### 14 Integers

One of the most fundamental datatypes in every programming language is the integer type. GAP is no exception.

GAP integers are entered as a sequence of decimal digits optionally preceded by a "+" sign for positive integers or a "-" sign for negative integers. The size of integers in GAP is only limited by the amount of available memory, so you can compute with integers having thousands of digits.

gap> -1234;
-1234
gap> 123456789012345678901234567890123456789012345678901234567890;
123456789012345678901234567890123456789012345678901234567890


Note that in a few places, only certain small integer values can be used. A small integer (also referred to as immediate integer) is an integer n satisfying INTOBJ_MIN ≤ n ≤ INTOBJ_MAX, where INTOBJ_MIN and INTOBJ_MAX equal either -2^28 and 2^28-1 (on 32-bit systems) or -2^60 and 2^60-1 (on 64-bit systems). For example, the elements of a range are restricted to small integers (see 21.22).

Many more functions that are mainly related to the prime residue group of integers modulo an integer are described in chapter 15, and functions dealing with combinatorics can be found in chapter 16.

#### 14.1 Integers: Global Variables

##### 14.1-1 Integers
 ‣ Integers ( global variable )
 ‣ PositiveIntegers ( global variable )
 ‣ NonnegativeIntegers ( global variable )

These global variables represent the ring of integers and the semirings of positive and nonnegative integers, respectively.

gap> Size( Integers ); 2 in Integers;
infinity
true


Integers is a subset of Rationals (17.1-1), which is a subset of Cyclotomics (18.1-2). See Chapter 18 for arithmetic operations and comparison of integers.

##### 14.1-2 IsIntegers
 ‣ IsIntegers( obj ) ( category )
 ‣ IsPositiveIntegers( obj ) ( category )
 ‣ IsNonnegativeIntegers( obj ) ( category )

are the defining categories for the domains Integers (14.1-1), PositiveIntegers (14.1-1), and NonnegativeIntegers (14.1-1).

gap> IsIntegers( Integers );  IsIntegers( Rationals );  IsIntegers( 7 );
true
false
false


#### 14.2 Elementary Operations for Integers

##### 14.2-1 IsInt
 ‣ IsInt( obj ) ( category )

Every rational integer lies in the category IsInt, which is a subcategory of IsRat (17.2-1).

##### 14.2-2 IsPosInt
 ‣ IsPosInt( obj ) ( category )

Every positive integer lies in the category IsPosInt.

##### 14.2-3 Int
 ‣ Int( elm ) ( attribute )

Int returns an integer int whose meaning depends on the type of elm. For example:

If elm is a rational number (see Chapter 17) then int is the integer part of the quotient of numerator and denominator of elm (see QuoInt (14.3-1)).

If elm is an element of a finite prime field (see Chapter 59) then int is the smallest nonnegative integer such that elm = int * One( elm ).

If elm is a string (see Chapter 27) consisting entirely of decimal digits '0', '1', ..., '9', and optionally a sign '-' (at the first position), then int is the integer described by this string. For all other strings, fail is returned. See Int (27.9-1).

The operation String (27.7-6) can be used to compute a string for rational integers, in fact for all cyclotomics.

gap> Int( 4/3 );  Int( -2/3 );
1
0
gap> int:= Int( Z(5) );  int * One( Z(5) );
2
Z(5)
gap> Int( "12345" );  Int( "-27" );  Int( "-27/3" );
12345
-27
fail


##### 14.2-4 IsEvenInt
 ‣ IsEvenInt( n ) ( function )

tests if the integer n is divisible by 2.

##### 14.2-5 IsOddInt
 ‣ IsOddInt( n ) ( function )

tests if the integer n is not divisible by 2.

##### 14.2-6 AbsInt
 ‣ AbsInt( n ) ( function )

AbsInt returns the absolute value of the integer n, i.e., n if n is positive, -n if n is negative and 0 if n is 0.

AbsInt is a special case of the general operation EuclideanDegree (56.6-2).

See also AbsoluteValue (18.1-8).

gap> AbsInt( 33 );
33
gap> AbsInt( -214378 );
214378
gap> AbsInt( 0 );
0


##### 14.2-7 SignInt
 ‣ SignInt( n ) ( function )

SignInt returns the sign of the integer n, i.e., 1 if n is positive, -1 if n is negative and 0 if n is 0.

gap> SignInt( 33 );
1
gap> SignInt( -214378 );
-1
gap> SignInt( 0 );
0


 ‣ LogInt( n, base ) ( function )

LogInt returns the integer part of the logarithm of the positive integer n with respect to the positive integer base, i.e., the largest positive integer e such that base^e ≤ n. The function LogInt will signal an error if either n or base is not positive.

For base = 2 this is very efficient because the internal binary representation of the integer is used.

gap> LogInt( 1030, 2 );
10
gap> 2^10;
1024
gap> LogInt( 1, 10 );
0


##### 14.2-9 RootInt
 ‣ RootInt( n[, k] ) ( function )

RootInt returns the integer part of the kth root of the integer n. If the optional integer argument k is not given it defaults to 2, i.e., RootInt returns the integer part of the square root in this case.

