# Integers

The default integer type in Nemo is provided by Flint. The associated ring of integers is represented by the constant parent object called FlintZZ.

For convenience we define

ZZ = FlintZZ

so that integers can be constructed using ZZ instead of FlintZZ. Note that this is the name of a specific parent object, not the name of its type.

The types of the integer ring parent objects and elements of the associated rings of integers are given in the following table according to the library providing them.

LibraryElement typeParent type
FlintZZRingElemZZRing

All integer element types belong directly to the abstract type RingElem and all the integer ring parent object types belong to the abstract type Ring.

A lot of code will want to accept both ZZRingElem integers and Julia integers, that is, subtypes of Base.Integer. Thus for convenience we define

IntegerUnion = Union{Integer,ZZRingElem}

## Integer functionality

Nemo integers provide all of the ring and Euclidean ring functionality of AbstractAlgebra.jl.

https://nemocas.github.io/AbstractAlgebra.jl/stable/ring

https://nemocas.github.io/AbstractAlgebra.jl/stable/euclidean_interface

Below, we describe the functionality that is specific to the Nemo/Flint integer ring.

### Constructors

ZZ(n::Integer)

Coerce a Julia integer value into the integer ring.

ZZ(n::String)

Parse the given string as an integer.

ZZ(n::Float64)
ZZ(n::Float32)
ZZ(n::Float16)
ZZ(n::BigFloat)

Coerce the given floating point number into the integer ring, assuming that it can be exactly represented as an integer.

### Basic manipulation

signMethod
sign(a::ZZRingElem)

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

sizeMethod
size(a::ZZRingElem)

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

fitsMethod
fits(::Type{UInt}, a::ZZRingElem)

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

fitsMethod
fits(::Type{Int}, a::ZZRingElem)

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

denominatorMethod
denominator(a::ZZRingElem)

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

numeratorMethod
numerator(a::ZZRingElem)

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

Examples

julia> a = ZZ(12)
12

julia> is_unit(a)
false

julia> sign(a)
1

julia> s = size(a)
1

julia> fits(Int, a)
true

julia> n = numerator(a)
12

julia> d = denominator(a)
1

### Euclidean division

Nemo also provides a large number of Euclidean division operations. Recall that for a dividend $a$ and divisor $b$, we can write $a = bq + r$ with $0 \leq |r| < |b|$. We call $q$ the quotient and $r$ the remainder.

We distinguish three cases. If $q$ is rounded towards zero, $r$ will have the same sign as $a$. If $q$ is rounded towards plus infinity, $r$ will have the opposite sign to $b$. Finally, if $q$ is rounded towards minus infinity, $r$ will have the same sign as $b$.

In the following table we list the division functions and their rounding behaviour. We also give the return value of the function, with $q$ representing return of the quotient and $r$ representing return of the remainder.

FunctionReturnRounding of the quotient
modrtowards minus infinity
remrtowards zero
divqtowards minus infinity
divrem(a::ZZRingElem, b::ZZRingElem)q, rtowards minus infinity
tdivrem(a::ZZRingElem, b::ZZRingElem)q, rtowards zero
fdivrem(a::ZZRingElem, b::ZZRingElem)q, rtowards minus infinity
cdivrem(a::ZZRingElem, b::ZZRingElem)q, rtowards plus infinity
ntdivrem(a::ZZRingElem, b::ZZRingElem)q, rnearest integer, ties toward zero
nfdivrem(a::ZZRingElem, b::ZZRingElem)q, rnearest integer, ties toward minus infinity
ncdivrem(a::ZZRingElem, b::ZZRingElem)q, rnearest integer, ties toward plus infinity

N.B: the internal definition of Nemo.div and Nemo.divrem are the same as fdiv and fdivrem. The definitions in the table are of Base.div and Base.divrem which agree with Julia's definitions of div and divrem.

Nemo also offers the following ad hoc division operators. The notation and description is as for the other Euclidean division functions.

