UDACA/Split-Isa-AF-MMA · Datasets at Hugging Face

Statement: stringlengths 7 24.3k
The normalization of the denominator of a fraction is the same as the denominator of the fraction.
If $p$ is a polynomial and $c$ is a rational number, then the polynomial $cp$ is equal to the rational number $c$ times the polynomial $p$.
The fractional polynomial of degree 0 is 0.
The fractional polynomial $1$ is equal to $1$.
The fractional part of a sum of two polynomials is the sum of the fractional parts of the polynomials.
The fractional part of a polynomial is equal to the fractional part of the difference of the polynomial and its integer part.
The fractional part of a sum is the sum of the fractional parts.
The fractional part of a product of two polynomials is equal to the product of the fractional parts of the two polynomials.
Two polynomials are equal if and only if their fractional parts are equal.
A fractional polynomial is zero if and only if its numerator is zero.
If $p$ divides $q$, then $\frac{p}{1}$ divides $\frac{q}{1}$.
The product of the polynomials $f(x)$ for $x$ in a multiset $A$ is equal to the product of the polynomials $f(x)$ for $x$ in the multiset $A$ with the coefficients converted to fractions.
A polynomial $p$ is a unit if and only if its fractional part is a unit and its content is 1.
If a polynomial $p$ divides $1$, then the fractional polynomial $p$ divides $1$.
If two polynomials are equal up to a constant factor, then the constant factor is a rational number and the polynomials are equal up to a constant factor that is a unit.
The primitive part of the fractional part of $0$ is $0$.
The content of a polynomial is zero if and only if the polynomial is zero.
If $p$ is a nonzero polynomial with rational coefficients, then the content of the primitive part of $p$ is $1$.
The content of a polynomial times the primitive part of the polynomial is the polynomial itself.
The fractional content of a fractional polynomial is equal to the content of the polynomial.
Every polynomial $p$ over a field $K$ can be written as $c p'$, where $c$ is a nonzero element of $K$ and $p'$ is a polynomial with integer coefficients whose content is $1$.
If $c$ is a prime element, then $[c]$ is a prime element.
A constant polynomial is a prime element if and only if its constant is a prime element.
If $p$ and $q$ are polynomials with integer coefficients and $p$ is primitive, then $p$ divides $q$ if and only if the fractional polynomial $p/\gcd(p,q)$ divides $q/\gcd(p,q)$.
If $p$ is an irreducible polynomial over a field, then $p$ is a prime element.
If $p$ is a nonzero polynomial over a field, then the product of its prime factors is equal to $p$ divided by its leading coefficient.
If a polynomial $p$ is a factor of a polynomial $x$ in the prime factorization of $x$, then $p$ is prime.
A non-constant polynomial $p$ is irreducible if and only if the fractional polynomial $\frac{p}{\text{content}(p)}$ is irreducible.
$1$ is not a prime number.
If $p$ is a prime number, then $p \geq 2$.
If $p$ is a prime number, then $p \geq 2$.
If $p$ is a prime number, then $p \geq 0$.
If $p$ is a prime number, then $p > 0$.
A prime number is greater than zero.
If $p$ is a prime number, then $p \geq 1$.
If $p$ is a prime number, then $p \geq 1$.
A prime number is greater than or equal to 1.
A prime number is greater than 1.
If $p$ is a prime number, then $p > 1$.
If $p$ is a prime number, then $p > 1$.
If $p$ is a natural number greater than or equal to 2, and if $p$ divides the product of any two natural numbers, then $p$ divides one of the two numbers, then $p$ is a prime number.
If $p$ is an integer greater than or equal to 2, and if $p$ divides the product of any two integers, then $p$ divides one of the integers. Then $p$ is a prime number.
A natural number $n$ is a prime element if and only if $n$ is a prime number.
An integer $k$ is a prime element if and only if $\bar{k}$ is a prime number.
A natural number is prime if and only if the corresponding integer is prime.
A natural number is prime if and only if the corresponding integer is prime.
A natural number $k$ is prime if and only if $k \geq 0$ and $k$ is prime as an integer.
If $p$ is a natural number greater than or equal to 2, and if $n$ divides $p$ then $n$ is either 1 or $p$, then $p$ is prime.
If $p$ is an integer greater than or equal to 2, and if $k$ divides $p$, then $\|k\| = 1$ or $\|k\| = p$.
A natural number $n$ is prime if and only if $n > 1$ and the only divisors of $n$ are $1$ and $n$.
An integer $n$ is prime if and only if $n > 1$ and the only positive divisors of $n$ are $1$ and $n$.
If $p$ is a prime number greater than $n$, then $p$ is not divisible by $n$.
If $p$ is a prime number greater than $n$ and $n$ is greater than $1$, then $n$ does not divide $p$.
If $p$ is a prime number greater than $2$, then $p$ is odd.
If $p$ is a prime number greater than $2$, then $p$ is odd.
A positive integer $p$ is prime if and only if $p$ is greater than 1 and for every positive integer $m$, if $m$ divides $p$, then $m$ is either 1 or $p$.
