sign(a::fmpz)

Return the sign of $a$, i.e. $+1$, $0$ or $-1$.

size(a::fmpz)

Return the number of limbs required to store the absolute value of $a$.

fits(::Type{UInt}, a::fmpz)

Return true if $a$ fits into a UInt, otherwise return false.

fits(::Type{Int}, a::fmpz)

Return true if $a$ fits into an Int, otherwise return false.

denominator(a::fmpz)

Return the denominator of $a$ thought of as a rational. Always returns $1$.

numerator(a::fmpz)

Return the numerator of $a$ thought of as a rational. Always returns $a$.

<<(x::fmpz, c::Int)

Return $2^cx$ where $c \geq 0$.

>>(x::fmpz, c::Int)

Return $x/2^c$, discarding any remainder, where $c \geq 0$.

sqrtmod(x::fmpz, m::fmpz)

Return a square root of $x (\mod m)$ if one exists. The remainder will be in the range $[0, m)$. We require that $m$ is prime, otherwise the algorithm may not terminate.

crt(r1::fmpz, m1::fmpz, r2::fmpz, m2::fmpz, signed=false)

Return $r$ such that $r \equiv r_1 (\mod m_1)$ and $r \equiv r_2 (\mod m_2)$. If signed = true, $r$ will be in the range $-m_1m_2/2 < r \leq m_1m_2/2$. If signed = false the value will be in the range $0 \leq r < m_1m_2$.

crt(r1::fmpz, m1::fmpz, r2::Int, m2::Int, signed=false)

Return $r$ such that $r \equiv r_1 (\mod m_1)$ and $r \equiv r_2 (\mod m_2)$. If signed = true, $r$ will be in the range $-m_1m_2/2 < r \leq m_1m_2/2$. If signed = false the value will be in the range $0 \leq r < m_1m_2$.

flog(x::fmpz, c::fmpz)

Return the floor of the logarithm of $x$ to base $c$.

flog(x::fmpz, c::Int)

Return the floor of the logarithm of $x$ to base $c$.

clog(x::fmpz, c::fmpz)

Return the ceiling of the logarithm of $x$ to base $c$.

clog(x::fmpz, c::Int)

Return the ceiling of the logarithm of $x$ to base $c$.

isqrt(x::fmpz)

Return the floor of the square root of $x$.

isqrtrem(x::fmpz)

Return a tuple $s, r$ consisting of the floor $s$ of the square root of $x$ and the remainder $r$, i.e. such that $x = s^2 + r$. We require $x \geq 0$.

root(x::fmpz, n::Int; check::Bool=true)

Return the $n$-the root of $x$. We require $n > 0$ and that $x \geq 0$ if $n$ is even. By default the function tests whether the input was a perfect $n$-th power and if not raises an exception. If check=false this check is omitted.

iroot(x::fmpz, n::Int)

Return the integer truncation of the $n$-the root of $x$ (round towards zero). We require $n > 0$ and that $x \geq 0$ if $n$ is even.

divisible(x::fmpz, y::Int)

Return true if $x$ is divisible by $y$, otherwise return false. We require $x \neq 0$.

divisible(x::fmpz, y::fmpz)

Return true if $x$ is divisible by $y$, otherwise return false. We require $x \neq 0$.

is_square(f::PolyElem{T}) where T <: RingElement

Return true if $f$ is a perfect square.

is_square(a::FracElem{T}) where T <: RingElem

Return true if $a$ is a square.

is_prime(x::fmpz)

Return true if $x$ is a prime number, otherwise return false.

is_probable_prime(x::fmpz)

Return true if $x$ is very probably a prime number, otherwise return false. No counterexamples are known to this test, but it is conjectured that infinitely many exist.

factor(a::fmpz)
factor(a::UInt)
factor(a::Int)

Return a factorisation of $a$ using a Fac struct (see the documentation on factorisation in Nemo).

divisor_lenstra(n::fmpz, r::fmpz, m::fmpz)

If $n$ has a factor which lies in the residue class $r (\mod m)$ for $0 < r < m < n$, this function returns such a factor. Otherwise it returns $0$. This is only efficient if $m$ is at least the cube root of $n$. We require gcd$(r, m) = 1$ and this condition is not checked.

factorial(x::fmpz)

Return the factorial of $x$, i.e. $x! = 1.2.3\ldots x$. We require $x \geq 0$.

rising_factorial(x::fmpz, n::fmpz)

Return the rising factorial of $x$, i.e. $x(x + 1)(x + 2)\cdots (x + n - 1)$. If $n < 0$ we throw a DomainError().

rising_factorial(x::fmpz, n::Int)

Return the rising factorial of $x$, i.e. $x(x + 1)(x + 2)\ldots (x + n - 1)$. If $n < 0$ we throw a DomainError().

rising_factorial(x::Int, n::Int)

Return the rising factorial of $x$, i.e. $x(x + 1)(x + 2)\ldots (x + n - 1)$. If $n < 0$ we throw a DomainError().

primorial(x::fmpz)

Return the primorial of $x$, i.e. the product of all primes less than or equal to $x$. If $x < 0$ we throw a DomainError().

primorial(x::Int)

Return the primorial of $x$, i.e. the product of all primes less than or equal to $x$. If $x < 0$ we throw a DomainError().

