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15 Number Theory

**GAP** provides a couple of elementary number theoretic functions. Most of these deal with the group of integers coprime to m, called the *prime residue group*. The order of this group is ?(m) (see?`Phi`

(15.2-2)), and λ(m) (see?`Lambda`

(15.2-3)) is its exponent. This group is cyclic if and only if m is 2, 4, an odd prime power p^n, or twice an odd prime power 2 p^n. In this case the generators of the group, i.e., elements of order ?(m), are called *primitive roots* (see?`PrimitiveRootMod`

(15.3-3)).

Note that neither the arguments nor the return values of the functions listed below are groups or group elements in the sense of **GAP**. The arguments are simply integers.

`‣ InfoNumtheor` | ( info class ) |

`InfoNumtheor`

is the info class (see?7.4) for the functions in the number theory chapter.

`‣ PrimeResidues` ( m ) | ( function ) |

`PrimeResidues`

returns the set of integers from the range `[ 0 .. Abs( `

that are coprime to the integer `m` )-1 ]`m`.

`Abs(`

must be less than 2^28, otherwise the set would probably be too large anyhow.`m`)

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

`‣ Phi` ( m ) | ( operation ) |

`Phi`

returns the number ?(`m`) of positive integers less than the positive integer `m` that are coprime to `m`.

Suppose that m = p_1^{e_1} p_2^{e_2} ? p_k^{e_k}. Then ?(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).

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

`‣ Lambda` ( m ) | ( operation ) |

`Lambda`

returns the exponent λ(`m`) of the group of prime residues modulo the integer `m`.

λ(`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^{?(`m`)} ≡ 1 mod `m`; thus λ(`m`) divides ?(`m`) (see?`Phi`

(15.2-2)).

Carmichael's theorem states that λ can be computed as follows: λ(2) = 1, λ(4) = 2 and λ(2^e) = 2^{e-2} if 3 ≤ e, λ(p^e) = (p-1) p^{e-1} (i.e. ?(m)) if p is an odd prime and λ(m*n) =`Lcm`

( λ(m), λ(n) ) if m, n are coprime.

Composites for which λ(m) divides m - 1 are called Carmichaels. If 6k+1, 12k+1 and 18k+1 are primes their product is such a number. There are only 1547 Carmichaels below 10^10 but 455052511 primes.

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

`‣ GeneratorsPrimeResidues` ( n ) | ( function ) |

Let `n` be a positive integer. `GeneratorsPrimeResidues`

returns a description of generators of the group of prime residues modulo `n`. The return value is a record with components

`primes`

:a list of the prime factors of

`n`,`exponents`

:a list of the exponents of these primes in the factorization of

`n`, and`generators`

:a list describing generators of the group of prime residues; for the prime factor 2, either a primitive root or a list of two generators is stored, for each other prime factor of

`n`, a primitive root is stored.

gap> GeneratorsPrimeResidues( 1 ); rec( exponents := [ ], generators := [ ], primes := [ ] ) gap> GeneratorsPrimeResidues( 4*3 ); rec( exponents := [ 2, 1 ], generators := [ 7, 5 ], primes := [ 2, 3 ] ) gap> GeneratorsPrimeResidues( 8*9*5 ); rec( exponents := [ 3, 2, 1 ], generators := [ [ 271, 181 ], 281, 217 ], primes := [ 2, 3, 5 ] )

`‣ OrderMod` ( n, m ) | ( function ) |

`OrderMod`

returns the multiplicative order of the integer `n` modulo the positive integer `m`. If `n` and `m` are not coprime 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`. If the group of prime residues modulo `m` is cyclic then each element of maximal order is called a primitive root modulo `m` (see?`IsPrimitiveRootMod`

(15.3-4)).

`OrderMod`

usually spends most of its time factoring `m` and ?(`m`) (see?`FactorsInt`

(14.4-7)).

