Second Supplement to Law of Quadratic Reciprocity
:$\paren {\dfrac 2 p} = \paren {-1}^{\paren {p^2 - 1} / 8} = \begin{cases} +1 & : p \equiv \pm 1 \pmod 8 \\ -1 & : p \equiv \pm 3 \pmod 8 \end{cases}$
First Supplement to Law of Quadratic Reciprocity
:$\paren {\dfrac {-1} p} = \paren {-1}^{\paren {p - 1} / 2} = \begin{cases} +1 & : p \equiv 1 \pmod 4 \\ -1 & : p \equiv 3 \pmod 4 \end{cases}$
Manipulation of Absolutely Convergent Series
Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be a real or complex series that is absolutely convergent.
Product of Sums
Let $\displaystyle \sum_{n \mathop \in A} a_n$ and $\displaystyle \sum_{n \mathop \in B} b_n$ be absolutely convergent sequences. Then: :$\displaystyle \left({ \sum_{i \mathop \in A} a_i }\right) \left({ \sum_{j \mathop \in B} b_j }\right) = \sum_{\left({i, j}\right) \mathop \in A \times B} a_i b_j$.
Green's Theorem
Let $\Gamma$ be a positively oriented piecewise smooth simple closed curve in  $\R^2$. Let $U = \map {\operatorname {Int} } \Gamma$, that is, the interior of $\Gamma$. Let $A$ and $B$ be functions of $\tuple {x, y}$ defined on an open region containing $U$ and have continuous partial derivatives in such a set. Then: :$\displaystyle \oint_\Gamma \paren {A \rd x + B \rd y} = \iint_U \paren {\frac {\partial B} {\partial x} - \frac {\partial A} {\partial y} } \rd x \rd y$
Gauss-Ostrogradsky Theorem
Let $U$ be a subset of $\R^3$ which is compact and has a piecewise smooth boundary $\partial U$. Let $\mathbf F: \R^3 \to \R^3$ be a smooth vector function defined on a neighborhood of $U$. Then: :$\displaystyle \iiint \limits_U \paren {\nabla \cdot \mathbf F} \rd V = \iint \limits_{\partial U} \mathbf F \cdot \mathbf n \rd S$ where $\mathbf n$ is the normal to $\partial U$.
Implications of Stokes' Theorem
Stokes' Theorem implies all of the following results: * Classical Stokes' Theorem * Green's Theorem * Gauss's Theorem * Fundamental Theorem of Calculus * Cauchy's Residue Theorem
Divergence Test
Let $\sequence {a_n}$ be a sequence in $\R$. If $\displaystyle \lim_{k \mathop \to \infty} a_k \ne 0$, then $\displaystyle \sum_{i \mathop = 1}^\infty a_n$ diverges.
Eisenstein's Lemma
Let $p$ be an odd prime. Let $a \in \Z$ be an odd integer such that $p \nmid a$. Let $\paren {\dfrac a p}$ be defined as the Legendre symbol. Then: :$\paren {\dfrac a p} = \paren {-1}^{\map \alpha {a, p} }$ where: :$\displaystyle \map \alpha {a, p} = \sum_{k \mathop = 1}^{\frac {p - 1} 2} \floor {\frac {k a} p}$ :$\floor {\dfrac {k a} p}$ is the floor function of $\dfrac {k a} p$.
Law of Quadratic Reciprocity
Let $p$ and $q$ be distinct odd primes. Then: :$\paren {\dfrac p q} \paren {\dfrac q p} = \paren {-1}^{\dfrac {\paren {p - 1} \paren {q - 1} } 4}$ where $\paren {\dfrac p q}$ and $\paren {\dfrac q p}$ are defined as the Legendre symbol. An alternative formulation is: $\paren {\dfrac p q} = \begin{cases} \quad \paren {\dfrac q p} & : p \equiv 1 \lor q \equiv 1 \pmod 4 \\ -\paren {\dfrac q p} & : p \equiv q \equiv 3 \pmod 4 \end{cases}$ The fact that these formulations are equivalent is immediate. This fact is known as the '''Law of Quadratic Reciprocity''', or '''LQR''' for short.
