11 Number Theory

The integers relatively prime to m form a group under multiplication modulo m, called the prime residue group. This chapter describes the functions that deal with this group.

The first section describes the function that computes the set of representatives of the group (see PrimeResidues).

The next sections describe the functions that compute the size and the exponent of the group (see Phi and Lambda).

The next section describes the function that computes the order of an element in the group (see OrderMod).

The next section describes the functions that test whether a residue generates the group or computes a generator of the group, provided it is cyclic (see IsPrimitiveRootMod, PrimitiveRootMod).

The next section describes the functions that test whether an element is a square in the group (see Jacobi and Legendre).

The next sections describe the functions that compute general roots in the group (see RootMod and RootsUnityMod).

All these functions are in the file LIBNAME/"numtheor.g".

Subsections

  1. PrimeResidues
  2. Phi
  3. Lambda
  4. OrderMod
  5. IsPrimitiveRootMod
  6. PrimitiveRootMod
  7. Jacobi
  8. Legendre
  9. RootMod
  10. RootsUnityMod

11.1 PrimeResidues

PrimeResidues( m )

PrimeResidues returns the set of integers from the range 0..Abs(m)-1 that are relatively prime to the integer m.

Abs(m) must be less than 2^{28}, otherwise the set would probably be too large anyhow.

The integers relatively prime to m form a group under multiplication modulo m, called the prime residue group. phi(m) (see Phi) is the order of this group, lambda(m) (see Lambda) the exponent. If and only if m is 2, 4, an odd prime power p^e, or twice an odd prime power 2 p^e, this group is cyclic. In this case the generators of the group, i.e., elements of order phi(m), are called primitive roots (see IsPrimitiveRootMod, PrimitiveRootMod).

    gap> PrimeResidues( 0 );
    [  ]
    gap> PrimeResidues( 1 );
    [ 0 ]
    gap> PrimeResidues( 20 );
    [ 1, 3, 7, 9, 11, 13, 17, 19 ] 

11.2 Phi

Phi( m )

Phi returns the value of the Euler totient function phi(m) for the integer m. phi(m) is defined as the number of positive integers less than or equal to m that are relatively prime to m.

Suppose that m = p_1^{e_1} p_2^{e_2} ... p_k^{e_k}. Then phi(m) is p_1^{e_1-1} (p_1-1) p_2^{e_2-1} (p_2-1) ... p_k^{e_k-1} (p_k-1). It follows that m is a prime if and only if phi(m) = m - 1.

The integers relatively prime to m form a group under multiplication modulo m, called the prime residue group. It can be computed with PrimeResidues (see PrimeResidues). phi(m) is the order of this group, lambda(m) (see Lambda) the exponent. If and only if m is 2, 4, an odd prime power p^e, or twice an odd prime power 2 p^e, this group is cyclic. In this case the generators of the group, i.e., elements of order phi(m), are called primitive roots (see IsPrimitiveRootMod, PrimitiveRootMod).

Phi usually spends most of its time factoring m (see FactorsInt).

    gap> Phi( 12 );
    4
    gap> Phi( 2^13-1 );
    8190        # which proves that $2^{13}-1$ is a prime
    gap> Phi( 2^15-1 );
    27000 

11.3 Lambda

Lambda( m )

Lambda returns the exponent of the group of relatively prime residues modulo the integer m.

lambda(m) is the smallest positive integer l such that for every a relatively prime to m we have a^l=1 mod m. Fermat's theorem asserts a^{phi(m)}=1 mod m, thus lambda(m) divides phi(m) (see Phi).

Carmichael's theorem states that lambda can be computed as follows lambda(2)=1, lambda(4)=2 and lambda(2^e) = 2^{e-2} if 3 <= e, lambda(p^e) = (p-1) p^{e-1} (= phi(p^e)) if p is an odd prime, and lambda(n m) = Lcm(lambda(n),lambda(m)) if n, m are relatively prime.

Composites for which lambda(m) divides m - 1 are called Carmichaels. If 6k+1, 12k+1 and 18k+1 are primes their product is such a number. It is believed but unproven that there are infinitely many Carmichaels. There are only 1547 Carmichaels below 10^{10} but 455052511 primes.