If $m$ is a composite number, then there exist two numbers $n$ and $k$ such that $n = m \cdot k$, $1 < m < n$, and $1 < k < n$.
If $p$ is a prime, $m$ is not divisible by $p$, and $m p^k = m' p^{k'}$ with $k < k'$, then this is impossible.
If $p$ is prime, $m$ and $m'$ are not divisible by $p$, and $m p^k = m' p^{k'}$, then $k = k'$.
If $p$ is a prime number, $m$ and $m'$ are not divisible by $p$, and $m p^k = m' p^{k'}$, then $m = m'$ and $k = k'$.
If $p$ is a prime number and $m$ is a positive integer, then there exist a positive integer $k$ and a positive integer $n$ that is not divisible by $p$ such that $m = n \cdot p^k$.
A natural number $p$ is prime if and only if $p > 1$ and $p$ is not divisible by any number in the set $\{2, 3, \ldots, p-1\}$.
A positive integer $p$ is prime if and only if $p$ is greater than $1$ and for all $n$ in the range $2 \leq n < p$, $n$ does not divide $p$.
The prime numbers are the same in $\mathbb{Z}$ and $\mathbb{N}$.
The number 2 is prime.
A natural number is prime if and only if it is greater than 1 and has no divisors in the set $\{2, 3, \ldots, n-1\}$.
The number 97 is prime.
The number 97 is prime.
If $n$ is a natural number greater than 1, then there exists a prime number $p$ that divides $n$.
If $k$ is an integer and $\|k\| \neq 1$, then there exists a prime number $p$ such that $p$ divides $k$.
For every natural number $n$, there exists a prime number $p$ such that $n < p \leq n! + 1$.
There exists a prime number greater than $n$.
There are infinitely many primes.
If $p$ and $q$ are natural numbers such that $pq$ is prime, then $p = 1$ or $q = 1$.
If $p$ is a prime number and $x$ and $y$ are natural numbers such that $x y = p^k$, then there exist natural numbers $i$ and $j$ such that $x = p^i$ and $y = p^j$.
If $p$ is a prime number, $n$ is a positive integer, and $x^n = p^k$, then $x = p^i$ for some integer $i$.
If $p$ is a prime number, then $d$ divides $p^k$ if and only if $d = p^i$ for some $i \leq k$.
For any two natural numbers $a$ and $b$, there exist two natural numbers $x$ and $y$ such that $a x - b y = \gcd(a, b)$ or $b x - a y = \gcd(a, b)$.
If $a$ and $b$ are relatively prime, then $a$ and $b$ divide $a x + b y$ for any $x$ and $y$.
If $a$ and $b$ are coprime, then there exist integers $q_1$ and $q_2$ such that $u + q_1 a = v + q_2 b$.
If $a$ and $b$ are coprime and $b \neq 1$, then there exist integers $x$ and $y$ such that $a x = b y + 1$.
If $p$ is a prime and $p$ does not divide $a$, then there exist integers $x$ and $y$ such that $a x = 1 + p y$.
If $p$ is a prime factor of $x$, then $p > 0$.
If $p$ is a prime factor of $x$, then $p > 0$.
If $p$ is a prime factor of $n$, then $p \geq 0$.
If $n$ is a positive integer, then the product of the prime factors of $n$ is $n$.
Every positive integer $n$ can be written as a product of primes.
The product of the prime factors of a positive integer is the integer itself.
Every positive integer $n$ can be written as a product of prime powers.
Every positive integer $n$ can be written as a product of prime powers.
If $n$ is a natural number and $S$ is the set of prime factors of $n$, then the function $f$ defined by $f(p) = \text{multiplicity}(p, n)$ is the unique function such that $n = \prod_{p \in S} p^{f(p)}$.
If $n$ is a positive integer and $S$ is a finite set of primes such that $n = \prod_{p \in S} p^{f(p)}$, then $S$ is the set of prime factors of $n$ and $f(p)$ is the multiplicity of $p$ in $n$.
If $S$ is a finite set of primes, and $n$ is the product of the $p$th powers of the primes in $S$, then $S$ is the set of prime factors of $n$.
If $f$ is a function from the set of primes to the set of positive integers such that the set of primes $p$ with $f(p) > 0$ is finite, then the set of prime factors of $\prod_{p \in \mathbb{P}} p^{f(p)}$ is exactly the set of primes $p$ with $f(p) > 0$.
If $n$ is a positive integer and $S$ is a finite set of primes such that $n = \prod_{p \in S} p^{f(p)}$, then $S$ is the set of prime factors of $n$.
The absolute value of the product of a set of numbers is equal to the product of the absolute values of the numbers.
If $f$ is a function from the positive integers to the positive integers such that the set of primes $p$ such that $f(p) > 0$ is finite, then the prime factors of $\prod_{p \geq 0, f(p) > 0} p^{f(p)}$ are exactly the primes $p$ such that $f(p) > 0$.
If $S$ is the set of primes $p$ such that $f(p) > 0$, and $n$ is the product of the $p^{f(p)}$ for $p \in S$, then the multiplicity of $p$ in $n$ is $f(p)$.
If $f$ is a function from the natural numbers to the natural numbers such that $f(p) > 0$ for only finitely many primes $p$, and if $f(p) > 0$ implies that $p$ is prime, then the multiplicity of $p$ in the product $\prod_{p \in \mathbb{N}} p^{f(p)}$ is $f(p)$.
If $S$ is a finite set of primes, $n$ is the product of the $p$-th powers of the primes in $S$, and $p$ is a prime in $S$, then the multiplicity of $p$ in $n$ is the power of $p$ in the product.