Rational Numbers and SFCFs are Equivalent
Every simple finite continued fraction has a rational value. Conversely, every rational number can be expressed as a simple finite continued fraction.
Properties of Convergents of Continued Fractions
Let $n \in \N \cup \set \infty$ be an extended natural number. Let $\sqbrk {a_0, a_1, a_2, \ldots}$ be a continued fraction in $\R$ of length $n$. Let $p_0, p_1, p_2, \ldots$ and $q_0, q_1, q_2, \ldots$ be its numerators and denominators. Let $C_0, C_1, C_2, \ldots$ be the convergents of $\sqbrk {a_0, a_1, a_2, \ldots}$. Then the following results apply:
Manipulation of Exterior Derivative
For the exterior derivative, the following statements are true: :$\map \d {a b} = a \rd b + \paren {\d a} b$ :$\map \d {a \wedge b} = \d a \wedge b - a \wedge \d b$ where $a \wedge b$ is the wedge product.
General Stokes' Theorem
Let $\omega$ be a smooth $\paren {n - 1}$-form with compact support on a smooth $n$-dimensional oriented manifold $X$. Let the boundary of $X$ be $\partial X$. Then: :$\displaystyle \int_{\partial X} \omega = \int_X \rd \omega$ where $\d \omega$ is the exterior derivative of $\omega$.
Kelvin-Stokes Theorem
Let $S$ be some orientable smooth surface with boundary in $\R^3$. Let $\mathbf F:\R^3 \to \R^3$ be a vector-valued function with Euclidean coordinate expression: :$F = f_1 \mathbf i + f_2 \mathbf j + f_3 \mathbf k$ where $f_i: \R^3 \to \R$. Then: :$\displaystyle \oint_{\partial S} f_1 \rd x + f_2 \rd y + f_3 \rd z = \iint_S \paren {\nabla \times \mathbf F} \cdot \mathbf n \rd A$ where $\mathbf n$ is the unit normal to $S$ and $\d A$ is the area element on the surface.
Even Convergent of Simple Continued Fraction is Strictly Smaller than Odd Convergent
Every even convergent is strictly smaller than every odd convergent.
Denominators of Simple Continued Fraction are Strictly Increasing
Let $N \in \N \cup \set \infty$ be an extended natural number. Let $\sqbrk {a_0, a_1, a_2, \ldots}$ be a simple continued fraction in $\R$ of length $N$. Let $q_0, q_1, q_2, \ldots$ be its denominators. Then with the possible exception of $q_0 = q_1$, the sequence $\sequence {q_n}$ is strictly increasing.
Simple Infinite Continued Fraction Converges to Irrational Number
The value of any simple infinite continued fraction in $\R$ is irrational.
Uniqueness of Analytic Continuation
Let $U \subset V \subset \C$ be open subsets of the complex plane. Let $V$ be connected. Suppose: :$(1): \quad F_1, F_2$ are functions defined on $V$ :$(2): \quad f$ is a function defined on $U$. Let $F_1$ and $F_2$ be analytic continuations of $f$ to $V$. Then $F_1 = F_2$.
Zeroes of Analytic Function are Isolated
Let $U \subset \C$ be some open set and let $f$ be an analytic function defined on $U$. Then either $f$ is a constant function, or the set $\set {z \in U: \map f z = 0}$ is totally disconnected.
Irrational Number is Limit of Unique Simple Infinite Continued Fraction
Let $x$ be an irrational number. Then the continued fraction expansion of $x$ is the unique simple infinite continued fraction that converges to $x$.
Analytic Continuation of Riemann Zeta Function
The Riemann zeta function is meromorphic on $\C$.