The integers relatively prime to m form a group under multiplication modulo m, called the prime residue group. It can be computed with PrimeResidues (see PrimeResidues). phi(m) (see Phi) is the order of this group, lambda(m) the exponent. If and only if m is 2, 4, an odd prime power p^e, or twice an odd prime power 2 p^e, this group is cyclic. In this case the generators of the group, i.e., elements of order phi(m), are called primitive roots (see IsPrimitiveRootMod, PrimitiveRootMod).

Lambda usually spends most of its time factoring m (see FactorsInt).

    gap> Lambda( 10 );
    4
    gap> Lambda( 30 );
    4
    gap> Lambda( 561 );
    80        # 561 is the smallest Carmichael number 

11.4 OrderMod

OrderMod( n, m )

OrderMod returns the multiplicative order of the integer n modulo the positive integer m. If n is less than 0 or larger than m it is replaced by its remainder. If n and m are not relatively prime the order of n is not defined and OrderMod will return 0.

If n and m are relatively prime the multiplicative order of n modulo m is the smallest positive integer i such that n^i = 1 mod m. Elements of maximal order are called primitive roots (see Phi).

OrderMod usually spends most of its time factoring m and phi(m) (see FactorsInt).

    gap> OrderMod( 2, 7 );
    3
    gap> OrderMod( 3, 7 );
    6        # 3 is a primitive root modulo 7 

11.5 IsPrimitiveRootMod

IsPrimitiveRootMod( r, m )

IsPrimitiveRootMod returns true if the integer r is a primitive root modulo the positive integer m and false otherwise. If r is less than 0 or larger than m it is replaced by its remainder.

The integers relatively prime to m form a group under multiplication modulo m, called the prime residue group. It can be computed with PrimeResidues (see PrimeResidues). phi(m) (see Phi) is the order of this group, lambda(m) (see Lambda) the exponent. If and only if m is 2, 4, an odd prime power p^e, or twice an odd prime power 2 p^e, this group is cyclic. In this case the generators of the group, i.e., elements of order phi(m), are called primitive roots (see also PrimitiveRootMod).

    gap> IsPrimitiveRootMod( 2, 541 );
    true
    gap> IsPrimitiveRootMod( -539, 541 );
    true        # same computation as above
    gap> IsPrimitiveRootMod( 4, 541 );
    false
    gap> ForAny( [1..29], r -> IsPrimitiveRootMod( r, 30 ) );
    false        # there does not exist a primitive root modulo 30 

11.6 PrimitiveRootMod

PrimitiveRootMod( m )
PrimitiveRootMod( m, start )

PrimitiveRootMod returns the smallest primitive root modulo the positive integer m and false if no such primitive root exists. If the optional second integer argument start is given PrimitiveRootMod returns the smallest primitive root that is strictly larger than start.

The integers relatively prime to m form a group under multiplication modulo m, called the prime residue group. It can be computed with PrimeResidues (see PrimeResidues). phi(m) (see Phi) is the order of this group, lambda(m) (see Lambda) the exponent. If and only if m is 2, 4, an odd prime power p^e, or twice an odd prime power 2 p^e, this group is cyclic. In this case the generators of the group, i.e., elements of order phi(m), are called primitive roots (see also IsPrimitiveRootMod).

    gap> PrimitiveRootMod( 409 );
    21        # largest primitive root for a prime less than 2000
    gap> PrimitiveRootMod( 541, 2 );
    10
    gap> PrimitiveRootMod( 337, 327 );
    false        # 327 is the largest primitive root mod 337
    gap> PrimitiveRootMod( 30 );
    false        # the exists no primitive root modulo 30 

11.7 Jacobi

Jacobi( n, m )

Jacobi returns the value of the Jacobi symbol of the integer n modulo the integer m.

Suppose that m = p_1 p_2 .. p_k as a product of primes, not necessarily distinct. Then for n relatively prime to m the Jacobi symbol is defined by J(n/m) = L(n/p_1) L(n/p_2) .. L(n/p_k), where L(n/p) is the Legendre symbol (see Legendre). By convention J(n/1) = 1. If the gcd of n and m is larger than 1 we define J(n/m) = 0.

If n is an quadratic residue modulo m, i.e., if there exists an r such that r^2 = n mod m then J(n/m) = 1. However J(n/m) = 1 implies the existence of such an r only if m is a prime.