fibonacci(x::Int)

Return the $x$-th Fibonacci number $F_x$. We define $F_1 = 1$, $F_2 = 1$ and $F_{i + 1} = F_i + F_{i - 1}$ for all integers $i$.

fibonacci(x::fmpz)

Return the $x$-th Fibonacci number $F_x$. We define $F_1 = 1$, $F_2 = 1$ and $F_{i + 1} = F_i + F_{i - 1}$ for all integers $i$.

bell(x::fmpz)

Return the Bell number $B_x$.

bell(x::Int)

Return the Bell number $B_x$.

binomial(n::fmpz, k::fmpz)

Return the binomial coefficient $\frac{n (n-1) \cdots (n-k+1)}{k!}$. If $k < 0$ we return $0$, and the identity binomial(n, k) == binomial(n - 1, k - 1) + binomial(n - 1, k) always holds for integers n and k.

binomial(n::UInt, k::UInt, ::FlintIntegerRing)

Return the binomial coefficient $\frac{n!}{(n - k)!k!}$ as an fmpz.

moebius_mu(x::Int)

Return the Moebius mu function of $x$ as an Int. The value returned is either $-1$, $0$ or $1$. If $x \leq 0$ we throw a DomainError().

moebius_mu(x::fmpz)

Return the Moebius mu function of $x$ as an Int. The value returned is either $-1$, $0$ or $1$. If $x \leq 0$ we throw a DomainError().

jacobi_symbol(x::Int, y::Int)

Return the value of the Jacobi symbol $\left(\frac{x}{y}\right)$. The modulus $y$ must be odd and positive, otherwise a DomainError is thrown.

jacobi_symbol(x::fmpz, y::fmpz)

Return the value of the Jacobi symbol $\left(\frac{x}{y}\right)$. The modulus $y$ must be odd and positive, otherwise a DomainError is thrown.

kronecker_symbol(x::fmpz, y::fmpz)
kronecker_symbol(x::Int, y::Int)

Return the value of the Kronecker symbol $\left(\frac{x}{y}\right)$. The definition is as per Henri Cohen's book, "A Course in Computational Algebraic Number Theory", Definition 1.4.8.

divisor_sigma(x::Int, y::Int)

Return the value of the sigma function, i.e. $\sum_{0 < d \;| x} d^y$. If $x \leq 0$ or $y < 0$ we throw a DomainError().

divisor_sigma(x::fmpz, y::Int)

Return the value of the sigma function, i.e. $\sum_{0 < d \;| x} d^y$. If $x \leq 0$ or $y < 0$ we throw a DomainError().

divisor_sigma(x::fmpz, y::fmpz)

Return the value of the sigma function, i.e. $\sum_{0 < d \;| x} d^y$. If $x \leq 0$ or $y < 0$ we throw a DomainError().

euler_phi(x::Int)

Return the value of the Euler phi function at $x$, i.e. the number of positive integers up to $x$ (inclusive) that are coprime with $x$. An exception is raised if $x \leq 0$.

euler_phi(x::fmpz)

Return the value of the Euler phi function at $x$, i.e. the number of positive integers up to $x$ (inclusive) that are coprime with $x$. An exception is raised if $x \leq 0$.

number_of_partitions(x::Int)

Return the number of partitions of $x$. This function is not available on Windows 64.

number_of_partitions(x::fmpz)

Return the number of partitions of $x$. This function is not available on Windows 64.

bin(n::fmpz)

Return $n$ as a binary string.

oct(n::fmpz)

Return $n$ as a octal string.

dec(n::fmpz)

Return $n$ as a decimal string.

hex(n::fmpz) = base(n, 16)

Return $n$ as a hexadecimal string.

base(n::fmpz, b::Integer)

Return $n$ as a string in base $b$. We require $2 \leq b \leq 62$.

ndigits(x::fmpz, b::Integer)

Return the number of digits of $x$ in the base $b$ (default is $b = 10$).

nbits(x::fmpz)

Return the number of binary bits of $x$. We return zero if $x = 0$.

popcount(x::fmpz)

Return the number of ones in the binary representation of $x$.

prevpow2(x::fmpz)

Return the previous power of $2$ up to including $x$.

nextpow2(x::fmpz)

Return the next power of $2$ that is at least $x$.

trailing_zeros(x::fmpz)

Return the number of trailing zeros in the binary representation of $x$.

clrbit!(x::fmpz, c::Int)

Clear bit $c$ of $x$, where the least significant bit is the $0$-th bit. Note that this function modifies its input in-place.

setbit!(x::fmpz, c::Int)

Set bit $c$ of $x$, where the least significant bit is the $0$-th bit. Note that this function modifies its input in-place.

combit!(x::fmpz, c::Int)

Complement bit $c$ of $x$, where the least significant bit is the $0$-th bit. Note that this function modifies its input in-place.

tstbit(x::fmpz, c::Int)

Return bit $i$ of x (numbered from 0) as true for 1 or false for 0.

rand_bits(::FlintIntegerRing, b::Int)

Return a random signed integer whose absolute value has $b$ bits.

rand_bits_prime(::FlintIntegerRing, n::Int, proved::Bool=true)

Return a random prime number with the given number of bits. If only a probable prime is required, one can pass proved=false.