Accuracy of Convergents of Continued Fraction Expansion of Irrational Number
Let $x$ be an irrational number. Let $(a_0, a_1, \ldots)$ be its continued fraction expansion. Let $\left \langle {C_n}\right \rangle_{n \geq 0}$ be its sequence of convergents. Let $p_0, p_1, p_2, \ldots$ and $q_0, q_1, q_2, \ldots$ be its numerators and denominators. Then: :$\forall k \ge 1: \left\vert{x - \dfrac {p_{k + 1} } {q_{k + 1} } }\right\vert < \dfrac 1 {q_{k + 1} q_{k + 2} } \le \dfrac 1 {2 q_k q_{k + 1} } < \left\vert{x - \dfrac {p_k} {q_k} }\right\vert$ Thus: :The  of the inequality gives an indication of how close each convergent gets to its true value. :The  gives a bound that limits its accuracy.
Trivial Zeroes of Riemann Zeta Function are Even Negative Integers
Let $\rho = \sigma + i t$ be a zero of the Riemann zeta function not contained in the critical strip: :$0 \le \map \Re s \le 1$ Then: :$s \in \set {-2, -4, -6, \ldots}$ These are called the '''trivial zeros''' of $\zeta$.
Poles of Gamma Function
The gamma function $\Gamma: \C \to \C$ is analytic throughout the complex plane except at $\set {0, -1, -2, -3, \dotsc}$ where it has simple poles.
Zeroes of Gamma Function
The Gamma function is never equal to $0$.
Gamma Difference Equation
:$\map \Gamma {z + 1} = z \, \map \Gamma z$
Convergents are Best Approximations
Let $x$ be an irrational number. Let $(p_n)_{n\geq0}$ and $(q_n)_{n\geq0}$ be the numerators and denominators of its continued fraction expansion. Let $\dfrac {p_n} {q_n}$ be the $n$th convergent. Let $\dfrac a b$ be any rational number such that $0 < b < q_{n+1}$. Then: :$\forall n > 1: \left\vert{q_n x - p_n}\right\vert \le \left\vert{b x - a}\right\vert$ The equality holds only if $a = p_n$ and $b = q_n$.
Condition for Rational to be a Convergent
Let $x$ be an irrational number. Let the rational number $\dfrac a b$ satisfy the inequality: :$\left\vert{x - \dfrac a b}\right\vert < \dfrac 1 {2 b^2}$ Then $\dfrac a b$ is a convergent of $x$.
Quadratic Irrational is Root of Quadratic Equation
Let $x$ be a quadratic irrational. Then $x$ is a solution to a quadratic equation with rational coefficients.
Irrational Number has Periodic Continued Fraction iff Quadratic
Let $x$ be an irrational number. Then $x$ is a quadratic irrational  its continued fraction expansion is periodic.
Continued Fraction Expansion of Irrational Square Root
Let $n \in \Z$ such that $n$ is not a square. Then the continued fraction expansion of $\sqrt n$ is of the form: :$\sqbrk {a_1 \sequence {b_1, b_2, \ldots, b_{m - 1}, b_m, b_{m - 1}, \ldots, b_2, b_1, 2 a_1} }$ or :$\sqbrk {a_1 \sequence {b_1, b_2, \ldots, b_{m - 1}, b_m, b_m, b_{m - 1}, \ldots, b_2, b_1, 2 a_1} }$ where $m \in \Z: m \ge 0$. That is, it has the form as follows: :It is periodic :It starts with an integer $a_1$ :Its cycle starts with a palindromic section, either: :::$b_1, b_2, \ldots, b_{m - 1}, b_m, b_{m - 1}, \ldots, b_2, b_1$ ::or: :::$b_1, b_2, \ldots, b_{m - 1}, b_m, b_m, b_{m - 1}, \ldots, b_2, b_1$ ::which may be of length zero :Its cycle ends with twice the first partial quotient.
Solution of Pell's Equation is a Convergent
Let $x = a, y = b$ be a positive solution to Pell's Equation $x^2 - n y^2 = 1$. Then $\dfrac a b$ is a convergent of $\sqrt n$.
