# mpz data type¶

The mpz data type can store integer numbers of arbitrary size, up to the maximum memory allowed by PostgreSQL data types (about 1GB).

Every PostgreSQL integer type (int16, int32, int64) can be converted to mpz. Not integer types (float4, float8, numeric) are truncated. Cast from integer types are automatic, whereas non integer require explicit cast (but are implicitly converted in assignment).

SELECT mpz(10000);      -- Cast using mpz as a function
SELECT 10000::mpz;      -- PostgreSQL-style cast


Casting mpz to PostgreSQL integer types and numeric works as expected and will overflow if the value is too big for their range. Casting to float4 and float8 works as well: in case of overflow the value will be Infinity.

mpz values can be compared using the regular PostgreSQL comparison operators. Indexes on mpz columns can be created using the btree or the hash method.

## mpz textual input/output¶

mpz(text)
mpz(text, base)

Convert a textual representation into an mpz number. The form text::mpz is equivalent to mpz(text).

White space is allowed in the string, and is simply ignored.

The base may vary from 2 to 62, or if base is 0 or omitted, then the leading characters are used: 0x and 0X for hexadecimal, 0b and 0B for binary, 0 for octal, or decimal otherwise.

For bases up to 36, case is ignored; upper-case and lower-case letters have the same value. For bases 37 to 62, upper-case letter represent the usual 10..35 while lower-case letter represent 36..61.

=# SELECT '0x10'::mpz AS "hex", '10'::mpz AS "dec",
-#        '010'::mpz AS "oct", '0b10'::mpz AS "bin";
hex | dec | oct | bin
-----+-----+-----+-----
16  | 10  | 8   | 2


Note

The maximum base accepted by GMP 4.1 is 36, not 62.

text(z)
text(z, base)

Convert the mpz z into a string. The form z::text is equivalent to text(z).

base may vary from 2 to 62 or from −2 to −36. For base in the range 2..36, digits and lower-case letters are used; for −2..−36, digits and upper-case letters are used; for 37..62, digits, upper-case letters, and lower-case letters (in that significance order) are used. If base is not specified, 10 is assumed.

Note

The maximum base accepted by GMP 4.1 is 36, not 62.

## Arithmetic Operators and Functions¶

These operators can either work on mpz arguments or take an integer argument that will be implicitly converted. Operators taking a $$2^n$$ argument always use an integer as right argument.

Note

GMP defines many structures in terms of long or unsigned long, whose definitions may vary across platforms. PostgreSQL instead offers data types with a defined number of bytes (e.g. int4, int8). For this reason, functions taking an integer as argument are defined as int8, but they may actually fail if the server is 32 bit and the argument doesn’t fit into an int4.

Arithmetic operators
Operator Description Example Return
- Unary minus - 5::mpz -5
+ Unary plus + 5::mpz 5
+ Addition 2::mpz + 3::mpz 5
- Subtraction 2::mpz - 3::mpz -1
* Multiplication 7::mpz * 3::mpz 21
<< Multiplication by $$2^n$$ 3::mpz << 2 12
^ Power (1) 3::mpz ^ 2 9

Notes:

abs(z)

Return the absolute value of z.

sgn(z)

Return +1 if z > 0, 0 if z = 0, and -1 if z < 0.

odd(z)
even(z)

Return true if z is odd or even, respectively, else false.

## Division Operators and Functions¶

For all the division-related operators $$n \oslash d$$, $$q$$ and $$r$$ will satisfy $$n = q \cdot d + r$$, and $$r$$ will satisfy $$0 \le |r| \lt |d|$$.

Note

Only the truncating division and reminder (/ and %) have the correct precedence respect to addition, subtraction and multiplication. See the PostgreSQL precedence table for further details.