If n is positive, RootInt returns the largest positive integer r such that r^kn. If n is negative and k is odd RootInt returns -RootInt( -n, k ). If n is negative and k is even RootInt will cause an error. RootInt will also cause an error if k is 0 or negative.

gap> RootInt( 361 );
19
gap> RootInt( 2 * 10^12 );
1414213
gap> RootInt( 17000, 5 );
7
gap> 7^5;
16807


##### 14.2-10 SmallestRootInt
 ‣ SmallestRootInt( n ) ( function )

SmallestRootInt returns the smallest root of the integer n.

The smallest root of an integer n is the integer r of smallest absolute value for which a positive integer k exists such that n = r^k.

gap> SmallestRootInt( 2^30 );
2
gap> SmallestRootInt( -(2^30) );
-4


Note that (-2)^30 = +(2^30).

gap> SmallestRootInt( 279936 );
6
gap> LogInt( 279936, 6 );
7
gap> SmallestRootInt( 1001 );
1001


##### 14.2-11 IsSquareInt
 ‣ IsSquareInt( n ) ( function )

IsSquareInt tests whether the integer n is the square of an integer or not. This test is much faster than the simpler RootInt(n)^2=n because it first tests whether n is a square residue modulo some small integers.

##### 14.2-12 ListOfDigits
 ‣ ListOfDigits( n ) ( function )

For a positive integer n this function returns a list l, consisting of the digits of n in decimal representation.

gap> ListOfDigits(3142);
[ 3, 1, 4, 2 ]


##### 14.2-13 Random
 ‣ Random( Integers ) ( method )

Random for integers returns pseudo random integers between -10 and 10 distributed according to a binomial distribution. To generate uniformly distributed integers from a range, use the construction Random( [ low .. high ] )  (see Random (30.7-1)).

#### 14.3 Quotients and Remainders

##### 14.3-1 QuoInt
 ‣ QuoInt( n, m ) ( function )

QuoInt returns the integer part of the quotient of its integer operands.

If n and m are positive, QuoInt returns the largest positive integer q such that q * mn. If n or m or both are negative the absolute value of the integer part of the quotient is the quotient of the absolute values of n and m, and the sign of it is the product of the signs of n and m.

QuoInt is used in a method for the general operation EuclideanQuotient (56.6-3).

gap> QuoInt(5,3);  QuoInt(-5,3);  QuoInt(5,-3);  QuoInt(-5,-3);
1
-1
-1
1


##### 14.3-2 BestQuoInt
 ‣ BestQuoInt( n, m ) ( function )

BestQuoInt returns the best quotient q of the integers n and m. This is the quotient such that n-q*m has minimal absolute value. If there are two quotients whose remainders have the same absolute value, then the quotient with the smaller absolute value is chosen.

gap> BestQuoInt( 5, 3 );  BestQuoInt( -5, 3 );
2
-2


##### 14.3-3 RemInt
 ‣ RemInt( n, m ) ( function )

RemInt returns the remainder of its two integer operands.

If m is not equal to zero, RemInt returns n - m * QuoInt( n, m ). Note that the rules given for QuoInt (14.3-1) imply that the return value of RemInt has the same sign as n and its absolute value is strictly less than the absolute value of m. Note also that the return value equals n mod m when both n and m are nonnegative. Dividing by 0 signals an error.

RemInt is used in a method for the general operation EuclideanRemainder (56.6-4).

gap> RemInt(5,3);  RemInt(-5,3);  RemInt(5,-3);  RemInt(-5,-3);
2
-2
2
-2


##### 14.3-4 GcdInt
 ‣ GcdInt( m, n ) ( function )

GcdInt returns the greatest common divisor of its two integer operands m and n, i.e., the greatest integer that divides both m and n. The greatest common divisor is never negative, even if the arguments are. We define GcdInt( m, 0 ) = GcdInt( 0, m ) = AbsInt( m ) and GcdInt( 0, 0 ) = 0.

GcdInt is a method used by the general function Gcd (56.7-1).

gap> GcdInt( 123, 66 );
3


##### 14.3-5 Gcdex
 ‣ Gcdex( m, n ) ( function )

returns a record g describing the extended greatest common divisor of m and n. The component gcd is this gcd, the components coeff1 and coeff2 are integer cofactors such that g.gcd = g.coeff1 * m + g.coeff2 * n, and the components coeff3 and coeff4 are integer cofactors such that 0 = g.coeff3 * m + g.coeff4 * n.

If m and n both are nonzero, AbsInt( g.coeff1 ) is less than or equal to AbsInt(n) / (2 * g.gcd), and AbsInt( g.coeff2 ) is less than or equal to AbsInt(m) / (2 * g.gcd).

If m or n or both are zero coeff3 is -n / g.gcd and coeff4 is m / g.gcd.

The coefficients always form a unimodular matrix, i.e., the determinant g.coeff1 * g.coeff4 - g.coeff3 * g.coeff2 is 1 or -1.

gap> Gcdex( 123, 66 );
rec( coeff1 := 7, coeff2 := -13, coeff3 := -22, coeff4 := 41,
gcd := 3 )


This means 3 = 7 * 123 - 13 * 66, 0 = -22 * 123 + 41 * 66.

gap> Gcdex( 0, -3 );
rec( coeff1 := 0, coeff2 := -1, coeff3 := 1, coeff4 := 0, gcd := 3 )
gap> Gcdex( 0, 0 );
rec( coeff1 := 1, coeff2 := 0, coeff3 := 0, coeff4 := 1, gcd := 0 )


GcdRepresentation (56.7-3) provides similar functionality over arbitrary Euclidean rings.

