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
Many more functions that are mainly related to the prime residue group of integers modulo an integer are described in chapter Number Theory, and functions dealing with combinatorics can be found in chapter Combinatorics.
Integers V
PositiveIntegers V
NonnegativeIntegers V
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
IsIntegers( obj ) C
IsPositiveIntegers( obj ) C
IsNonnegativeIntegers( obj ) C
are the defining categories for the domains Integers,
PositiveIntegers, and NonnegativeIntegers.
gap> IsIntegers( Integers ); IsIntegers( Rationals ); IsIntegers( 7 ); true false false
Integers is a subset of Rationals, which is a subset of Cyclotomics.
See Chapter Cyclotomic Numbers for arithmetic operations and comparison of
integers.
IsInt( obj ) C
Every rational integer lies in the category IsInt,
which is a subcategory of IsRat (see Rationals).
IsPosInt( obj ) C
Every positive integer lies in the category IsPosInt.
Int( elm ) A
Int returns an integer int whose meaning depends on the type
of elm.
If elm is a rational number then int is the integer part of the quotient of numerator and denominator of elm (see QuoInt).
If elm is an element of a finite prime field then int is the smallest
nonnegative integer such that elm = int * One( elm ).
If elm is a string consisting of digits '0', '1', ¼,
'9' and '-' (at the first position) then int is the
integer described by this string.
The operation String (see String) 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
IsEvenInt( n ) F
tests if the integer n is divisible by 2.
IsOddInt( n ) F
tests if the integer n is not divisible by 2.
AbsInt( n ) F
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
see EuclideanDegree).
gap> AbsInt( 33 ); 33 gap> AbsInt( -214378 ); 214378 gap> AbsInt( 0 ); 0
SignInt( n ) F
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 ) F
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 exp such that baseexp £ n. LogInt
will signal an error if either n or base is not positive.
LogInt is a method for the general operation Log.
gap> LogInt( 1030, 2 ); 10 # 2^10 = 1024 gap> LogInt( 1, 10 ); 0
RootInt( n ) F
RootInt( n, k ) F
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 rk £ n. 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 # 7^5 = 16807
SmallestRootInt( n ) F
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 = rk.
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
QuoInt( n, m ) F
QuoInt returns the integer part of the quotient of its integer
operands.
If n and m are positive QuoInt( n, m ) is the largest
positive integer q such that q \* m £ n .
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 (see EuclideanQuotient).
gap> QuoInt(5,3); QuoInt(-5,3); QuoInt(5,-3); QuoInt(-5,-3); 1 -1 -1 1
BestQuoInt( n, m ) F
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
RemInt( n, m ) F
RemInt returns the remainder of its two integer operands.
If m is not equal to zero
RemInt( n, m ) = n - m * QuoInt( n, m ).
Note that the rules given for QuoInt imply that RemInt( n, m )
has the same sign as n and its absolute value is strictly less than the
absolute value of m.
Note also that RemInt( n, m ) = 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 (see EuclideanRemainder).
gap> RemInt(5,3); RemInt(-5,3); RemInt(5,-3); RemInt(-5,-3); 2 -2 2 -2
GcdInt( a, b ) F
GcdInt returns the greatest common divisor of its two integer operands.
GcdInt is a method used by the general function Gcd (see Gcd).
gap> GcdInt( 123, 66 ); 3
Gcdex( m, n ) F
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(
gcd := 3,
coeff1 := 7,
coeff2 := -13,
coeff3 := -22,
coeff4 := 41 )
# 3 = 7*123 - 13*66, 0 = -22*123 + 41*66
gap> Gcdex( 0, -3 );
rec(
gcd := 3,
coeff1 := 0,
coeff2 := -1,
coeff3 := 1,
coeff4 := 0 )
gap> Gcdex( 0, 0 );
rec(
gcd := 0,
coeff1 := 1,
coeff2 := 0,
coeff3 := 0,
coeff4 := 1 )
LcmInt( m, n ) F
returns the least common multiple of the integers m and n.
LcmInt is a method used by the general function Lcm.
gap> LcmInt( 123, 66 ); 2706
CoefficientsQadic( i, q ) F
returns the q-adic representation of the integer i as a list l of coefficients where i = åj = 0 qj ·l[j+1].
CoefficientsMultiadic( ints, int ) F
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.
ChineseRem( moduli, residues ) F
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
PowerModInt( r, e, m ) F
returns re mod m for integers r,e and m ( e ³ 0).
Note that using r ^ e mod m will generally be slower,
because it can not reduce intermediate results the way PowerModInt
does but would compute r^e first and then reduce the result
afterwards.
PowerModInt is a method for the general operation PowerMod.
Primes V
Primes is a strictly sorted list of the 168 primes less than 1000.