Division operators
Operator Description Example Return
/

Division quotient

Rounding towards zero

7::mpz / 3::mpz

-7::mpz / 3::mpz

2

-2

%

Division reminder

Rounding towards zero

7::mpz % 3::mpz

-7::mpz % 3::mpz

1

-1

+/

Division quotient

Rounding towards +infinity

7::mpz +/ 3::mpz

-7::mpz +/ 3::mpz

3

-2

+%

Division reminder

Rounding towards +infinity

7::mpz +% 3::mpz

-7::mpz +% 3::mpz

-2

-1

-/

Division quotient

Rounding towards -infinity

7::mpz -/ 3::mpz

-7::mpz -/ 3::mpz

2

-3

-%

Division reminder

Rounding towards -infinity

7::mpz -% 3::mpz

-7::mpz -% 3::mpz

1

2

/? Divisible (1) 21::mpz /? 7::mpz true
/! Exact division (2) 21::mpz /! 7::mpz 3

Notes:

2. The exact division operator (/!) produces correct results only when it is known in advance that $$d$$ divides $$n$$. The operator is much faster than the other division operators, and is the best choice when exact division is known to occur, for example reducing a rational to lowest terms.
Division operators for powers of 2
Operator Description Example Return
>>

Quotient of division by $$2^n$$

Rounding towards zero

1027::mpz >> 3

-1027::mpz >> 3

128

-128

%>

Remainder of division by $$2^n$$

Rounding towards zero

1027::mpz %> 3

-1027::mpz %> 3

3

-3

+>>

Quotient of division by $$2^n$$

Rounding towards +infinity

1027::mpz +>> 3

-1027::mpz +>> 3

129

-128

+%>

Remainder of division by $$2^n$$

Rounding towards +infinity

1027::mpz +%> 3

-1027::mpz +%> 3

-5

-3

->>

Quotient of division by $$2^n$$

Rounding towards -infinity

1027::mpz ->> 3

-1027::mpz ->> 3

128

-129

-%>

Remainder of division by $$2^n$$

Rounding towards -infinity

1027::mpz -%> 3

-1027::mpz -%> 3

3

5

>>? Divisible by $$2^n$$ (1) 64::mpz >>? 3 true
tdiv_qr(n, d)

Return a tuple containing quotient q and remainder r of the division, rounding towards 0.

cdiv_qr(n, d)

Return a tuple containing quotient q and remainder r of the division, rounding towards +infinity (ceil).

fdiv_qr(n, d)

Return a tuple containing quotient q and remainder r of the division, rounding towards -infinity (floor).

divisible(n, d)
divisible_2exp(n, b)

Return true if n is exactly divisible by d, or in the case of divisible_2exp() by $$2^b$$.

$$n$$ is divisible by $$d$$ if there exists an integer $$q$$ satisfying $$n = q \cdot d$$. Unlike the other division operators, d=0 is accepted and following the rule it can be seen that only 0 is considered divisible by 0.

The operators /? and >>? are aliases for divisible() and divisible_2exp().

congruent(n, c, d)
congruent_2exp(n, c, b)

Return true if n is congruent to c modulo d, or in the case of congruent_2exp() modulo $$2^b$$.

$$n$$ is congruent to $$c \mod d$$ if there exists an integer $$q$$ satisfying $$n = c + q \cdot d$$. Unlike the other division operators, d=0 is accepted and following the rule it can be seen that n and c are considered congruent mod 0 only when exactly equal.

## Exponentiation Functions¶

pow(base, exp)

Return base raised to exp.

exp is defined as int8 but must fit into a long as defined on the server.

The function is an alias for the ^ operator.

powm(base, exp, mod)

Return (base raised to exp) modulo mod.

Negative exp is supported if an inverse base^-1 mod mod exists (see invert() function). If an inverse doesn’t exist then a divide by zero is raised.

## Root Extraction Functions¶

root(op, n)

Return the truncated integer part of the nth root of op.

n is defined as int8 but must fit into a long as defined on the server.

rootrem(op, n)

Return a tuple of 2 elements with the truncated integer part of the nth root of op and the remainder (i.e. op - root ^ n).

=# select * from rootrem(28, 3);
root | rem
------+-----
3    | 1


Note

The function is not available on GMP version < 4.2.

sqrt(op)

Return the truncated integer part of the square root of op.

sqrtrem(op)

Return a tuple of 2 elements with the truncated integer part of the square root of op and the remainder (i.e. op - root * root).

=# select * from sqrtrem(83);
root | rem
------+-----
9    | 2

perfect_power(op)

Return true if op is a perfect power, i.e., if there exist integers $$a$$ and $$b$$, with $$b>1$$, such that op equals $$a^b$$.

Under this definition both 0 and 1 are considered to be perfect powers. Negative values of op are accepted, but of course can only be odd perfect powers.

perfect_square(op)

Return true if op is a perfect square, i.e., if the square root of op is an integer. Under this definition both 0 and 1 are considered to be perfect squares.