##### 14.3-6 LcmInt
 ‣ LcmInt( m, n ) ( function )

returns the least common multiple of the integers m and n.

LcmInt is a method used by the general operation Lcm (56.7-6).

gap> LcmInt( 123, 66 );
2706


 ‣ CoefficientsQadic( i, q ) ( operation )

returns the q-adic representation of the integer i as a list l of coefficients satisfying the equality i = ∑_{j = 0} q^j ⋅ l[j+1] for an integer q > 1.

gap> l:=CoefficientsQadic(462,3);
[ 0, 1, 0, 2, 2, 1 ]


 ‣ CoefficientsMultiadic( ints, int ) ( function )

returns the multiadic expansion of the integer int modulo the integers given in ints (in ascending order). It returns a list of coefficients in the reverse order to that in ints.

##### 14.3-9 ChineseRem
 ‣ ChineseRem( moduli, residues ) ( function )

ChineseRem returns the combination of the residues modulo the moduli, i.e., the unique integer c from [0..Lcm(moduli)-1] such that c = residues[i] modulo moduli[i] for all i, if it exists. If no such combination exists ChineseRem signals an error.

Such a combination does exist if and only if residues[i] = residues[k] mod Gcd( moduli[i], moduli[k] ) for every pair i, k. Note that this implies that such a combination exists if the moduli are pairwise relatively prime. This is called the Chinese remainder theorem.

gap> ChineseRem( [ 2, 3, 5, 7 ], [ 1, 2, 3, 4 ] );
53
gap> ChineseRem( [ 6, 10, 14 ], [ 1, 3, 5 ] );
103

gap> ChineseRem( [ 6, 10, 14 ], [ 1, 2, 3 ] );
Error, the residues must be equal modulo 2 called from
<function>( <arguments> ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> gap>


##### 14.3-10 PowerModInt
 ‣ PowerModInt( r, e, m ) ( function )

returns r^e mod m for integers r, e and m.

Note that PowerModInt can reduce intermediate results and thus will generally be faster than using r^e mod m, which would compute r^e first and reduces the result afterwards.

PowerModInt is a method for the general operation PowerMod (56.7-9).

#### 14.4 Prime Integers and Factorization

##### 14.4-1 Primes
 ‣ Primes ( global variable )

Primes is a strictly sorted list of the 168 primes less than 1000.

This is used in IsPrimeInt (14.4-2) and FactorsInt (14.4-7) to cast out small primes quickly.

gap> Primes;
2
gap> Primes;
541


##### 14.4-2 IsPrimeInt
 ‣ IsPrimeInt( n ) ( function )
 ‣ IsProbablyPrimeInt( n ) ( function )

IsPrimeInt returns false if it can prove that the integer n is composite and true otherwise. By convention IsPrimeInt(0) = IsPrimeInt(1) = false and we define IsPrimeInt(-n) = IsPrimeInt(n).

IsPrimeInt will return true for every prime n. IsPrimeInt will return false for all composite n < 10^18 and for all composite n that have a factor p < 1000. So for integers n < 10^18, IsPrimeInt is a proper primality test. It is conceivable that IsPrimeInt may return true for some composite n > 10^18, but no such n is currently known. So for integers n > 10^18, IsPrimeInt is a probable-primality test. IsPrimeInt will issue a warning when its argument is probably prime but not a proven prime. (The function IsProbablyPrimeInt will do a similar calculation but not issue a warning.) The warning can be switched off by SetInfoLevel( InfoPrimeInt, 0 );, the default level is 1 (also see SetInfoLevel (7.4-3) ).

If composites that fool IsPrimeInt do exist, they would be extremely rare, and finding one by pure chance might be less likely than finding a bug in GAP. We would appreciate being informed about any example of a composite number n for which IsPrimeInt returns true.

IsPrimeInt is a deterministic algorithm, i.e., the computations involve no random numbers, and repeated calls will always return the same result. IsPrimeInt first does trial divisions by the primes less than 1000. Then it tests that n is a strong pseudoprime w.r.t. the base 2. Finally it tests whether n is a Lucas pseudoprime w.r.t. the smallest quadratic nonresidue of n. A better description can be found in the comment in the library file primality.gi.

The time taken by IsPrimeInt is approximately proportional to the third power of the number of digits of n. Testing numbers with several hundreds digits is quite feasible.

IsPrimeInt is a method for the general operation IsPrime (56.5-8).

Remark: In future versions of GAP we hope to change the definition of IsPrimeInt to return true only for proven primes (currently, we lack a sufficiently good primality proving function). In applications, use explicitly IsPrimeInt or IsProbablyPrimeInt with this change in mind.

gap> IsPrimeInt( 2^31 - 1 );
true
gap> IsPrimeInt( 10^42 + 1 );
false


##### 14.4-3 PrimalityProof
 ‣ PrimalityProof( n ) ( function )

Construct a machine verifiable proof of the primality of (the probable prime) n, following the ideas of [BLS75]. The proof consists of various Fermat and Lucas pseudoprimality tests, which taken as a whole prove the primality. The proof is represented as a list of witnesses of two kinds. The first kind, [ "F", divisor, base ], indicates a successful Fermat pseudoprimality test, where n is a strong pseudoprime at base with order not divisible by (n-1)/divisor. The second kind, [ "L", divisor, discriminant, P ] indicates a successful Lucas pseudoprimality test, for a quadratic form of given discriminant and middle term P with an extra check at (n+1)/divisor.

