  
  [1X14 [33X[0;0YIntegers[133X[101X
  
  [33X[0;0YOne  of  the most fundamental datatypes in every programming language is the
  integer type. [5XGAP[105X is no exception.[133X
  
  [33X[0;0Y[5XGAP[105X integers are entered as a sequence of decimal digits optionally preceded
  by  a  [21X[10X+[110X[121X  sign  for positive integers or a [21X[10X-[110X[121X sign for negative integers. The
  size  of  integers in [5XGAP[105X is only limited by the amount of available memory,
  so you can compute with integers having thousands of digits.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X-1234;[127X[104X
    [4X[28X-1234[128X[104X
    [4X[25Xgap>[125X [27X123456789012345678901234567890123456789012345678901234567890;[127X[104X
    [4X[28X123456789012345678901234567890123456789012345678901234567890[128X[104X
  [4X[32X[104X
  
  [33X[0;0YMany  more  functions  that are mainly related to the prime residue group of
  integers  modulo  an  integer  are  described  in  chapter [14X15[114X, and functions
  dealing with combinatorics can be found in chapter [14X16[114X.[133X
  
  
  [1X14.1 [33X[0;0YIntegers: Global Variables[133X[101X
  
  [1X14.1-1 Integers[101X
  
  [29X[2XIntegers[102X[32X global variable
  [29X[2XPositiveIntegers[102X[32X global variable
  [29X[2XNonnegativeIntegers[102X[32X global variable
  
  [33X[0;0YThese  global  variables represent the ring of integers and the semirings of
  positive and nonnegative integers, respectively.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSize( Integers ); 2 in Integers;[127X[104X
    [4X[28Xinfinity[128X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[2XIntegers[102X is a subset of [2XRationals[102X ([14X17.1-1[114X), which is a subset of [2XCyclotomics[102X
  ([14X18.1-2[114X).  See  Chapter [14X18[114X  for  arithmetic  operations  and  comparison  of
  integers.[133X
  
  [1X14.1-2 IsIntegers[101X
  
  [29X[2XIsIntegers[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsPositiveIntegers[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsNonnegativeIntegers[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0Yare   the   defining   categories   for   the   domains  [2XIntegers[102X  ([14X14.1-1[114X),
  [2XPositiveIntegers[102X ([14X14.1-1[114X), and [2XNonnegativeIntegers[102X ([14X14.1-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsIntegers( Integers );  IsIntegers( Rationals );  IsIntegers( 7 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[28Xfalse[128X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  
  [1X14.2 [33X[0;0YElementary Operations for Integers[133X[101X
  
  [1X14.2-1 IsInt[101X
  
  [29X[2XIsInt[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YEvery rational integer lies in the category [2XIsInt[102X, which is a subcategory of
  [2XIsRat[102X ([14X17.2-1[114X).[133X
  
  [1X14.2-2 IsPosInt[101X
  
  [29X[2XIsPosInt[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YEvery positive integer lies in the category [2XIsPosInt[102X.[133X
  
  [1X14.2-3 Int[101X
  
  [29X[2XInt[102X( [3Xelm[103X ) [32X attribute
  
  [33X[0;0Y[2XInt[102X returns an integer [10Xint[110X whose meaning depends on the type of [3Xelm[103X.[133X
  
  [33X[0;0YIf [3Xelm[103X is a rational number (see Chapter [14X17[114X) then [10Xint[110X is the integer part of
  the quotient of numerator and denominator of [3Xelm[103X (see [2XQuoInt[102X ([14X14.3-1[114X)).[133X
  
  [33X[0;0YIf  [3Xelm[103X  is  an element of a finite prime field (see Chapter [14X59[114X) then [10Xint[110X is
  the smallest nonnegative integer such that [10X[3Xelm[103X[10X = int * One( [3Xelm[103X[10X )[110X.[133X
  
  [33X[0;0YIf  [3Xelm[103X is a string (see Chapter [14X27[114X) consisting of digits [10X'0'[110X, [10X'1'[110X, [22X...[122X, [10X'9'[110X
  and  [10X'-'[110X  (at  the first position) then [10Xint[110X is the integer described by this
  string.  The  operation  [2XString[102X ([14X27.6-6[114X) can be used to compute a string for
  rational integers, in fact for all cyclotomics.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XInt( 4/3 );  Int( -2/3 );[127X[104X
    [4X[28X1[128X[104X
    [4X[28X0[128X[104X
    [4X[25Xgap>[125X [27Xint:= Int( Z(5) );  int * One( Z(5) );[127X[104X
    [4X[28X2[128X[104X
    [4X[28XZ(5)[128X[104X
    [4X[25Xgap>[125X [27XInt( "12345" );  Int( "-27" );  Int( "-27/3" );[127X[104X
    [4X[28X12345[128X[104X
    [4X[28X-27[128X[104X
    [4X[28Xfail[128X[104X
  [4X[32X[104X
  