The normalization of the denominator of a fraction is the same as the denominator of the fraction.

If $p$ is a polynomial and $c$ is a rational number, then the polynomial $cp$ is equal to the rational number $c$ times the polynomial $p$.

The fractional polynomial of degree 0 is 0.

The fractional polynomial $1$ is equal to $1$.

The fractional part of a sum of two polynomials is the sum of the fractional parts of the polynomials.

The fractional part of a polynomial is equal to the fractional part of the difference of the polynomial and its integer part.

The fractional part of a sum is the sum of the fractional parts.

The fractional part of a product of two polynomials is equal to the product of the fractional parts of the two polynomials.

Two polynomials are equal if and only if their fractional parts are equal.

A fractional polynomial is zero if and only if its numerator is zero.

If $p$ divides $q$, then $\frac{p}{1}$ divides $\frac{q}{1}$.

The product of the polynomials $f(x)$ for $x$ in a multiset $A$ is equal to the product of the polynomials $f(x)$ for $x$ in the multiset $A$ with the coefficients converted to fractions.

A polynomial $p$ is a unit if and only if its fractional part is a unit and its content is 1.

If a polynomial $p$ divides $1$, then the fractional polynomial $p$ divides $1$.

If two polynomials are equal up to a constant factor, then the constant factor is a rational number and the polynomials are equal up to a constant factor that is a unit.

The primitive part of the fractional part of $0$ is $0$.

The content of a polynomial is zero if and only if the polynomial is zero.

If $p$ is a nonzero polynomial with rational coefficients, then the content of the primitive part of $p$ is $1$.

The content of a polynomial times the primitive part of the polynomial is the polynomial itself.

The fractional content of a fractional polynomial is equal to the content of the polynomial.