##### 14.4-4 IsPrimePowerInt
 ‣ IsPrimePowerInt( n ) ( function )

IsPrimePowerInt returns true if the integer n is a prime power and false otherwise.

An integer n is a prime power if there exists a prime p and a positive integer i such that p^i = n. If n is negative the condition is that there must exist a negative prime p and an odd positive integer i such that p^i = n. The integers 1 and -1 are not prime powers.

Note that IsPrimePowerInt uses SmallestRootInt (14.2-10) and a probable-primality test (see IsPrimeInt (14.4-2)).

gap> IsPrimePowerInt( 31^5 );
true
gap> IsPrimePowerInt( 2^31-1 );  # 2^31-1 is actually a prime
true
gap> IsPrimePowerInt( 2^63-1 );
false
gap> Filtered( [-10..10], IsPrimePowerInt );
[ -8, -7, -5, -3, -2, 2, 3, 4, 5, 7, 8, 9 ]


##### 14.4-5 NextPrimeInt
 ‣ NextPrimeInt( n ) ( function )

NextPrimeInt returns the smallest prime which is strictly larger than the integer n.

Note that NextPrimeInt uses a probable-primality test (see IsPrimeInt (14.4-2)).

gap> NextPrimeInt( 541 ); NextPrimeInt( -1 );
547
2


##### 14.4-6 PrevPrimeInt
 ‣ PrevPrimeInt( n ) ( function )

PrevPrimeInt returns the largest prime which is strictly smaller than the integer n.

Note that PrevPrimeInt uses a probable-primality test (see IsPrimeInt (14.4-2)).

gap> PrevPrimeInt( 541 ); PrevPrimeInt( 1 );
523
-2


##### 14.4-7 FactorsInt
 ‣ FactorsInt( n ) ( function )
 ‣ FactorsInt( n: RhoTrials := trials ) ( function )

FactorsInt returns a list of factors of a given integer n such that Product( FactorsInt( n ) ) = n. If |n| ≤ 1 the list [n] is returned. Otherwise the result contains probable primes, sorted by absolute value. The entries will all be positive except for the first one in case of a negative n.

See PrimeDivisors (14.4-8) for a function that returns a set of (probable) primes dividing n.

Note that FactorsInt uses a probable-primality test (see IsPrimeInt (14.4-2)). Thus FactorsInt might return a list which contains composite integers. In such a case you will get a warning about the use of a probable prime. You can switch off these warnings by SetInfoLevel( InfoPrimeInt, 0 ); (also see SetInfoLevel (7.4-3)).

The time taken by FactorsInt is approximately proportional to the square root of the second largest prime factor of n, which is the last one that FactorsInt has to find, since the largest factor is simply what remains when all others have been removed. Thus the time is roughly bounded by the fourth root of n. FactorsInt is guaranteed to find all factors less than 10^6 and will find most factors less than 10^10. If n contains multiple factors larger than that FactorsInt may not be able to factor n and will then signal an error.

FactorsInt is used in a method for the general operation Factors (56.5-9).

In the second form, FactorsInt calls FactorsRho with a limit of trials on the number of trials it performs. The default is 8192. Note that Pollard's Rho is the fastest method for finding prime factors with roughly 5-10 decimal digits, but becomes more and more inferior to other factorization techniques like e.g. the Elliptic Curves Method (ECM) the bigger the prime factors are. Therefore instead of performing a huge number of Rho trials, it is usually more advisable to install the FactInt package and then simply to use the operation Factors (56.5-9). The factorization of the 8-th Fermat number by Pollard's Rho below takes already a while.

gap> FactorsInt( -Factorial(6) );
[ -2, 2, 2, 2, 3, 3, 5 ]
gap> Set( FactorsInt( Factorial(13)/11 ) );
[ 2, 3, 5, 7, 13 ]
gap> FactorsInt( 2^63 - 1 );
[ 7, 7, 73, 127, 337, 92737, 649657 ]
gap> FactorsInt( 10^42 + 1 );
[ 29, 101, 281, 9901, 226549, 121499449, 4458192223320340849 ]
gap> FactorsInt(2^256+1:RhoTrials:=100000000);
[ 1238926361552897,
93461639715357977769163558199606896584051237541638188580280321 ]


##### 14.4-8 PrimeDivisors
 ‣ PrimeDivisors( n ) ( attribute )

PrimeDivisors returns for a non-zero integer n a set of its positive (probable) primes divisors. In rare cases the result could contain a composite number which passed certain primality tests, see IsProbablyPrimeInt (14.4-2) and FactorsInt (14.4-7) for more details.

gap> PrimeDivisors(-12);
[ 2, 3 ]
gap> PrimeDivisors(1);
[  ]


##### 14.4-9 PartialFactorization
 ‣ PartialFactorization( n[, effort] ) ( operation )

PartialFactorization returns a partial factorization of the integer n. No assertions are made about the primality of the factors, except of those mentioned below.