  [1X14.2-4 IsEvenInt[101X
  
  [29X[2XIsEvenInt[102X( [3Xn[103X ) [32X function
  
  [33X[0;0Ytests if the integer [3Xn[103X is divisible by 2.[133X
  
  [1X14.2-5 IsOddInt[101X
  
  [29X[2XIsOddInt[102X( [3Xn[103X ) [32X function
  
  [33X[0;0Ytests if the integer [3Xn[103X is not divisible by 2.[133X
  
  [1X14.2-6 AbsInt[101X
  
  [29X[2XAbsInt[102X( [3Xn[103X ) [32X function
  
  [33X[0;0Y[2XAbsInt[102X  returns  the  absolute  value  of  the  integer  [3Xn[103X,  i.e., [3Xn[103X if [3Xn[103X is
  positive, -[3Xn[103X if [3Xn[103X is negative and 0 if [3Xn[103X is 0.[133X
  
  [33X[0;0Y[2XAbsInt[102X is a special case of the general operation [2XEuclideanDegree[102X ([14X56.6-2[114X).[133X
  
  [33X[0;0YSee also [2XAbsoluteValue[102X ([14X18.1-8[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAbsInt( 33 );[127X[104X
    [4X[28X33[128X[104X
    [4X[25Xgap>[125X [27XAbsInt( -214378 );[127X[104X
    [4X[28X214378[128X[104X
    [4X[25Xgap>[125X [27XAbsInt( 0 );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  [1X14.2-7 SignInt[101X
  
  [29X[2XSignInt[102X( [3Xn[103X ) [32X function
  
  [33X[0;0Y[2XSignInt[102X returns the sign of the integer [3Xn[103X, i.e., 1 if [3Xn[103X is positive, -1 if [3Xn[103X
  is negative and 0 if [3Xn[103X is 0.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSignInt( 33 );[127X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XSignInt( -214378 );[127X[104X
    [4X[28X-1[128X[104X
    [4X[25Xgap>[125X [27XSignInt( 0 );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  [1X14.2-8 LogInt[101X
  
  [29X[2XLogInt[102X( [3Xn[103X, [3Xbase[103X ) [32X function
  
  [33X[0;0Y[2XLogInt[102X  returns  the integer part of the logarithm of the positive integer [3Xn[103X
  with  respect  to  the  positive  integer  [3Xbase[103X,  i.e., the largest positive
  integer  [22Xe[122X such that [22X[3Xbase[103X^e ≤ [3Xn[103X[122X. The function [2XLogInt[102X will signal an error if
  either [3Xn[103X or [3Xbase[103X is not positive.[133X
  
  [33X[0;0YFor   [3Xbase[103X   [22X=  2[122X  this  is  very  efficient  because  the  internal  binary
  representation of the integer is used.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLogInt( 1030, 2 );[127X[104X
    [4X[28X10[128X[104X
    [4X[25Xgap>[125X [27X2^10;[127X[104X
    [4X[28X1024[128X[104X
    [4X[25Xgap>[125X [27XLogInt( 1, 10 );[127X[104X
    [4X[28X0[128X[104X
  [4X[32X[104X
  
  [1X14.2-9 RootInt[101X
  
  [29X[2XRootInt[102X( [3Xn[103X[, [3Xk[103X] ) [32X function
  
  [33X[0;0Y[2XRootInt[102X  returns  the  integer part of the [3Xk[103Xth root of the integer [3Xn[103X. If the
  optional  integer  argument  [3Xk[103X  is not given it defaults to 2, i.e., [2XRootInt[102X
  returns the integer part of the square root in this case.[133X
  
  [33X[0;0YIf  [3Xn[103X  is positive, [2XRootInt[102X returns the largest positive integer [22Xr[122X such that
  [22Xr^[3Xk[103X ≤ [3Xn[103X[122X. If [3Xn[103X is negative and [3Xk[103X is odd [2XRootInt[102X returns [10X-RootInt( -[3Xn[103X[10X, [3Xk[103X[10X )[110X. If
  [3Xn[103X  is  negative and [3Xk[103X is even [2XRootInt[102X will cause an error. [2XRootInt[102X will also
  cause an error if [3Xk[103X is 0 or negative.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRootInt( 361 );[127X[104X
    [4X[28X19[128X[104X
    [4X[25Xgap>[125X [27XRootInt( 2 * 10^12 );[127X[104X
    [4X[28X1414213[128X[104X
    [4X[25Xgap>[125X [27XRootInt( 17000, 5 );[127X[104X
    [4X[28X7[128X[104X
    [4X[25Xgap>[125X [27X7^5;[127X[104X
    [4X[28X16807[128X[104X
  [4X[32X[104X
  