FunctionReturnRounding
mod(a::ZZRingElem, b::Int)rtowards minus infinity
rem(a::ZZRingElem, b::Int)rtowards zero
div(a::ZZRingElem, b::Int)qtowards zero
tdiv(a::ZZRingElem, b::Int)qtowards zero
fdiv(a::ZZRingElem, b::Int)qtowards minus infinity
cdiv(a::ZZRingElem, b::Int)qtowards plus infinity

N.B: the internal definition of Nemo.div is the same as fdiv. The definition in the table is Base.div which agrees with Julia's definition of div.

The following functions are also available, for the case where one is dividing by a power of $2$. In other words, for Euclidean division of the form $a = b2^{d} + r$. These are useful for bit twiddling.

FunctionReturnRounding
tdivpow2(a::ZZRingElem, d::Int)qtowards zero
fdivpow2(a::ZZRingElem, d::Int)qtowards minus infinity
fmodpow2(a::ZZRingElem, d::Int)rtowards minus infinity
cdivpow2(a::ZZRingElem, d::Int)qtowards plus infinity

Examples

julia> a = ZZ(12)
12

julia> b = ZZ(5)
5

julia> q, r = divrem(a, b)
(2, 2)

julia> c = cdiv(a, b)
3

julia> d = fdiv(a, b)
2

julia> f = tdivpow2(a, 2)
3

julia> g = fmodpow2(a, 3)
4

### Comparison

Instead of isless we implement a function cmp(a, b) which returns a positive value if $a > b$, zero if $a == b$ and a negative value if $a < b$. We then implement all the other operators, including == in terms of cmp.

For convenience we also implement a cmpabs(a, b) function which returns a positive value if $|a| > |b|$, zero if $|a| == |b|$ and a negative value if $|a| < |b|$. This can be slightly faster than a call to cmp or one of the comparison operators when comparing nonnegative values for example.

Here is a list of the comparison functions implemented, with the understanding that cmp provides all of the comparison operators listed above.

Function
cmp(a::ZZRingElem, b::ZZRingElem)
cmpabs(a::ZZRingElem, b::ZZRingElem)

We also provide the following ad hoc comparisons which again provide all of the comparison operators mentioned above.

Function
cmp(a::ZZRingElem, b::Int)
cmp(a::Int, b::ZZRingElem)
cmp(a::ZZRingElem, b::UInt)
cmp(a::UInt, b::ZZRingElem)

Examples

julia> a = ZZ(12)
12

julia> b = ZZ(3)
3

julia> a < b
false

julia> a != b
true

julia> a > 4
true

julia> 5 <= b
false

julia> cmpabs(a, b)
1

### Shifting

<<Method
<<(x::ZZRingElem, c::Int)

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

>>Method
>>(x::ZZRingElem, c::Int)

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

Examples

julia> a = ZZ(12)
12

julia> a << 3
96

julia> a >> 5
0

### Modular arithmetic

sqrtmodMethod
sqrtmod(x::ZZRingElem, m::ZZRingElem)

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.

Examples

julia> sqrtmod(ZZ(12), ZZ(13))
5
crtFunction
crt(r1::ZZRingElem, m1::ZZRingElem, r2::ZZRingElem, m2::ZZRingElem, signed=false; check::Bool=true)
crt(r1::ZZRingElem, m1::ZZRingElem, r2::Union{Int, UInt}, m2::Union{Int, UInt}, signed=false; check::Bool=true)
crt(r::Vector{ZZRingElem}, m::Vector{ZZRingElem}, signed=false; check::Bool=true)
crt_with_lcm(r1::ZZRingElem, m1::ZZRingElem, r2::ZZRingElem, m2::ZZRingElem, signed=false; check::Bool=true)
crt_with_lcm(r1::ZZRingElem, m1::ZZRingElem, r2::Union{Int, UInt}, m2::Union{Int, UInt}, signed=false; check::Bool=true)
crt_with_lcm(r::Vector{ZZRingElem}, m::Vector{ZZRingElem}, signed=false; check::Bool=true)

As per the AbstractAlgebra crt interface, with the following option. If signed = true, the solution is the range $(-m/2, m/2]$, otherwise it is in the range $[0,m)$, where $m$ is the least common multiple of the moduli.