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

`‣ LogMod` ( n, r, m ) | ( function ) |

`‣ LogModShanks` ( n, r, m ) | ( function ) |

computes the discrete `r`-logarithm of the integer `n` modulo the integer `m`. It returns a number `l` such that `r`^`l` ≡ `n` mod `m` if such a number exists. Otherwise `fail`

is returned.

`LogModShanks`

uses the Baby Step - Giant Step Method of Shanks (see for example [Coh93, section 5.4.1]) and in general requires more memory than a call to `LogMod`

.

gap> l:= LogMod( 2, 5, 7 ); 5^l mod 7 = 2; 4 true gap> LogMod( 1, 3, 3 ); LogMod( 2, 3, 3 ); 0 fail

`‣ PrimitiveRootMod` ( m[, start] ) | ( function ) |

`PrimitiveRootMod`

returns the smallest primitive root modulo the positive integer `m` and `fail`

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`.

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

`‣ IsPrimitiveRootMod` ( r, m ) | ( function ) |

`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.

gap> IsPrimitiveRootMod( 2, 541 ); true gap> IsPrimitiveRootMod( -539, 541 ); # same computation as above; true gap> IsPrimitiveRootMod( 4, 541 ); false gap> ForAny( [1..29], r -> IsPrimitiveRootMod( r, 30 ) ); false gap> # there is no a primitive root modulo 30

`‣ Jacobi` ( n, m ) | ( function ) |

`Jacobi`

returns the value of the *Kronecker-Jacobi symbol* J(`n`,`m`) of the integer `n` modulo the integer `m`. It is defined as follows:

If n and m are not coprime then J(n,m) = 0. Furthermore, J(n,1) = 1 and J(n,-1) = -1 if m < 0 and +1 otherwise. And for odd n it is J(n,2) = (-1)^k with k = (n^2-1)/8. For odd primes m which are coprime to n the Kronecker-Jacobi symbol has the same value as the Legendre symbol (see?`Legendre`

(15.4-2)).

For the general case suppose that m = p_1 ? p_2 ? p_k is a product of -1 and of primes, not necessarily distinct, and that n is coprime to m. Then J(n,m) = J(n,p_1) ? J(n,p_2) ? J(n,p_k).

Note that the Kronecker-Jacobi symbol coincides with the Jacobi symbol that is defined for odd m in many number theory books. For odd primes m and n coprime to m it coincides with the Legendre symbol.

`Jacobi`

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

(56.7-1)).

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

`‣ Legendre` ( n, m ) | ( function ) |

`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`

(15.4-3).

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`

(15.4-1)) 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`

(14.4-7)).

A description of the Jacobi symbol, the Legendre symbol, and related topics can be found in [Bak84].

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

`‣ RootMod` ( n[, k], m ) | ( function ) |

`RootMod`

computes a `k`th root of the integer `n` modulo the positive integer `m`, i.e., a r such that r^`k` ≡ `n` mod `m`. If no such root exists `RootMod`

returns `fail`

. If only the arguments `n` and `m` are given, the default value for `k` is 2.

A square root of `n` exists only if `Legendre(`

(see?`n`,`m`) = 1`Legendre`

(15.4-2)). If `m` has r different prime factors then there are 2^r different roots of `n` mod `m`. It is unspecified which one `RootMod`

returns. You can, however, use `RootsMod`

(15.4-4) to compute the full set of roots.

`RootMod`

is efficient even for large values of `m`, in fact the most time is usually spent factoring `m` (see `FactorsInt`

(14.4-7)).