  [1X14.2-10 SmallestRootInt[101X
  
  [29X[2XSmallestRootInt[102X( [3Xn[103X ) [32X function
  
  [33X[0;0Y[2XSmallestRootInt[102X returns the smallest root of the integer [3Xn[103X.[133X
  
  [33X[0;0YThe  smallest  root  of  an  integer [3Xn[103X is the integer [22Xr[122X of smallest absolute
  value for which a positive integer [22Xk[122X exists such that [22X[3Xn[103X = r^k[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSmallestRootInt( 2^30 );[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XSmallestRootInt( -(2^30) );[127X[104X
    [4X[28X-4[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote that [22X(-2)^30 = +(2^30)[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSmallestRootInt( 279936 );[127X[104X
    [4X[28X6[128X[104X
    [4X[25Xgap>[125X [27XLogInt( 279936, 6 );[127X[104X
    [4X[28X7[128X[104X
    [4X[25Xgap>[125X [27XSmallestRootInt( 1001 );[127X[104X
    [4X[28X1001[128X[104X
  [4X[32X[104X
  
  [1X14.2-11 ListOfDigits[101X
  
  [29X[2XListOfDigits[102X( [3Xn[103X ) [32X function
  
  [33X[0;0YFor  a  positive integer [3Xn[103X this function returns a list [3Xl[103X, consisting of the
  digits of [3Xn[103X in decimal representation.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XListOfDigits(3142);   [127X[104X
    [4X[28X[ 3, 1, 4, 2 ][128X[104X
  [4X[32X[104X
  
  [1X14.2-12 Random[101X
  
  [29X[2XRandom[102X( [3XIntegers[103X ) [32X function
  
  [33X[0;0Y[2XRandom[102X  for  integers  returns  pseudo  random  integers  between [22X-10[122X and [22X10[122X
  distributed  according  to  a  binomial  distribution. To generate uniformly
  distributed  integers  from  a  range, use the construction [10XRandom( [ [3Xlow[103X[10X ..
  [3Xhigh[103X[10X ] )[110X  (see [2XRandom[102X ([14X30.7-1[114X)).[133X
  
  
  [1X14.3 [33X[0;0YQuotients and Remainders[133X[101X
  
  [1X14.3-1 QuoInt[101X
  
  [29X[2XQuoInt[102X( [3Xn[103X, [3Xm[103X ) [32X function
  
  [33X[0;0Y[2XQuoInt[102X returns the integer part of the quotient of its integer operands.[133X
  
  [33X[0;0YIf  [3Xn[103X and [3Xm[103X are positive, [2XQuoInt[102X returns the largest positive integer [22Xq[122X such
  that  [22Xq  *  [3Xm[103X  ≤ [3Xn[103X[122X. If [3Xn[103X or [3Xm[103X or both are negative the absolute value of the
  integer part of the quotient is the quotient of the absolute values of [3Xn[103X and
  [3Xm[103X, and the sign of it is the product of the signs of [3Xn[103X and [3Xm[103X.[133X
  
  [33X[0;0Y[2XQuoInt[102X  is  used  in  a  method  for the general operation [2XEuclideanQuotient[102X
  ([14X56.6-3[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XQuoInt(5,3);  QuoInt(-5,3);  QuoInt(5,-3);  QuoInt(-5,-3);[127X[104X
    [4X[28X1[128X[104X
    [4X[28X-1[128X[104X
    [4X[28X-1[128X[104X
    [4X[28X1[128X[104X
  [4X[32X[104X
  
  [1X14.3-2 BestQuoInt[101X
  
  [29X[2XBestQuoInt[102X( [3Xn[103X, [3Xm[103X ) [32X function
  
  [33X[0;0Y[2XBestQuoInt[102X  returns the best quotient [22Xq[122X of the integers [3Xn[103X and [3Xm[103X. This is the
  quotient  such  that  [22X[3Xn[103X-q*[3Xm[103X[122X  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.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XBestQuoInt( 5, 3 );  BestQuoInt( -5, 3 );[127X[104X
    [4X[28X2[128X[104X
    [4X[28X-2[128X[104X
  [4X[32X[104X
  