Convergence of P-Series
Let $p \in \C$ be a complex number.
Square Modulo 4
Let $x \in \Z$ be an integer. Then $x$ is: :even  $x^2 \equiv 0 \pmod 4$ :odd  $x^2 \equiv 1 \pmod 4$
Parity of Smaller Elements of Primitive Pythagorean Triple
Let $\left({x, y, z}\right)$ be a Pythagorean triple, that is, integers such that $x^2 + y^2 = z^2$. Then $x$ and $y$ are of opposite parity.
Dirichlet Series is Analytic
Let $(a_n)$ be sequence of complex numbers. Let :$\displaystyle f \left({z}\right) = \sum_{n \mathop = 1}^\infty \frac {a_n} {n^z}$ be the associated Dirichlet Series, which is defined at the points where the series converges. Then $f$ is analytic in every open set such that the sum converges in the set.
Consecutive Integers are Coprime
$\forall h \in \Z$, $h$ and $h + 1$ have only two common factors: $1$ and $-1$. That is, consecutive integers are always coprime.
Elements of Primitive Pythagorean Triple are Pairwise Coprime
Let $\tuple {x, y, z}$ be a primitive Pythagorean triple. Then: :$x \perp y$ :$y \perp z$ :$x \perp z$ That is, all elements of $\tuple {x, y, z}$ are pairwise coprime.
Odd Square Modulo 8
Let $x \in \Z$ be an odd square. Then $x \equiv 1 \pmod 8$.
Sum of Consecutive Triangular Numbers is Square
The sum of two consecutive triangular numbers is a square number.
If n is Triangular then so is 9n + 1
Let $n$ be a triangular number. Then $9 n + 1$ is also triangular.
Sum of Arithmetic Sequence
Let $\sequence {a_k}$ be an arithmetic sequence defined as: :$a_k = a + k d$ for $n = 0, 1, 2, \ldots, n - 1$ Then its closed-form expression is: {{begin-eqn}} {{eqn | l = \sum_{k \mathop = 0}^{n - 1} \paren {a + k d}       | r = n \paren {a + \frac {n - 1} 2 d}       | c =  }} {{eqn | r = \frac {n \paren {a + l} } 2       | c = where $l$ is the last term of $\sequence {a_k}$ }} {{end-eqn}}
Odd Square is Eight Triangles Plus One
Let $n \in \Z$ be an odd integer. Then $n$ is square  $n = 8 m + 1$ where $m$ is triangular.
Triangular Number Modulo 3 and 9
Let $n$ be a triangular number. Then one of the following two conditions applies: : $n \equiv 0 \pmod 3$ : $n \equiv 1 \pmod 9$
Closed Form for Polygonal Numbers
Let $\map P {k, n}$ be the $n$th $k$-gonal number. The closed-form expression for $\map P {k, n}$ is given by: {{begin-eqn}} {{eqn | l = \map P {k, n}       | r = \frac n 2 \paren {\paren {k - 2} n - k + 4} }} {{eqn | r = \frac {k - 2} 2 \paren {n^2 - n} + n }} {{end-eqn}}
Difference Between Adjacent Polygonal Numbers is Triangular Number
Let $\map P {k, n}$ be the $n$th $k$-gonal number. Then: :$\map P {k + 1, n} - \map P {k, n} = T_{n - 1}$ where $T_n$ is the $n$th triangular number.
Method of Infinite Descent
Let $P \left({n_\alpha}\right)$ be a propositional function depending on $n_\alpha \in \N$. Let $P \left({n_\alpha}\right) \implies P \left({n_\beta}\right)$ such that $0 < n_\beta < n_\alpha$. Then we may deduce that $P \left({n}\right)$ is false for all $n \in \N$. That is, suppose that by assuming the truth of $P \left({n_\alpha}\right)$ for ''any'' natural number $n_\alpha$, we may deduce that there always exists ''some'' number $n_\beta$ strictly less than $n_\alpha$ for which $P \left({n_\beta}\right)$ is also true, then $P \left({n_\alpha}\right)$ can not be true after all. This technique is known as the '''method of infinite descent'''. The process of deducing the truth of $P \left({n_\beta}\right)$ from $P \left({n_\alpha}\right)$ such that $0 < n_\beta < n_\alpha$ is known as the '''descent step'''.