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

`‣ RootsMod` ( n[, k], m ) | ( function ) |

`RootsMod`

computes the set of `k`th roots of the integer `n` modulo the positive integer `m`, i.e., the list of all r such that r^`k` ≡ `n` mod `m`. If only the arguments `n` and `m` are given, the default value for `k` is 2.

gap> RootsMod( 1, 7*31 ); # the same as `RootsUnityMod( 7*31 )' [ 1, 92, 125, 216 ] gap> RootsMod( 7, 7*31 ); [ 21, 196 ] gap> RootsMod( 5, 7*31 ); [ ] gap> RootsMod( 1, 5, 7*31 ); [ 1, 8, 64, 78, 190 ]

`‣ RootsUnityMod` ( [k, ]m ) | ( function ) |

`RootsUnityMod`

returns the set of `k`-th roots of unity modulo the positive integer `m`, i.e., the list of all solutions r of r^`k` ≡ `n` mod `m`. If only the argument `m` is given, the default value for `k` is 2.

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.

In the current implementation `k` must be a prime.

gap> RootsUnityMod( 7*31 ); RootsUnityMod( 3, 7*31 ); [ 1, 92, 125, 216 ] [ 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 ); # set of all square roots of 64 mod 1009 [ 1001, 8 ]

`‣ Sigma` ( n ) | ( operation ) |

`Sigma`

returns the sum of the positive divisors of the nonzero integer `n`.

`Sigma`

is a multiplicative arithmetic function, i.e., if n and m are relatively prime we have that σ(n ? m) = σ(n) σ(m).

Together with the formula σ(p^k) = (p^{k+1}-1) / (p-1) this allows us to compute σ(`n`).

Integers `n` for which σ(`n`) = 2 `n` are called perfect. Even perfect integers are exactly of the form 2^{`n`-1}(2^`n`-1) where 2^`n`-1 is prime. Primes of the form 2^`n`-1 are called *Mersenne primes*, and 42 among the known Mersenne primes are obtained for `n` = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583 and 25964951. Please find more up to date information about Mersenne primes at https://www.mersenne.org. It is not known whether odd perfect integers exist, however?[BC89] show that any such integer must have at least 300 decimal digits.

`Sigma`

usually spends most of its time factoring `n` (see?`FactorsInt`

(14.4-7)).

gap> Sigma( 1 ); 1 gap> Sigma( 1009 ); # 1009 is a prime 1010 gap> Sigma( 8128 ) = 2*8128; # 8128 is a perfect number true

`‣ Tau` ( n ) | ( operation ) |

`Tau`

returns the number of the positive divisors of the nonzero integer `n`.

`Tau`

is a multiplicative arithmetic function, i.e., if n and m are relative prime we have τ(n ? m) = τ(n) τ(m). Together with the formula τ(p^k) = k+1 this allows us to compute τ(`n`).

`Tau`

usually spends most of its time factoring `n` (see?`FactorsInt`

(14.4-7)).

gap> Tau( 1 ); 1 gap> Tau( 1013 ); # thus 1013 is a prime 2 gap> Tau( 8128 ); 14 gap> # result is odd if and only if argument is a perfect square: gap> Tau( 36 ); 9

`‣ MoebiusMu` ( n ) | ( function ) |

`MoebiusMu`

computes the value of Moebius inversion function for the nonzero integer `n`. This is 0 for integers which are not squarefree, i.e., which are divided by a square r^2. Otherwise it is 1 if `n` has a even number and -1 if `n` has an odd number of prime factors.

The importance of μ stems from the so called inversion formula. Suppose f is a multiplicative arithmetic function defined on the positive integers and let g(n) = ∑_{d ∣ n} f(d). Then f(n) = ∑_{d ∣ n} μ(d) g(n/d). As a special case we have ?(n) = ∑_{d ∣ n} μ(d) n/d since n = ∑_{d ∣ n} ?(d) (see?`Phi`

(15.2-2)).