  [1X14.3-3 RemInt[101X
  
  [29X[2XRemInt[102X( [3Xn[103X, [3Xm[103X ) [32X function
  
  [33X[0;0Y[2XRemInt[102X returns the remainder of its two integer operands.[133X
  
  [33X[0;0YIf  [3Xm[103X is not equal to zero, [2XRemInt[102X returns [10X[3Xn[103X[10X - [3Xm[103X[10X * QuoInt( [3Xn[103X[10X, [3Xm[103X[10X )[110X. Note that
  the  rules  given  for [2XQuoInt[102X ([14X14.3-1[114X) imply that the return value of [2XRemInt[102X
  has  the  same  sign  as  [3Xn[103X and its absolute value is strictly less than the
  absolute  value  of  [3Xm[103X.  Note also that the return value equals [10X[3Xn[103X[10X mod [3Xm[103X[10X[110X when
  both [3Xn[103X and [3Xm[103X are nonnegative. Dividing by [10X0[110X signals an error.[133X
  
  [33X[0;0Y[2XRemInt[102X  is  used  in  a  method for the general operation [2XEuclideanRemainder[102X
  ([14X56.6-4[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XRemInt(5,3);  RemInt(-5,3);  RemInt(5,-3);  RemInt(-5,-3);[127X[104X
    [4X[28X2[128X[104X
    [4X[28X-2[128X[104X
    [4X[28X2[128X[104X
    [4X[28X-2[128X[104X
  [4X[32X[104X
  
  [1X14.3-4 GcdInt[101X
  
  [29X[2XGcdInt[102X( [3Xm[103X, [3Xn[103X ) [32X function
  
  [33X[0;0Y[2XGcdInt[102X returns the greatest common divisor of its two integer operands [3Xm[103X and
  [3Xn[103X, i.e., the greatest integer that divides both [3Xm[103X and [3Xn[103X. The greatest common
  divisor is never negative, even if the arguments are. We define [10XGcdInt( [3Xm[103X[10X, 0
  ) = GcdInt( 0, [3Xm[103X[10X ) = AbsInt( [3Xm[103X[10X )[110X and [10XGcdInt( 0, 0 ) = 0[110X.[133X
  
  [33X[0;0Y[2XGcdInt[102X is a method used by the general function [2XGcd[102X ([14X56.7-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGcdInt( 123, 66 );[127X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
  [1X14.3-5 Gcdex[101X
  
  [29X[2XGcdex[102X( [3Xm[103X, [3Xn[103X ) [32X function
  
  [33X[0;0Yreturns  a record [10Xg[110X describing the extended greatest common divisor of [3Xm[103X and
  [3Xn[103X.  The  component  [10Xgcd[110X  is  this  gcd, the components [10Xcoeff1[110X and [10Xcoeff2[110X are
  integer  cofactors  such  that  [10Xg.gcd = g.coeff1 * [3Xm[103X[10X + g.coeff2 * [3Xn[103X[10X[110X, and the
  components  [10Xcoeff3[110X and [10Xcoeff4[110X are integer cofactors such that [10X0 = g.coeff3 *
  [3Xm[103X[10X + g.coeff4 * [3Xn[103X[10X[110X.[133X
  
  [33X[0;0YIf  [3Xm[103X  and  [3Xn[103X  both are nonzero, [10XAbsInt( g.coeff1 )[110X is less than or equal to
  [10XAbsInt([3Xn[103X[10X)  /  (2  *  g.gcd)[110X, and [10XAbsInt( g.coeff2 )[110X is less than or equal to
  [10XAbsInt([3Xm[103X[10X) / (2 * g.gcd)[110X.[133X
  
  [33X[0;0YIf [3Xm[103X or [3Xn[103X or both are zero [10Xcoeff3[110X is [10X-[3Xn[103X[10X / g.gcd[110X and [10Xcoeff4[110X is [10X[3Xm[103X[10X / g.gcd[110X.[133X
  
  [33X[0;0YThe  coefficients  always  form  a  unimodular matrix, i.e., the determinant
  [10Xg.coeff1 * g.coeff4 - g.coeff3 * g.coeff2[110X is [22X1[122X or [22X-1[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGcdex( 123, 66 );[127X[104X
    [4X[28Xrec( coeff1 := 7, coeff2 := -13, coeff3 := -22, coeff4 := 41, [128X[104X
    [4X[28X  gcd := 3 )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YThis means [22X3 = 7 * 123 - 13 * 66[122X, [22X0 = -22 * 123 + 41 * 66[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGcdex( 0, -3 );[127X[104X
    [4X[28Xrec( coeff1 := 0, coeff2 := -1, coeff3 := 1, coeff4 := 0, gcd := 3 )[128X[104X
    [4X[25Xgap>[125X [27XGcdex( 0, 0 );[127X[104X
    [4X[28Xrec( coeff1 := 1, coeff2 := 0, coeff3 := 0, coeff4 := 1, gcd := 0 )[128X[104X
  [4X[32X[104X
  