Every polynomial $p$ over a field $K$ can be written as $c p'$, where $c$ is a nonzero element of $K$ and $p'$ is a polynomial with integer coefficients whose content is $1$.

If $c$ is a prime element, then $[c]$ is a prime element.

A constant polynomial is a prime element if and only if its constant is a prime element.

If $p$ and $q$ are polynomials with integer coefficients and $p$ is primitive, then $p$ divides $q$ if and only if the fractional polynomial $p/\gcd(p,q)$ divides $q/\gcd(p,q)$.

If $p$ is an irreducible polynomial over a field, then $p$ is a prime element.

If $p$ is a nonzero polynomial over a field, then the product of its prime factors is equal to $p$ divided by its leading coefficient.

If a polynomial $p$ is a factor of a polynomial $x$ in the prime factorization of $x$, then $p$ is prime.

A non-constant polynomial $p$ is irreducible if and only if the fractional polynomial $\frac{p}{\text{content}(p)}$ is irreducible.

$1$ is not a prime number.

If $p$ is a prime number, then $p \geq 2$.

If $p$ is a prime number, then $p \geq 0$.

If $p$ is a prime number, then $p > 0$.

A prime number is greater than zero.

If $p$ is a prime number, then $p \geq 1$.

A prime number is greater than or equal to 1.

A prime number is greater than 1.

If $p$ is a prime number, then $p > 1$.

If $p$ is a natural number greater than or equal to 2, and if $p$ divides the product of any two natural numbers, then $p$ divides one of the two numbers, then $p$ is a prime number.

If $p$ is an integer greater than or equal to 2, and if $p$ divides the product of any two integers, then $p$ divides one of the integers. Then $p$ is a prime number.

A natural number $n$ is a prime element if and only if $n$ is a prime number.

An integer $k$ is a prime element if and only if $\bar{k}$ is a prime number.

A natural number is prime if and only if the corresponding integer is prime.

A natural number $k$ is prime if and only if $k \geq 0$ and $k$ is prime as an integer.

If $p$ is a natural number greater than or equal to 2, and if $n$ divides $p$ then $n$ is either 1 or $p$, then $p$ is prime.

If $p$ is an integer greater than or equal to 2, and if $k$ divides $p$, then $|k| = 1$ or $|k| = p$.

A natural number $n$ is prime if and only if $n > 1$ and the only divisors of $n$ are $1$ and $n$.

An integer $n$ is prime if and only if $n > 1$ and the only positive divisors of $n$ are $1$ and $n$.

If $p$ is a prime number greater than $n$, then $p$ is not divisible by $n$.

If $p$ is a prime number greater than $n$ and $n$ is greater than $1$, then $n$ does not divide $p$.

If $p$ is a prime number greater than $2$, then $p$ is odd.

A positive integer $p$ is prime if and only if $p$ is greater than 1 and for every positive integer $m$, if $m$ divides $p$, then $m$ is either 1 or $p$.

If $m$ is a composite number, then there exist two numbers $n$ and $k$ such that $n = m \cdot k$, $1 < m < n$, and $1 < k < n$.

If $p$ is a prime, $m$ is not divisible by $p$, and $m p^k = m' p^{k'}$ with $k < k'$, then this is impossible.

If $p$ is prime, $m$ and $m'$ are not divisible by $p$, and $m p^k = m' p^{k'}$, then $k = k'$.

If $p$ is a prime number, $m$ and $m'$ are not divisible by $p$, and $m p^k = m' p^{k'}$, then $m = m'$ and $k = k'$.

If $p$ is a prime number and $m$ is a positive integer, then there exist a positive integer $k$ and a positive integer $n$ that is not divisible by $p$ such that $m = n \cdot p^k$.

A natural number $p$ is prime if and only if $p > 1$ and $p$ is not divisible by any number in the set $\{2, 3, \ldots, p-1\}$.

A positive integer $p$ is prime if and only if $p$ is greater than $1$ and for all $n$ in the range $2 \leq n < p$, $n$ does not divide $p$.

The prime numbers are the same in $\mathbb{Z}$ and $\mathbb{N}$.

The number 2 is prime.

A natural number is prime if and only if it is greater than 1 and has no divisors in the set $\{2, 3, \ldots, n-1\}$.