The argument effort, if given, specifies how intensively the function should try to determine factors of n. The default is effort = 5.

• If effort = 0, trial division by the primes below 100 is done. Returned factors below 10^4 are guaranteed to be prime.

• If effort = 1, trial division by the primes below 1000 is done. Returned factors below 10^6 are guaranteed to be prime.

• If effort = 2, additionally trial division by the numbers in the lists Primes2 and ProbablePrimes2 is done, and perfect powers are detected. Returned factors below 10^6 are guaranteed to be prime.

• If effort = 3, additionally FactorsRho (Pollard's Rho) with RhoTrials = 256 is used.

• If effort = 4, as above, but RhoTrials = 2048.

• If effort = 5, as above, but RhoTrials = 8192. Returned factors below 10^12 are guaranteed to be prime, and all prime factors below 10^6 are guaranteed to be found.

• If effort = 6 and the package FactInt is loaded, in addition to the above quite a number of special cases are handled.

• If effort = 7 and the package FactInt is loaded, the only thing which is not attempted to obtain a full factorization into Baillie-Pomerance-Selfridge-Wagstaff pseudoprimes is the application of the MPQS to a remaining composite with more than 50 decimal digits.

Increasing the value of the argument effort by one usually results in an increase of the runtime requirements by a factor of (very roughly!) 3 to 10. (Also see CheapFactorsInt (EDIM: CheapFactorsInt)).

gap> List([0..5],i->PartialFactorization(97^35-1,i));
[ [ 2, 2, 2, 2, 2, 3, 11, 31, 43,
2446338959059521520901826365168917110105972824229555319002965029 ],
[ 2, 2, 2, 2, 2, 3, 11, 31, 43, 967,
2529823122088440042297648774735177983563570655873376751812787 ],
[ 2, 2, 2, 2, 2, 3, 11, 31, 43, 967,
2529823122088440042297648774735177983563570655873376751812787 ],
[ 2, 2, 2, 2, 2, 3, 11, 31, 43, 967, 39761, 262321,
242549173950325921859769421435653153445616962914227 ],
[ 2, 2, 2, 2, 2, 3, 11, 31, 43, 967, 39761, 262321, 687121,
352993394104278463123335513593170858474150787 ],
[ 2, 2, 2, 2, 2, 3, 11, 31, 43, 967, 39761, 262321, 687121,
20241187, 504769301, 34549173843451574629911361501 ] ]


##### 14.4-10 PrintFactorsInt
 ‣ PrintFactorsInt( n ) ( function )

prints the prime factorization of the integer n in human-readable form. See also StringPP (27.7-9).

gap> PrintFactorsInt( Factorial( 7 ) ); Print( "\n" );
2^4*3^2*5*7


##### 14.4-11 PrimePowersInt
 ‣ PrimePowersInt( n ) ( function )

returns the prime factorization of the integer n as a list [ p_1, e_1, ..., p_k, e_k ] with n = p_1^{e_1} ⋅ p_2^{e_2} ⋅ ... ⋅ p_k^{e_k}.

For negative integers, the absolute value is taken. Zero is not allowed as input.

gap> PrimePowersInt( Factorial( 7 ) );
[ 2, 4, 3, 2, 5, 1, 7, 1 ]
gap> PrimePowersInt( 1 );
[  ]


##### 14.4-12 DivisorsInt
 ‣ DivisorsInt( n ) ( function )

DivisorsInt returns a list of all divisors of the integer n. The list is sorted, so that it starts with 1 and ends with n. We define that DivisorsInt( -n ) = DivisorsInt( n ).

Since the set of divisors of 0 is infinite calling DivisorsInt( 0 ) causes an error.

DivisorsInt may call FactorsInt (14.4-7) to obtain the prime factors. Sigma (15.5-1) and Tau (15.5-2) compute the sum and the number of positive divisors, respectively.

gap> DivisorsInt( 1 ); DivisorsInt( 20 ); DivisorsInt( 541 );
[ 1 ]
[ 1, 2, 4, 5, 10, 20 ]
[ 1, 541 ]


#### 14.5 Residue Class Rings

ZmodnZ (14.5-2) returns a residue class ring of Integers (14) modulo an ideal. These residue class rings are rings, thus all operations for rings (see Chapter 56) apply. See also Chapters 59 and 15.

##### 14.5-1 \mod
 ‣ \mod( r/s, n ) ( operation )

If r, s and n are integers, r / s as a reduced fraction is p/q, where q and n are coprime, then r / s mod n is defined to be the product of p and the inverse of q modulo n. See Section 4.14 for more details and definitions.

With the above definition, 4 / 6 mod 32 equals 2 / 3 mod 32 and hence exists (and is equal to 22), despite the fact that 6 has no inverse modulo 32.

##### 14.5-2 ZmodnZ
 ‣ ZmodnZ( n ) ( function )
 ‣ ZmodpZ( p ) ( function )
 ‣ ZmodpZNC( p ) ( function )

ZmodnZ returns a ring R isomorphic to the residue class ring of the integers modulo the ideal generated by n. The element corresponding to the residue class of the integer i in this ring can be obtained by i * One( R ), and a representative of the residue class corresponding to the element x ∈ R can be computed by Int( x ).