## Number Theoretic Functions¶

probab_prime(n, reps)

Determine whether n is prime. Return 2 if n is definitely prime, return 1 if n is probably prime (without being certain), or return 0 if n is definitely composite.

This function does some trial divisions, then some Miller-Rabin probabilistic primality tests. reps controls how many such tests are done, 5 to 10 is a reasonable number, more will reduce the chances of a composite being returned as “probably prime”.

Miller-Rabin and similar tests can be more properly called compositeness tests. Numbers which fail are known to be composite but those which pass might be prime or might be composite. Only a few composites pass, hence those which pass are considered probably prime.

Primality test

nextprime(op)

Return the next prime greater than op.

This function uses a probabilistic algorithm to identify primes. For practical purposes it’s adequate, the chance of a composite passing will be extremely small.

gcd(a, b)

Return the greatest common divisor of a and b. The result is always positive even if one or both input operands are negative.

gcdext(a, b)

Return the greatest common divisor g of a and b, and in addition coefficients s and t satisfying $$a \cdot s + b \cdot t = g$$. The value g is always positive, even if one or both of a and b are negative. The values s and t are chosen such that $$|s| \le |b| \hspace{0em}$$ and $$|t| \le |a| \hspace{0em}$$.

=# select * from  gcdext(6, 15);
g | s  | t
---+----+---
3 | -2 | 1

lcm(a, b)

Return the least common multiple of a and b. The value returned is always positive, irrespective of the signs of a and b. The return will be zero if either a or b is zero.

fac(op)

Return op!, the factorial of op.

bin(n, k)

Return the binomial coefficient $${n \choose k}$$. Negative values of n are supported, using the identity $${-n \choose k} = (-1)^k {n+k-1 \choose k}$$.

fib(n)
fib2(n)

fib() returns $$F_n$$, the nth Fibonacci number. fib2() returns $$F_n$$ and $$F_{n-1}$$.

These functions are designed for calculating isolated Fibonacci numbers. When a sequence of values is wanted it’s best to start with fib2() and iterate the defining $$F_{n+1}=F_n+F_{n-1}$$ or similar.

lucnum(n)
lucnum2(n)

lucnum() returns $$L_n$$, the nth Lucas number. lucnum2() returns $$L_n$$ and $$L_{n-1}$$.

These functions are designed for calculating isolated Lucas numbers. When a sequence of values is wanted it’s best to start with lucnum2() and iterate the defining $$L_{n+1}=L_n+L_{n-1}$$ or similar.

The Fibonacci numbers and Lucas numbers are related sequences, so it’s never necessary to call both fib2() and lucnum2(). The formulas for going from Fibonacci to Lucas can be found in Lucas Numbers Algorithm, the reverse is straightforward too.

invert(a, b)

Return the inverse of a modulo b if exists. The return value r will satisfy $$0 \le r \lt b$$. If an inverse doesn’t exist return NULL.

jacobi(a, b)

Calculate the Jacobi symbol $$(\frac{a}{b})$$. This is defined only for b odd.

legendre(a, p)

Calculate the Legendre symbol $$(\frac{a}{p})$$. This is defined only for p an odd positive prime, and for such p it’s identical to the Jacobi symbol.

kronecker(a, b)

Calculate the Jacobi symbol $$(\frac{a}{b})$$ with the Kronecker extension $$(\frac{a}{2})=(\frac{2}{a})$$ when a odd, or $$(\frac{a}{2})=0$$ when a even.

Section 1.4.2, Henri Cohen, “A Course in Computational Algebraic Number Theory”, Graduate Texts in Mathematics number 138, Springer-Verlag, 1993. http://www.math.u-bordeaux.fr/~cohen/

## Logical and Bit Manipulation Functions¶

These functions behave as if twos complement arithmetic were used (although sign-magnitude is the actual implementation). The least significant bit is number 0.