Brahmagupta-Fibonacci Identity
Let $a, b, c, d$ be numbers. Then: :$\paren {a^2 + b^2} \paren {c^2 + d^2} = \paren {a c + b d}^2 + \paren {a d - b c}^2$
Fermat's Two Squares Theorem
Let $p$ be a prime number. Then $p$ can be expressed as the sum of two squares  either: : $p = 2$ or: : $p \equiv 1 \pmod 4$ The expression of a prime of the form $4 k + 1$ as the sum of two squares is unique except for the order of the two summands.
Integer as Sum of Two Squares
Let $n$ be a positive integer. Then: : $n$ can be expressed as the sum of two squares : :each of its prime divisors of the form $4 k + 3$ (if any) occur to an even power.
Square Modulo 8
Let $x \in \Z$ be an integer. : If $x$ is even then: ::: $x^2 \equiv 0 \pmod 8$ or $x^2 \equiv 4 \pmod 8$ : If $x$ is odd then: ::: $x^2 \equiv 1 \pmod 8$
Integer as Sum of Three Squares
Let $r$ be a positive integer. Then $r$ can be expressed as the sum of three squares  it is not of the form: :$4^n \paren {8 m + 7}$ for some $m, n \in \Z_{\ge 0}$.
Product of Sums of Four Squares
Let $a, b, c, d, w, x, y, z$ be numbers. Then: {{begin-eqn}} {{eqn | o =        | r = \left({a^2 + b^2 + c^2 + d^2}\right) \left({w^2 + x^2 + y^2 + z^2}\right)       | c =  }} {{eqn | l = =       | o =        | r = \left({a w + b x + c y + d z}\right)^2       | c =  }} {{eqn | o = +       | r = \left({a x - b w + c z - d y}\right)^2       | c =  }} {{eqn | o = +       | r = \left({a y - b z - c w + d x}\right)^2       | c =  }} {{eqn | o = +       | r = \left({a z + b y - c x - d w}\right)^2       | c =  }} {{end-eqn}}
Lagrange's Four Square Theorem
Every positive integer can be expressed as a sum of four squares.
Hilbert-Waring Theorem
For each $k \in \Z: k \ge 2$, there exists a positive integer $\map g k$ such that every positive integer can be expressed as a sum of at most $\map g k$ $k$th powers.
Necessary Condition for Existence of BIBD
Let there exist be a BIBD with parameters $v, b, r, k, \lambda$. Then the following are true: : $(1): \quad b k = r v$ : $(2): \quad \lambda \paren {v - 1} = r \paren {k - 1}$ : $(3): \quad b \dbinom k 2 = \lambda \dbinom v 2$ : $(4): \quad k < v$ : $(5): \quad r > \lambda$ All of $v, b, r, k, \lambda$ are integers. Some sources prefer to report $(3)$ as: :$b = \dfrac {\dbinom v 2} {\dbinom k 2} \lambda$ which is less appealing visually, and typographically horrendous.
Mapping from Singleton is Injection
Let $f: S \to T$ be a mapping. Let $S$ be a singleton. Then $f$ is an injection.
Mapping to Singleton is Surjection
Let $S$ be a non-empty set. Let $f: S \to T$ be a mapping. Let $T$ be a singleton. Then $f$ is a surjection.
Generating Function for Constant Sequence
Let $\sequence {a_n}$ be the sequence defined as: : $\forall n \in \N: a_n = r$ for some $r \in \R$. Then the generating function for $\sequence {a_n}$ is given as: :$\map G z = \dfrac r {1 - z}$ for $\size z < 1$
Area of Circle
The area $A$ of a circle is given by: : $A = \pi r^2$ where $r$ is the radius of the circle.