ZmodnZ( n ) is equal to Integers mod n.

ZmodpZ does the same if the argument p is a prime integer, additionally the result is a field. ZmodpZNC omits the check whether p is a prime.

Each ring returned by these functions contains the whole family of its elements if n is not a prime, and is embedded into the family of finite field elements of characteristic n if n is a prime.

##### 14.5-3 ZmodnZObj
 ‣ ZmodnZObj( Fam, r ) ( operation )
 ‣ ZmodnZObj( r, n ) ( operation )

If the first argument is a residue class family Fam then ZmodnZObj returns the element in Fam whose coset is represented by the integer r.

If the two arguments are an integer r and a positive integer n then ZmodnZObj returns the element in ZmodnZ( n ) (see ZmodnZ (14.5-2)) whose coset is represented by the integer r.

gap> r:= ZmodnZ(15);
(Integers mod 15)
gap> fam:=ElementsFamily(FamilyObj(r));;
gap> a:= ZmodnZObj(fam,9);
ZmodnZObj( 9, 15 )
gap> a+a;
ZmodnZObj( 3, 15 )
gap> Int(a+a);
3


##### 14.5-4 IsZmodnZObj
 ‣ IsZmodnZObj( obj ) ( category )
 ‣ IsZmodnZObjNonprime( obj ) ( category )
 ‣ IsZmodpZObj( obj ) ( category )
 ‣ IsZmodpZObjSmall( obj ) ( category )
 ‣ IsZmodpZObjLarge( obj ) ( category )

The elements in the rings Z / n Z are in the category IsZmodnZObj. If n is a prime then the elements are of course also in the category IsFFE (59.1-1), otherwise they are in IsZmodnZObjNonprime. IsZmodpZObj is an abbreviation of IsZmodnZObj and IsFFE. This category is the disjoint union of IsZmodpZObjSmall and IsZmodpZObjLarge, the former containing all elements with n at most MAXSIZE_GF_INTERNAL.

The reasons to distinguish the prime case from the nonprime case are

• that objects in IsZmodnZObjNonprime have an external representation (namely the residue in the range [ 0, 1, ..., n-1 ]),

• that the comparison of elements can be defined as comparison of the residues, and

• that the elements lie in a family of type IsZmodnZObjNonprimeFamily (note that for prime n, the family must be an IsFFEFamily).

The reasons to distinguish the small and the large case are that for small n the elements must be compatible with the internal representation of finite field elements, whereas we are free to define comparison as comparison of residues for large n.

Note that we cannot claim that every finite field element of degree 1 is in IsZmodnZObj, since finite field elements in internal representation may not know that they lie in the prime field.

#### 14.6 Check Digits

##### 14.6-1 CheckDigitISBN
 ‣ CheckDigitISBN( n ) ( function )
 ‣ CheckDigitISBN13( n ) ( function )
 ‣ CheckDigitPostalMoneyOrder( n ) ( function )
 ‣ CheckDigitUPC( n ) ( function )

These functions can be used to compute, or check, check digits for some everyday items. In each case what is submitted as input is either the number with check digit (in which case the function returns true or false), or the number without check digit (in which case the function returns the missing check digit). The number can be specified as integer, as string (for example in case of leading zeros) or as a sequence of arguments, each representing a single digit. The check digits tested are the 10-digit ISBN (International Standard Book Number) using CheckDigitISBN (since arithmetic is module 11, a digit 11 is represented by an X); the newer 13-digit ISBN-13 using CheckDigitISBN13; the numbers of 11-digit US postal money orders using CheckDigitPostalMoneyOrder; and the 12-digit UPC bar code found on groceries using CheckDigitUPC.

gap> CheckDigitISBN("052166103");
Check Digit is 'X'
'X'
gap> CheckDigitISBN("052166103X");
Checksum test satisfied
true
gap> CheckDigitISBN(0,5,2,1,6,6,1,0,3,1);
Checksum test failed
false
gap> CheckDigitISBN(0,5,2,1,6,6,1,0,3,'X'); # note single quotes!
Checksum test satisfied
true
gap> CheckDigitISBN13("9781420094527");
Checksum test satisfied
true
gap> CheckDigitUPC("07164183001");
Check Digit is 1
1
gap> CheckDigitPostalMoneyOrder(16786457155);
Checksum test satisfied
true


##### 14.6-2 CheckDigitTestFunction
 ‣ CheckDigitTestFunction( l, m, f ) ( function )

This function creates check digit test functions such as CheckDigitISBN (14.6-1) for check digit schemes that use the inner products with a fixed vector modulo a number. The scheme creates will use strings of l digits (including the check digits), the check consists of taking the standard product of the vector of digits with the fixed vector f modulo m; the result needs to be 0. The function returns a function that then can be used for testing or determining check digits.

gap> isbntest:=CheckDigitTestFunction(10,11,[1,2,3,4,5,6,7,8,9,-1]);
function( arg... ) ... end
gap> isbntest("038794680");
Check Digit is 2
2


#### 14.7 Random Sources

GAP provides Random (30.7-1) methods for many collections of objects. On a lower level these methods use random sources which provide random integers and random choices from lists.

