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## 5.9 Number Theoretic Functions

Function: int mpz_probab_prime_p (const mpz_t n, int 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. The argument reps controls how many such tests are done; a higher value will reduce the chances of a composite being returned as “probably prime”. 25 is a reasonable number; a composite number will then be identified as a prime with a probability of less than 2^(-50).

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.

Function: void mpz_nextprime (mpz_t rop, const mpz_t op)

Set rop to 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.

Function: void mpz_gcd (mpz_t rop, const mpz_t op1, const mpz_t op2)

Set rop to the greatest common divisor of op1 and op2. The result is always positive even if one or both input operands are negative. Except if both inputs are zero; then this function defines gcd(0,0) = 0.

Function: unsigned long int mpz_gcd_ui (mpz_t rop, const mpz_t op1, unsigned long int op2)

Compute the greatest common divisor of op1 and op2. If rop is not `NULL`, store the result there.

If the result is small enough to fit in an `unsigned long int`, it is returned. If the result does not fit, 0 is returned, and the result is equal to the argument op1. Note that the result will always fit if op2 is non-zero.

Function: void mpz_gcdext (mpz_t g, mpz_t s, mpz_t t, const mpz_t a, const mpz_t b)

Set g to the greatest common divisor of a and b, and in addition set s and t to coefficients satisfying a*s + b*t = g. The value in g is always positive, even if one or both of a and b are negative (or zero if both inputs are zero). The values in s and t are chosen such that normally, abs(s) < abs(b) / (2 g) and abs(t) < abs(a) / (2 g), and these relations define s and t uniquely. There are a few exceptional cases:

If abs(a) = abs(b), then s = 0, t = sgn(b).

Otherwise, s = sgn(a) if b = 0 or abs(b) = 2 g, and t = sgn(b) if a = 0 or abs(a) = 2 g.

In all cases, s = 0 if and only if g = abs(b), i.e., if b divides a or a = b = 0.

If t is `NULL` then that value is not computed.

Function: void mpz_lcm (mpz_t rop, const mpz_t op1, const mpz_t op2)
Function: void mpz_lcm_ui (mpz_t rop, const mpz_t op1, unsigned long op2)

Set rop to the least common multiple of op1 and op2. rop is always positive, irrespective of the signs of op1 and op2. rop will be zero if either op1 or op2 is zero.

Function: int mpz_invert (mpz_t rop, const mpz_t op1, const mpz_t op2)

Compute the inverse of op1 modulo op2 and put the result in rop. If the inverse exists, the return value is non-zero and rop will satisfy 0 < rop < abs(op2). If an inverse doesn’t exist the return value is zero and rop is undefined. The behaviour of this function is undefined when op2 is zero.

Function: int mpz_jacobi (const mpz_t a, const mpz_t b)

Calculate the Jacobi symbol (a/b). This is defined only for b odd.

Function: int mpz_legendre (const mpz_t a, const mpz_t p)

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

Function: int mpz_kronecker (const mpz_t a, const mpz_t b)
Function: int mpz_kronecker_si (const mpz_t a, long b)
Function: int mpz_kronecker_ui (const mpz_t a, unsigned long b)
Function: int mpz_si_kronecker (long a, const mpz_t b)
Function: int mpz_ui_kronecker (unsigned long a, const mpz_t b)

Calculate the Jacobi symbol (a/b) with the Kronecker extension (a/2)=(2/a) when a odd, or (a/2)=0 when a even.

When b is odd the Jacobi symbol and Kronecker symbol are identical, so `mpz_kronecker_ui` etc can be used for mixed precision Jacobi symbols too.

For more information see Henri Cohen section 1.4.2 (see section References), or any number theory textbook. See also the example program ‘demos/qcn.c’ which uses `mpz_kronecker_ui`.

Function: mp_bitcnt_t mpz_remove (mpz_t rop, const mpz_t op, const mpz_t f)

Remove all occurrences of the factor f from op and store the result in rop. The return value is how many such occurrences were removed.

Function: void mpz_fac_ui (mpz_t rop, unsigned long int n)
Function: void mpz_2fac_ui (mpz_t rop, unsigned long int n)
Function: void mpz_mfac_uiui (mpz_t rop, unsigned long int n, unsigned long int m)

Set rop to the factorial of n: `mpz_fac_ui` computes the plain factorial n!, `mpz_2fac_ui` computes the double-factorial n!!, and `mpz_mfac_uiui` the m-multi-factorial n!^(m).

Function: void mpz_primorial_ui (mpz_t rop, unsigned long int n)

Set rop to the primorial of n, i.e. the product of all positive prime numbers <=n.

Function: void mpz_bin_ui (mpz_t rop, const mpz_t n, unsigned long int k)
Function: void mpz_bin_uiui (mpz_t rop, unsigned long int n, unsigned long int k)

Compute the binomial coefficient n over k and store the result in rop. Negative values of n are supported by `mpz_bin_ui`, using the identity bin(-n,k) = (-1)^k * bin(n+k-1,k), see Knuth volume 1 section 1.2.6 part G.

Function: void mpz_fib_ui (mpz_t fn, unsigned long int n)
Function: void mpz_fib2_ui (mpz_t fn, mpz_t fnsub1, unsigned long int n)

`mpz_fib_ui` sets fn to to F[n], the n’th Fibonacci number. `mpz_fib2_ui` sets fn to F[n], and fnsub1 to 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 `mpz_fib2_ui` and iterate the defining F[n+1]=F[n]+F[n-1] or similar.

Function: void mpz_lucnum_ui (mpz_t ln, unsigned long int n)
Function: void mpz_lucnum2_ui (mpz_t ln, mpz_t lnsub1, unsigned long int n)

`mpz_lucnum_ui` sets ln to to L[n], the n’th Lucas number. `mpz_lucnum2_ui` sets ln to L[n], and lnsub1 to 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 `mpz_lucnum2_ui` 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 `mpz_fib2_ui` and `mpz_lucnum2_ui`. The formulas for going from Fibonacci to Lucas can be found in Lucas Numbers, the reverse is straightforward too.

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