Logical Operators
Operator Description Example Return
& Bitwise and '0b10001'::mpz & '0b01001'::mpz '0b1'::mpz
| Bitwise inclusive-or '0b10001'::mpz | '0b01001'::mpz '0b11001'::mpz
# Bitwise exclusive-or '0b10001'::mpz # '0b01001'::mpz '0b11000'::mpz
com(op)

Return the ones’ complement of op.

popcount(op)

If op>=0, return the population count of op, which is the number of 1 bits in the binary representation. If op<0, the number of 1s is infinite, and the return value is the largest possible, represented by gmp_max_bitcnt().

hamdist(op1, op2)

If op1 and op2 are both >=0 or both <0, return the Hamming distance between the two operands, which is the number of bit positions where op1 and op2 have different bit values. If one operand is >=0 and the other <0 then the number of bits different is infinite, and the return value is the largest possible, represented by gmp_max_bitcnt().

scan0(op, starting_bit)
scan1(op, starting_bit)

Scan op, starting from bit starting_bit, towards more significant bits, until the first 0 or 1 bit (respectively) is found. Return the index of the found bit.

If the bit at starting_bit is already what’s sought, then starting_bit is returned.

If there’s no bit found, then the largest possible bit count is returned (represented by gmp_max_bitcnt()). This will happen in mpz_scan0() past the end of a negative number, or mpz_scan1() past the end of a nonnegative number.

setbit(op, bit_index)

Return op with bit bit_index set.

clrbit(op, bit_index)

Return op with bit bit_index cleared.

combit(op, bit_index)

Return op with bit bit_index complemented.

Note

The function is not available on GMP version < 4.2.

tstbit(op, bit_index)

Test bit bit_index in op and return 0 or 1 accordingly.

## Random number functions¶

Sequences of pseudo-random numbers are generated using an internal per-session variable, which holds an algorithm selection and a current state. Such a variable must be initialized by a call to one of the randinit*() functions, and can be seeded with the randseed() function.

randinit()

Initialize the session random state with a default algorithm. This will be a compromise between speed and randomness, and is recommended for applications with no special requirements. Currently this is randinit_mt().

randinit_mt()

Initialize the session random state for a Mersenne Twister algorithm. This algorithm is fast and has good randomness properties.

Note

The function is not available on GMP version < 4.2.

randinit_lc_2exp(a, c, e)

Initialize the session random state with a linear congruential algorithm $$X = (a \cdot X + c) \mod 2^e$$.

The low bits of X in this algorithm are not very random. The least significant bit will have a period no more than 2, and the second bit no more than 4, etc. For this reason only the high half of each X is actually used.

When a random number of more than $$e/2$$ bits is to be generated, multiple iterations of the recurrence are used and the results concatenated.

randinit_lc_2exp_size(s)

Initialize the session random state for a linear congruential algorithm as per randinit_lc_2exp(). a, c and e are selected from a table, chosen so that size bits (or more) of each X will be used, ie. $$e/2 \ge s$$.

The function fails if s is bigger than the table data provides. The maximum size currently supported is 128.

randseed(seed)

Set an initial seed value into session random state.

The size of a seed determines how many different sequences of random numbers is possible to generate. The “quality” of the seed is the randomness of a given seed compared to the previous seed used, and this affects the randomness of separate number sequences. The method for choosing a seed is critical if the generated numbers are to be used for important applications, such as generating cryptographic keys.

Traditionally the system time has been used to seed, but care needs to be taken with this. If an application seeds often and the resolution of the system clock is low, then the same sequence of numbers might be repeated. Also, the system time is quite easy to guess, so if unpredictability is required then it should definitely not be the only source for the seed value. On some systems there’s a special device /dev/random which provides random data better suited for use as a seed.

urandomb(n)

Generate a uniformly distributed random integer in the range $$0$$ to $$2^n−1$$, inclusive.

The session state must be initialized by calling one of the randinit() functions before invoking this function.

urandomm(n)

Generate a uniformly distributed random integer in the range 0 to n−1, inclusive.

The session state must be initialized by calling one of the randinit() functions before invoking this function.

urandomb(n)

Generate a random integer with long strings of zeros and ones in the binary representation. Useful for testing functions and algorithms, since this kind of random numbers have proven to be more likely to trigger corner-case bugs. The random number will be in the range $$0$$ to $$2^n−1$$, inclusive.

The session state must be initialized by calling one of the randinit() functions before invoking this function.

## Aggregation functions¶

sum(z)

Return the sum of z across all input values.

prod(z)

Return the product of z across all input values.

max(z)

Return the maximum value of z across all input values.

min(z)

Return the minimum value of z across all input values.

bit_and(z)

Return the bitwise and of z across all input values.

bit_or(z)

Return the bitwise inclusive-or of z across all input values.

bit_xor(z)

Return the bitwise exclusive-or of z across all input values.