`MoebiusMu`

usually spends all of its time factoring `n` (see `FactorsInt`

(14.4-7)).

gap> MoebiusMu( 60 ); MoebiusMu( 61 ); MoebiusMu( 62 ); 0 -1 1

`‣ ContinuedFractionExpansionOfRoot` ( f, n ) | ( function ) |

The first `n` terms of the continued fraction expansion of the only positive real root of the polynomial `f` with integer coefficients. The leading coefficient of `f` must be positive and the value of `f` at 0 must be negative. If the degree of `f` is 2 and `n` = 0, the function computes one period of the continued fraction expansion of the root in question. Anything may happen if `f` has three or more positive real roots.

gap> x := Indeterminate(Integers);; gap> ContinuedFractionExpansionOfRoot(x^2-7,20); [ 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1 ] gap> ContinuedFractionExpansionOfRoot(x^2-7,0); [ 2, 1, 1, 1, 4 ] gap> ContinuedFractionExpansionOfRoot(x^3-2,20); [ 1, 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3 ] gap> ContinuedFractionExpansionOfRoot(x^5-x-1,50); [ 1, 5, 1, 42, 1, 3, 24, 2, 2, 1, 16, 1, 11, 1, 1, 2, 31, 1, 12, 5, 1, 7, 11, 1, 4, 1, 4, 2, 2, 3, 4, 2, 1, 1, 11, 1, 41, 12, 1, 8, 1, 1, 1, 1, 1, 9, 2, 1, 5, 4 ]

`‣ ContinuedFractionApproximationOfRoot` ( f, n ) | ( function ) |

The `n`th continued fraction approximation of the only positive real root of the polynomial `f` with integer coefficients. The leading coefficient of `f` must be positive and the value of `f` at 0 must be negative. Anything may happen if `f` has three or more positive real roots.

gap> ContinuedFractionApproximationOfRoot(x^2-2,10); 3363/2378 gap> 3363^2-2*2378^2; 1 gap> z := ContinuedFractionApproximationOfRoot(x^5-x-1,20); 499898783527/428250732317 gap> z^5-z-1; 486192462527432755459620441970617283/ 14404247382319842421697357558805709031116987826242631261357

`‣ PValuation` ( n, p ) | ( function ) |

For an integer `n` and a prime `p` this function returns the `p`-valuation of `n`, that is the exponent e such that p^e is the largest power of `p` that divides `n`. The valuation of zero is infinity.

gap> PValuation(100,2); 2 gap> PValuation(100,3); 0

`‣ TwoSquares` ( n ) | ( function ) |

`TwoSquares`

returns a list of two integers x ≤ y such that the sum of the squares of x and y is equal to the nonnegative integer `n`, i.e., n = x^2 + y^2. If no such representation exists `TwoSquares`

will return `fail`

. `TwoSquares`

will return a representation for which the gcd of x and y is as small as possible. It is not specified which representation `TwoSquares`

returns if there is more than one.

Let a be the product of all maximal powers of primes of the form 4k+3 dividing `n`. A representation of `n` as a sum of two squares exists if and only if a is a perfect square. Let b be the maximal power of 2 dividing `n` or its half, whichever is a perfect square. Then the minimal possible gcd of x and y is the square root c of a ? b. The number of different minimal representation with x ≤ y is 2^{l-1}, where l is the number of different prime factors of the form 4k+1 of `n`.

The algorithm first finds a square root r of -1 modulo `n` / (a ? b), which must exist, and applies the Euclidean algorithm to r and `n`. The first residues in the sequence that are smaller than sqrt{`n`/(a ? b)} times c are a possible pair x and y.

Better descriptions of the algorithm and related topics can be found in [Wag90] and [Zag90].

gap> TwoSquares( 5 ); [ 1, 2 ] gap> TwoSquares( 11 ); # there is no representation fail gap> TwoSquares( 16 ); [ 0, 4 ] gap> # 3 is the minimal possible gcd because 9 divides 45: gap> TwoSquares( 45 ); [ 3, 6 ] gap> # it is not [5,10] because their gcd is not minimal: gap> TwoSquares( 125 ); [ 2, 11 ] gap> # [10,11] would be the other possible representation: gap> TwoSquares( 13*17 ); [ 5, 14 ] gap> TwoSquares( 848654483879497562821 ); # argument is prime [ 6305894639, 28440994650 ]

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