Distance Formula
The distance $d$ between two points $A = \tuple {x_1, y_1}$ and $B = \tuple {x_2, y_2}$ on a Cartesian plane is: :$d = \sqrt {\paren {x_1 - x_2}^2 + \paren {y_1 - y_2}^2}$
Equation of Circle
The equation of a circle with radius $R$ and center $\tuple {a, b}$ can be expressed in the following forms:
Null URM Program Computes Identity Function
The null URM program computes the '''identity function''' $I_\N: \N \to \N$, defined as: :$\forall n \in \N: \map {I_\N} n = n$
Composition of One-Variable URM Computable Functions
Let $f: \N \to \N$ and $g: \N \to \N$ be URM computable functions of one variable. Let $f \circ g$ be the composition of $f$ and $g$. Then $f \circ g: \N \to \N$ is a URM computable function.
Concatenation of URM Programs is Associative
Let $P, Q, R$ be one-variable URM programs. Then the concatenated URM programs $P * \left({Q * R}\right)$ and $\left({P * Q}\right) * R$ are the same.
Primitive of Cosine Function
:$\ds \int \cos x \rd x = \sin x + C$
Primitive of Sine Function
:$\ds \int \sin x \rd x = -\cos x + C$
Function Obtained by Substitution from URM Computable Functions
Let the functions $f: \N^t \to \N, g_1: \N^k \to \N, g_2: \N^k \to \N, \ldots, g_t: \N^k \to \N$ all be URM computable functions. Let $h: \N^k \to \N$ be defined from $f, g_1, g_2, \ldots, g_t$ by substitution. Then $h$ is also URM computable.
Normalized URM Program
Let $P$ be a URM program. Let $l = \map \lambda P$ be the number of basic instructions in $P$. Let $u = \map \rho P$ be the number of registers used by $P$. Then $P$ can be modified as follows: :Every <tt>Jump</tt> of the form $\map J {m, n, q}$ where $q > l$ may be replaced by $\map J {m, n, l + 1}$ :If $u > 0$, a Clear Registers Program $\map Z {2, u}$ can be appended to the end of $P$ at lines $l + 1$ to $l + u - 1$. The new program $P'$ that results from the above modifications produces exactly the same output as $P$ for each input. Note now though that $\map \lambda {P'} = l + u - 1$. Such a program as $P'$ is called a '''normalized URM program'''. The point of doing this is so as to make programs easier to concatenate. Once the above have been done, each program has a well-defined exit line which can be used as the start line of the program that immediately follows it.
Block Copy Program
Let $k, m, n \in \N$ be natural numbers such that: * $k \ge 1$; * $\left|{m - n}\right| \ge k$. The URM program defined as: {| |- ! align="right" | Line !! ! align="left" | Command |- | align="right" | $1$ || | align="left" | $C \left({m, n}\right)$ |- | align="right" | $2$ || | align="left" | $C \left({m+1, n+1}\right)$ |- | align="right" | $\vdots$ || | align="left" | $\vdots$ |- | align="right" | $k$ || | align="left" | $C \left({m+k-1, n+k-1}\right)$ |} is called a '''block copy program'''. It is abbreviated $C \left({m, n, k}\right)$. It has the effect of copying the contents of registers $R_m, R_{m+1}, \ldots, R_{m+k-1}$ into the registers $R_n, R_{n+1}, \ldots, R_{n+k-1}$ respectively. It has length defined as $\lambda \left({C \left({m, n, k}\right)}\right) = k$.
Function Obtained by Primitive Recursion from URM Computable Functions
Let the functions $f: \N^k \to \N$ and $g: \N^{k+2} \to \N$ be URM computable functions. Let $h: \N^{k+1} \to \N$ be obtained from $f$ and $g$ by primitive recursion. Then $h$ is also URM computable.