See IsRandomSource (14.7-1) for the user interface for random sources, and Section 14.7-6 for information about developing new kinds of random sources.

##### 14.7-1 IsRandomSource
 ‣ IsRandomSource( obj ) ( category )

This is the category of random source objects. The user interface for these objects consists of the following functions.

RandomSource (14.7-5) creates a new random source rs, say.

Random( rs, list ) yields a random element of the list list, and Random( rs, low, high ) yields a random integer between low and high (inclusive), see Random (14.7-2).

If rs supports resetting (see State (14.7-3)) then State( rs ) yields a copy state, say, of the current state of rs such that Reset( rs, state ) resets rs to the given state.

One idea behind providing several independent (pseudo) random sources is to make algorithms which use some sort of random choices deterministic. They can use their own new random source created with a fixed seed and so do exactly the same in different calls.

Random source objects lie in the family RandomSourcesFamily.

##### 14.7-2 Random
 ‣ Random( rs, list ) ( operation )
 ‣ Random( rs, coll ) ( operation )
 ‣ Random( rs, low, high ) ( operation )

This operation returns a random element from the dense, nonempty list list or the nonempty collection coll, or an integer in the range from the given (possibly large) integers low to high, respectively.

The choice should only depend on the random source rs and have no effect on other random sources.

It is not defined what happens if list or coll is empty, list is not dense, or low is larger than high.

gap> mysource := RandomSource(IsMersenneTwister, 42);;
gap> Random(mysource, 1, 10^60);
999331861769949319194941485000557997842686717712198687315183


##### 14.7-3 State and Reset for Random Sources
 ‣ State( rs ) ( operation )
 ‣ Reset( rs[, seed] ) ( operation )

These are the basic operations for random sources (see IsRandomSource (14.7-1)).

State returns a data structure which admits recovering the state of the random source such that a sequence of random calls using this random source can be reproduced. If a random source cannot be reset (say, it uses truly random physical data) then State returns fail.

Reset( rs, seed ) resets the random source rs to a state described by seed, if the random source can be reset; otherwise it does nothing. Here seed can be an output of State and then rs gets reset to that state. For historical reasons, random sources accept integer values as seed. We recommend that new code should not rely on this; always use the output of a prior call to State as seed, or omit it. Without the seed argument a fixed default seed is used. Reset returns the state of rs before the call.

Most methods for Random (30.7-1) in the GAP library that do not take a random source as argument use the GlobalMersenneTwister (14.7-4) as random source. It can be reset into a known state as in the following example.

gap> seed := Reset(GlobalMersenneTwister);;
gap> seed = State(GlobalMersenneTwister);
true
gap> List([1..10],i->Random(Integers));
[ -3, 2, -1, -2, -1, -1, 1, -4, 1, 0 ]
gap> List([1..10],i->Random(Integers));
[ -1, -1, -1, 1, -1, 1, -2, -1, -2, 0 ]
gap> Reset(GlobalMersenneTwister, seed);;
gap> List([1..10],i->Random(Integers));
[ -3, 2, -1, -2, -1, -1, 1, -4, 1, 0 ]


##### 14.7-4 Kinds of Random Sources
 ‣ IsMersenneTwister( rs ) ( category )
 ‣ IsGAPRandomSource( rs ) ( category )
 ‣ IsGlobalRandomSource( rs ) ( category )
 ‣ GlobalMersenneTwister ( global variable )
 ‣ GlobalRandomSource ( global variable )

Currently, the GAP library provides three types of random sources, distinguished by the three listed categories.

IsMersenneTwister are random sources which use a fast random generator of 32 bit numbers, called the Mersenne twister. The pseudo random sequence has a period of 2^19937-1 and the numbers have a 623-dimensional equidistribution. For more details and the origin of the code used in the GAP kernel, see: http://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html.

Use the Mersenne twister if possible, in particular for generating many large random integers.

There is also a predefined global random source GlobalMersenneTwister which is used as the default random source by those library methods for Random (30.7-1) that do not take a random source as an argument.

IsGAPRandomSource uses the same number generator as IsGlobalRandomSource, but you can create several of these random sources which generate their random numbers independently of all other random sources.

IsGlobalRandomSource gives access to the classical global random generator which was used by GAP in former releases. You do not need to construct new random sources of this kind which would all use the same global data structure. Just use the existing random source GlobalRandomSource. This uses the additive random number generator described in [Knu98] (Algorithm A in 3.2.2 with lag 30).

Other kinds of random sources are implemented by GAP packages.

##### 14.7-5 RandomSource
 ‣ RandomSource( cat[, seed] ) ( operation )

This operation is used to create new random sources. The first argument cat is the category describing the type of the random generator, for example one of the categories listed in Section 14.7-4.