Examples

julia> crt(ZZ(5), ZZ(13), ZZ(7), ZZ(37), true)
44

julia> crt(ZZ(5), ZZ(13), 7, 37, true)
44

### Integer logarithm

flogMethod
flog(x::ZZRingElem, c::ZZRingElem)
flog(x::ZZRingElem, c::Int)

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

Examples

julia> flog(ZZ(12), ZZ(2))
3

julia> flog(ZZ(12), 3)
2

clogMethod
clog(x::ZZRingElem, c::ZZRingElem)
clog(x::ZZRingElem, c::Int)

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

Examples

julia> clog(ZZ(12), ZZ(2))
4

julia> clog(ZZ(12), 3)
3


### Integer roots

isqrtMethod
isqrt(x::ZZRingElem)

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

Examples

julia> isqrt(ZZ(13))
3

isqrtremMethod
isqrtrem(x::ZZRingElem)

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 $x \geq 0$.

Examples

julia> isqrtrem(ZZ(13))
(3, 4)

rootMethod
root(x::ZZRingElem, 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.

Examples

julia> root(ZZ(27), 3; check=true)
3
irootMethod
iroot(x::ZZRingElem, 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.

Examples

julia> iroot(ZZ(13), 3)
2

### Number theoretic functionality

divisibleMethod
divisible(x::ZZRingElem, y::Int)

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

divisibleMethod
divisible(x::ZZRingElem, y::ZZRingElem)

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

is_squareMethod
is_square(f::PolyRingElem{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_primeMethod
is_prime(x::ZZRingElem)
is_prime(x::Int)

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

Examples

julia> is_prime(ZZ(13))
true
is_probable_primeMethod
is_probable_prime(x::ZZRingElem)

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.

factorMethod
factor(a::ZZRingElem)
factor(a::UInt)
factor(a::Int)

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

Examples

julia> factor(ZZ(12))
1 * 2^2 * 3

julia> factor(UInt(12))
1 * 2^2 * 3

julia> factor(12)
1 * 2^2 * 3

divisor_lenstraMethod
divisor_lenstra(n::ZZRingElem, r::ZZRingElem, m::ZZRingElem)

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.

factorialMethod
factorial(x::ZZRingElem)

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

Examples

julia> factorial(ZZ(100))
93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000
rising_factorialMethod
rising_factorial(x::ZZRingElem, n::ZZRingElem)

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_factorialMethod
rising_factorial(x::ZZRingElem, 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_factorialMethod
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().

primorialMethod
primorial(x::ZZRingElem)

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().

primorialMethod
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().

fibonacciMethod
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 $i$.

fibonacciMethod
fibonacci(x::ZZRingElem)

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 $i$.

bellMethod
bell(x::ZZRingElem)

Return the Bell number $B_x$.

bellMethod
bell(x::Int)

Return the Bell number $B_x$.

binomialMethod
binomial(n::ZZRingElem, k::ZZRingElem)

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.

binomialMethod
binomial(n::UInt, k::UInt, ::ZZRing)

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

moebius_muMethod
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_muMethod
moebius_mu(x::ZZRingElem)

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_symbolMethod
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_symbolMethod
jacobi_symbol(x::ZZRingElem, y::ZZRingElem)

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_symbolMethod
kronecker_symbol(x::ZZRingElem, y::ZZRingElem)
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_sigmaMethod
divisor_sigma(x::ZZRingElem, y::Int)
divisor_sigma(x::ZZRingElem, y::ZZRingElem)
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().

Examples

julia> divisor_sigma(ZZ(32), 10)
1127000493261825

julia> divisor_sigma(ZZ(32), ZZ(10))
1127000493261825

julia> divisor_sigma(32, 10)
1127000493261825
euler_phiMethod
euler_phi(x::ZZRingElem)
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 $x \leq 0$.