Jacobi is very efficient, even for large values of n and m, it is about as fast as the Euclidean algorithm (see Gcd).

    gap> Jacobi( 11, 35 );
    1         # $9^2 = 11$ mod $35$
    gap> Jacobi( 6, 35 );
    -1        # thus there is no $r$ such that $r^2 = 6$ mod $35$
    gap> Jacobi( 3, 35 );
    1         # even though there is no $r$ with $r^2 = 3$ mod $35$ 

11.8 Legendre

Legendre( n, m )

Legendre returns the value of the Legendre symbol of the integer n modulo the positive integer m.

The value of the Legendre symbol L(n/m) is 1 if n is a quadratic residue modulo m, i.e., if there exists an integer r such that r^2 = n mod m and -1 otherwise.

If a root of n exists it can be found by RootMod (see RootMod).

While the value of the Legendre symbol usually is only defined for m a prime, we have extended the definition to include composite moduli too. The Jacobi symbol (see Jacobi) is another generalization of the Legendre symbol for composite moduli that is much cheaper to compute, because it does not need the factorization of m (see FactorsInt).

    gap> Legendre( 5, 11 );
    1         # $4^2 = 5$ mod $11$
    gap> Legendre( 6, 11 );
    -1        # thus there is no $r$ such that $r^2 = 6$ mod $11$
    gap> Legendre( 3, 35 );
    -1        # thus there is no $r$ such that $r^2 = 3$ mod $35$ 

11.9 RootMod

RootMod( n, m )
RootMod( n, k, m )

In the first form RootMod computes a square root of the integer n modulo the positive integer m, i.e., an integer r such that r^2 = n mod m. If no such root exists RootMod returns false.

A root of n exists only if Legendre(n,m) = 1 (see Legendre). If m has k different prime factors then there are 2^k different roots of n mod m. It is unspecified which one RootMod returns. You can, however, use RootsUnityMod (see RootsUnityMod) to compute the full set of roots.

In the second form RootMod computes a kth root of the integer n modulo the positive integer m, i.e., an integer r such that r^k = n mod m. If no such root exists RootMod returns false.

In the current implementation k must be a prime.

RootMod is efficient even for large values of m, actually most time is usually spent factoring m (see FactorsInt).

    gap> RootMod( 64, 1009 );
    1001        # note 'RootMod' does not return 8 in this case but -8
    gap> RootMod( 64, 3, 1009 );
    518
    gap> RootMod( 64, 5, 1009 );
    656
    gap> List( RootMod( 64, 1009 ) * RootsUnityMod( 1009 ),
    >          x -> x mod 1009 );
    [ 1001, 8 ]        # set of all square roots of 64 mod 1009 

11.10 RootsUnityMod

RootsUnityMod( m )
RootsUnityMod( k, m )

In the first form RootsUnityMod computes the square roots of 1 modulo the integer m, i.e., the set of all positive integers r less than n such that r^2 = 1 mod m.

In the second form RootsUnityMod computes the kth roots of 1 modulo the integer m, i.e., the set of all positive integers r less than n such that r^k = 1 mod m.

In general there are k^n such roots if the modulus m has n different prime factors p such that p = 1 mod k. If k^2 divides m then there are k^{n+1} such roots; and especially if k = 2 and 8 divides m there are 2^{n+2} such roots.

If you are interested in the full set of roots of another number instead of 1 use RootsUnityMod together with RootMod (see RootMod).

In the current implementation k must be a prime.

RootsUnityMod is efficient even for large values of m, actually most time is usually spent factoring m (see FactorsInt).

    gap> RootsUnityMod(7*31);
    [ 1, 92, 125, 216 ]
    gap> RootsUnityMod(3,7*31);
    [ 1, 25, 32, 36, 67, 149, 156, 191, 211 ]
    gap> RootsUnityMod(5,7*31);
    [ 1, 8, 64, 78, 190 ]
    gap> List( RootMod( 64, 1009 ) * RootsUnityMod( 1009 ),
    >          x -> x mod 1009 );
    [ 1001, 8 ]        # set of all square roots of 64 mod 1009 

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GAP 3.4.4
April 1997