This is used in IsPrimeInt and FactorsInt to cast out small primes
quickly.
gap> Primes[1]; 2 gap> Primes[100]; 541
IsPrimeInt( n ) F
IsProbablyPrimeInt( n ) F
IsPrimeInt returns false if it can prove that 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 < 1013 and for all composite n
that have a factor p < 1000. So for integers n < 1013,
IsPrimeInt is a proper primality test. It is conceivable that
IsPrimeInt may return true for some composite n > 1013, but no
such n is currently known. So for integers n > 1013, IsPrimeInt
is a probable-primality test. Therefore IsPrimeInt will issue a
warning when called with an argument > 1013. (The function
IsProbablyPrimeInt will do the same calculations but not issue a
warning.)
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 integer.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.
gap> IsPrimeInt( 2^31 - 1 ); true gap> IsPrimeInt( 10^42 + 1 ); false
IsPrimePowerInt( n ) F
IsPrimePowerInt returns true if the integer n is a prime power and
false otherwise.
n is a prime power if there exists a prime p and a positive integer i such that pi = n. If n is negative the condition is that there must exist a negative prime p and an odd positive integer i such that pi = n. 1 and -1 are not prime powers.
Note that IsPrimePowerInt uses SmallestRootInt (see
SmallestRootInt) and a probable-primality test (see IsPrimeInt).
gap> IsPrimePowerInt( 31^5 );
true
gap> IsPrimePowerInt( 2^31-1 );
true # $2^{31}-1$ is actually a prime
gap> IsPrimePowerInt( 2^63-1 );
false
gap> Filtered( [-10..10], IsPrimePowerInt );
[ -8, -7, -5, -3, -2, 2, 3, 4, 5, 7, 8, 9 ]
NextPrimeInt( n ) F
NextPrimeInt returns the smallest prime which is strictly larger than
the integer n.
Note that NextPrimeInt uses a probable-primality test (see
IsPrimeInt).
gap> NextPrimeInt( 541 ); NextPrimeInt( -1 ); 547 2
PrevPrimeInt( n ) F
PrevPrimeInt returns the largest prime which is strictly smaller than
the integer n.
Note that PrevPrimeInt uses a probable-primality test (see
IsPrimeInt).
gap> PrevPrimeInt( 541 ); PrevPrimeInt( 1 ); 523 -2
FactorsInt( n ) F
FactorsInt( n:RhoTrials:=trials ) F
FactorsInt returns a list of prime factors of the integer n.
If the ith power of a prime divides n this prime appears i times.
The list is sorted, that is the smallest prime factors come first.
The first element has the same sign as n, the others are positive.
For any integer n it holds that Product( FactorsInt( n ) ) = n.
Note that FactorsInt uses a probable-primality test (see IsPrimeInt).
Thus FactorsInt might return a list which contains composite integers.
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 106 and will find most factors less than
1010. 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.
In the second form, FactorsInt calls FactorsRho with a limit of trials on the number of trials is performs. The default is 8192.
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 ]
PrintFactorsInt( n ) F
prints the prime factorization of the integer n in human-readable form.
gap> PrintFactorsInt( Factorial( 7 ) ); Print( "\n" ); 2^4*3^2*5*7
PrimePowersInt( n ) F
returns the prime factorization of the integer n as a list [ p1, e1, ¼, pn, en ] with n = Õi = 1n piei.
gap> PrimePowersInt( Factorial( 7 ) ); [ 2, 4, 3, 2, 5, 1, 7, 1 ]
DivisorsInt( n ) F
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 Divisors( -n ) = Divisors( n ).
Since the set of divisors of 0 is infinite calling DivisorsInt( 0 )
causes an error.
DivisorsInt may call FactorsInt to obtain the prime factors.
Sigma and Tau (see Sigma and Tau) 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 ]
ZmodnZ( n ) F
ZmodpZ( p ) F
ZmodpZNC( p ) F
ZmodnZ returns a ring R isomorphic to the residue class ring of the
integers modulo the positive integer 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 Î R can be computed by
Int( x ).
ZmodnZ( n ) is equivalent 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.
ZmodnZObj( Fam, i ) O
creates an object in the residue class family Fam whose coset is represented by integer i.
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
IsZmodnZObj( obj ) C
IsZmodnZObjNonprime( obj ) C
IsZmodpZObj( obj ) C
IsZmodpZObjSmall( obj ) C
IsZmodpZObjLarge( obj ) C
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 (see IsFFE), 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
IsZmodnZObjNonprime have an external representation
(namely the modulus in the range [ 0, 1, ... n-1 ]),
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.
The residue class rings are rings, thus all operations for rings (see Chapter Rings) apply. See also Chapters Finite fields and Number theory.
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GAP 4 manual