Examples

julia> euler_phi(ZZ(12480))
3072

julia> euler_phi(12480)
3072
number_of_partitionsMethod
number_of_partitions(x::Int)
number_of_partitions(x::ZZRingElem)

Return the number of partitions of $x$.

Examples

julia> number_of_partitions(100)
190569292

julia> number_of_partitions(ZZ(1000))
24061467864032622473692149727991
is_perfect_powerMethod
is_perfect_power(a::IntegerUnion)

Returns whether $a$ is a perfect power, that is, whether $a = m^r$ for some integer $m$ and $r > 1$.

is_prime_powerMethod
is_prime_power(q::IntegerUnion) -> Bool

Returns whether $q$ is a prime power.

is_prime_power_with_dataMethod
is_prime_power_with_data(q::IntegerUnion) -> Bool, ZZRingElem, Int

Returns a flag indicating whether $q$ is a prime power and integers $p, e$ such that $q = p^e$. If $q$ is a prime power, than $p$ is a prime.

### Digits and bases

binMethod
bin(n::ZZRingElem)

Return $n$ as a binary string.

Examples

julia> bin(ZZ(12))
"1100"
octMethod
oct(n::ZZRingElem)

Return $n$ as a octal string.

Examples

julia> oct(ZZ(12))
"14"
decMethod
dec(n::ZZRingElem)

Return $n$ as a decimal string.

Examples

julia> dec(ZZ(12))
"12"
hexMethod
hex(n::ZZRingElem) = base(n, 16)

Return $n$ as a hexadecimal string.

Examples

julia> hex(ZZ(12))
"c"
baseMethod
base(n::ZZRingElem, b::Integer)

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

Examples

julia> base(ZZ(12), 13)
"c"
ndigitsMethod
ndigits(x::ZZRingElem, b::Integer)

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

Examples

julia> ndigits(ZZ(12), 3)
3
nbitsMethod
nbits(x::ZZRingElem)

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

Examples

julia> nbits(ZZ(12))
4

### Bit twiddling

popcountMethod
popcount(x::ZZRingElem)

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

Examples

julia> popcount(ZZ(12))
2
prevpow2Method
prevpow2(x::ZZRingElem)

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

nextpow2Method
nextpow2(x::ZZRingElem)

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

Examples

julia> nextpow2(ZZ(12))
16
trailing_zerosMethod
trailing_zeros(x::ZZRingElem)

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

clrbit!Method
clrbit!(x::ZZRingElem, 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.

Examples

julia> a = ZZ(12)
12

julia> clrbit!(a, 3)

julia> a
4
setbit!Method
setbit!(x::ZZRingElem, 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.

Examples

julia> a = ZZ(12)
12

julia> setbit!(a, 0)

julia> a
13
combit!Method
combit!(x::ZZRingElem, 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.

Examples

julia> a = ZZ(12)
12

julia> combit!(a, 2)

julia> a
8
tstbitMethod
tstbit(x::ZZRingElem, c::Int)

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

Examples

julia> a = ZZ(12)
12

julia> tstbit(a, 0)
false

julia> tstbit(a, 2)
true

### Random generation

rand_bitsMethod
rand_bits(::ZZRing, b::Int)

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

rand_bits_primeMethod
rand_bits_prime(::ZZRing, 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.

Examples

a = rand_bits(ZZ, 23)
b = rand_bits_prime(ZZ, 7)

# Complex Integers

The Gaussian integer type in Nemo is provided by a pair of Flint integers. The associated ring of integers and the fraction field can be retrieved by Nemo.GaussianIntegers() and Nemo.GaussianRationals().

Examples

julia> ZZi = Nemo.GaussianIntegers()
ZZ[im]

julia> a = ZZ(5)*im
5*im

julia> b = ZZi(3, 4)
3 + 4*im

julia> is_unit(a)
false

julia> factor(a)
im * (2 - im) * (2 + im)

julia> a//b
4//5 + 3//5*im

julia> abs2(a//b)
1