The number 97 is prime.

If $n$ is a natural number greater than 1, then there exists a prime number $p$ that divides $n$.

If $k$ is an integer and $|k| \neq 1$, then there exists a prime number $p$ such that $p$ divides $k$.

For every natural number $n$, there exists a prime number $p$ such that $n < p \leq n! + 1$.

There exists a prime number greater than $n$.

There are infinitely many primes.

If $p$ and $q$ are natural numbers such that $pq$ is prime, then $p = 1$ or $q = 1$.

If $p$ is a prime number and $x$ and $y$ are natural numbers such that $x y = p^k$, then there exist natural numbers $i$ and $j$ such that $x = p^i$ and $y = p^j$.

If $p$ is a prime number, $n$ is a positive integer, and $x^n = p^k$, then $x = p^i$ for some integer $i$.

If $p$ is a prime number, then $d$ divides $p^k$ if and only if $d = p^i$ for some $i \leq k$.

For any two natural numbers $a$ and $b$, there exist two natural numbers $x$ and $y$ such that $a x - b y = \gcd(a, b)$ or $b x - a y = \gcd(a, b)$.

If $a$ and $b$ are relatively prime, then $a$ and $b$ divide $a x + b y$ for any $x$ and $y$.

If $a$ and $b$ are coprime, then there exist integers $q_1$ and $q_2$ such that $u + q_1 a = v + q_2 b$.

If $a$ and $b$ are coprime and $b \neq 1$, then there exist integers $x$ and $y$ such that $a x = b y + 1$.

If $p$ is a prime and $p$ does not divide $a$, then there exist integers $x$ and $y$ such that $a x = 1 + p y$.

If $p$ is a prime factor of $x$, then $p > 0$.

If $p$ is a prime factor of $n$, then $p \geq 0$.

If $n$ is a positive integer, then the product of the prime factors of $n$ is $n$.

Every positive integer $n$ can be written as a product of primes.

The product of the prime factors of a positive integer is the integer itself.

Every positive integer $n$ can be written as a product of prime powers.

If $n$ is a natural number and $S$ is the set of prime factors of $n$, then the function $f$ defined by $f(p) = \text{multiplicity}(p, n)$ is the unique function such that $n = \prod_{p \in S} p^{f(p)}$.

If $n$ is a positive integer and $S$ is a finite set of primes such that $n = \prod_{p \in S} p^{f(p)}$, then $S$ is the set of prime factors of $n$ and $f(p)$ is the multiplicity of $p$ in $n$.

If $S$ is a finite set of primes, and $n$ is the product of the $p$th powers of the primes in $S$, then $S$ is the set of prime factors of $n$.

If $f$ is a function from the set of primes to the set of positive integers such that the set of primes $p$ with $f(p) > 0$ is finite, then the set of prime factors of $\prod_{p \in \mathbb{P}} p^{f(p)}$ is exactly the set of primes $p$ with $f(p) > 0$.

If $n$ is a positive integer and $S$ is a finite set of primes such that $n = \prod_{p \in S} p^{f(p)}$, then $S$ is the set of prime factors of $n$.

The absolute value of the product of a set of numbers is equal to the product of the absolute values of the numbers.

If $f$ is a function from the positive integers to the positive integers such that the set of primes $p$ such that $f(p) > 0$ is finite, then the prime factors of $\prod_{p \geq 0, f(p) > 0} p^{f(p)}$ are exactly the primes $p$ such that $f(p) > 0$.

If $S$ is the set of primes $p$ such that $f(p) > 0$, and $n$ is the product of the $p^{f(p)}$ for $p \in S$, then the multiplicity of $p$ in $n$ is $f(p)$.

If $f$ is a function from the natural numbers to the natural numbers such that $f(p) > 0$ for only finitely many primes $p$, and if $f(p) > 0$ implies that $p$ is prime, then the multiplicity of $p$ in the product $\prod_{p \in \mathbb{N}} p^{f(p)}$ is $f(p)$.

If $S$ is a finite set of primes, $n$ is the product of the $p$-th powers of the primes in $S$, and $p$ is a prime in $S$, then the multiplicity of $p$ in $n$ is the power of $p$ in the product.