An optional seed can be given to specify the initial state. For details, see Section 14.7-3.

gap> rs1 := RandomSource(IsMersenneTwister);
<RandomSource in IsMersenneTwister>
gap> l1 := List([1..10000], i-> Random(rs1, [1..6]));;
gap> state1 := State(rs1);;
gap> rs2 := RandomSource(IsMersenneTwister);;
gap> l2 := List([1..10000], i-> Random(rs2, [1..6]));;
gap> l1 = l2;
true
gap> l3 := List([1..10000], i-> Random(rs1, [1..6]));;
gap> l1 = l3;
false
gap> rs3 := RandomSource(IsMersenneTwister, state1);;
gap> l4 := List([1..10000], i-> Random(rs3, [1..6]));;
gap> l3 = l4;
true
gap> n := Random(rs1, 1, 2^220);
1077726777923092117987668044202944212469136000816111066409337432400


##### 14.7-6 Implementing new kinds of random sources

If one wants to implement a new kind of random sources then the first step is the declaration of a new category C, say, that implies IsRandomSource (14.7-1), analogous to the categories listed in Section 14.7-4, as follows.

DeclareCategory( "C", IsRandomSource );.

Then the following method installations are needed.

InstallMethod( Init, [ C, IsObject ], function( prers, seed ) ... end );

Here prers is an empty component object (which has already the filter C), and seed is an integer or a state value as returned by State (14.7-3) that describes the initial state of the random source. The function should fill in the actual data and then return the (now initialized) object prers. The default used for seed is the integer 1. A given state value need not be copied by the function.

InstallMethod( Random, [ C, IsInt, IsInt ], function( rs, low, high ) ... end );

Here rs is an already initialized random source object in the category C, and the function returns an integer between low and high (inclusive). It is not defined what happens when low is larger than high.

InstallMethod( State, [ C ], function( rs ) ... end );

If rs supports resetting then the function must return an object that describes the current state of rs. This object must be an independent copy, that is, calling Random (14.7-2) for rs must not change the object that was returned by State (14.7-3); otherwise ReturnFail (5.4-3) should be installed.

InstallMethod( Reset, [ C, IsObject ], function( rs, seed ) ... end );

If rs supports resetting then the function must reinitialize rs to the integer or State (14.7-3) value seed and must return the State (14.7-3) value of rs before these changes; if resetting is not supported then ReturnNothing (5.4-4) should be installed. Reset need not copy a given state. Note that the generic unary Reset (14.7-3) method uses the default seed 1.

Examples of implementations as described here are given by the random sources with defining filter IsMersenneTwister (14.7-4) or IsRealRandomSource. (For the latter, see RandomSource (IO: RandomSource) in the GAP package IO.)

#### 14.8 Bitfields

Bitfields are a low-level feature intended to support efficient subdivision of immediate integers into bitfields of various widths. This is typically useful in implementing space-efficient and/or cache-efficient data structures. This feature should be used with care because (inter alia) it has different limitations on 32-bit and 64-bit architectures.

##### 14.8-1 MakeBitfields
 ‣ MakeBitfields( width.... ) ( function )

This function sets up the machinery for a set of bitfields of the given widths. All bitfield values are treated as unsigned. The total of the widths must not exceed 60 bits on 64-bit architecture or 28 bits on a 32-bit architecture. For performance reasons some checks that one might wish to do are ommitted. In particular, the builder and setter functions do not check if the value[s] passed to them are negative or too large (unless GAP is specially compiled for debugging). Behaviour when such arguments are passed is undefined. You can tell which type of architecture you are running on by acccessing GAPInfo.BytesPerVariable which is 8 on 64-bits and 4 on 32. The return value when n widths are given is a record whose fields are

widths

a copy of the arguments, for convenience,

getters

a list of n functions of one argument each of which extracts one of the fields from an immediate integer

setters

a list of n functions each taking two arguments: a packed value and a new value for one of its fields and returning a new packed value. The ith function returned the new packed value in which the ith field has been replaced by the new value. Note that this does NOT modify the original packed value.

Two additional fields may be present if any of the field widths is one. Each is a list and only has entried bound in the positions corresponding to the width 1 fields.

booleanGetters

if the ith position of this list is set, it contains a function which extracts the ith field (which will have width one) and returns true if it contains 1 and false if it contains 0

booleanSetters

if the ith position of this list is set, it contains a function of two arguments. The first argument is a packed value, the second is true or false. It returns a new packed value in which the ith field is set to 1 if the second argument was true and 0 if it was false. Behaviour for any other value is undefined.

##### 14.8-2 BuildBitfields
 ‣ BuildBitfields( widths, vals... ) ( function )

This function takes one or more argument. Its first argument is a list of field widths, as found in the widths entry of a record returned by MakeBitfields. The remaining arguments are unsigned integer values, equal in number to the entries of the list of field widths. It returns a small integer in which those entries are packed into bitfields of the given widths. The first entry occupies the least significant bits. DeclareGlobalFunction("BuildBitfields");

gap> bf := MakeBitfields(1,2,3);
rec( booleanGetters := [ function( data ) ... end ],
booleanSetters := [ function( data, val ) ... end ],
getters := [ function( data ) ... end, function( data ) ... end,
function( data ) ... end ],
setters := [ function( data, val ) ... end, function( data, val ) ... end,
function( data, val ) ... end ], widths := [ 1, 2, 3 ] )
gap> x := BuildBitfields(bf.widths,0,3,5);
46
gap> bf.getters(x);
5
gap> y := bf.setters(x,1);
47
gap> x;
46
gap> bf.booleanGetters(x);
false
gap> bf.booleanGetters(y);
true


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