  [33X[0;0Y[2XGcdRepresentation[102X  ([14X56.7-3[114X)  provides  similar  functionality over arbitrary
  Euclidean rings.[133X
  
  [1X14.3-6 LcmInt[101X
  
  [29X[2XLcmInt[102X( [3Xm[103X, [3Xn[103X ) [32X function
  
  [33X[0;0Yreturns the least common multiple of the integers [3Xm[103X and [3Xn[103X.[133X
  
  [33X[0;0Y[2XLcmInt[102X is a method used by the general operation [2XLcm[102X ([14X56.7-6[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XLcmInt( 123, 66 );[127X[104X
    [4X[28X2706[128X[104X
  [4X[32X[104X
  
  [1X14.3-7 CoefficientsQadic[101X
  
  [29X[2XCoefficientsQadic[102X( [3Xi[103X, [3Xq[103X ) [32X operation
  
  [33X[0;0Yreturns  the  [3Xq[103X-adic  representation  of  the  integer  [3Xi[103X  as  a  list  [22Xl[122X of
  coefficients  satisfying  the  equality  [22X[3Xi[103X  =  ∑_{j = 0} [3Xq[103X^j ⋅ l[j+1][122X for an
  integer [22X[3Xq[103X > 1[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xl:=CoefficientsQadic(462,3);[127X[104X
    [4X[28X[ 0, 1, 0, 2, 2, 1 ][128X[104X
  [4X[32X[104X
  
  [1X14.3-8 CoefficientsMultiadic[101X
  
  [29X[2XCoefficientsMultiadic[102X( [3Xints[103X, [3Xint[103X ) [32X function
  
  [33X[0;0Yreturns the multiadic expansion of the integer [3Xint[103X modulo the integers given
  in  [3Xints[103X  (in  ascending  order).  It  returns a list of coefficients in the
  [13Xreverse[113X order to that in [3Xints[103X.[133X
  
  [1X14.3-9 ChineseRem[101X
  
  [29X[2XChineseRem[102X( [3Xmoduli[103X, [3Xresidues[103X ) [32X function
  
  [33X[0;0Y[2XChineseRem[102X  returns the combination of the [3Xresidues[103X modulo the [3Xmoduli[103X, i.e.,
  the  unique  integer  [10Xc[110X  from  [10X[0..Lcm([3Xmoduli[103X[10X)-1][110X  such that [10Xc = [3Xresidues[103X[10X[i][110X
  modulo  [10X[3Xmoduli[103X[10X[i][110X  for  all  [10Xi[110X,  if it exists. If no such combination exists
  [2XChineseRem[102X signals an error.[133X
  
  [33X[0;0YSuch  a  combination does exist if and only if [10X[3Xresidues[103X[10X[i] = [3Xresidues[103X[10X[k] mod
  Gcd( [3Xmoduli[103X[10X[i], [3Xmoduli[103X[10X[k] )[110X for every pair [10Xi[110X, [10Xk[110X. Note that this implies that
  such  a combination exists if the moduli are pairwise relatively prime. This
  is called the Chinese remainder theorem.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XChineseRem( [ 2, 3, 5, 7 ], [ 1, 2, 3, 4 ] );[127X[104X
    [4X[28X53[128X[104X
    [4X[25Xgap>[125X [27XChineseRem( [ 6, 10, 14 ], [ 1, 3, 5 ] );[127X[104X
    [4X[28X103[128X[104X
  [4X[32X[104X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XChineseRem( [ 6, 10, 14 ], [ 1, 2, 3 ] );[127X[104X
    [4X[28XError, the residues must be equal modulo 2 called from[128X[104X
    [4X[28X<function>( <arguments> ) called from read-eval-loop[128X[104X
    [4X[28XEntering break read-eval-print loop ...[128X[104X
    [4X[28Xyou can 'quit;' to quit to outer loop, or[128X[104X
    [4X[28Xyou can 'return;' to continue[128X[104X
    [4X[26Xbrk>[126X [27Xgap> [127X[104X
  [4X[32X[104X
  
  [1X14.3-10 PowerModInt[101X
  
  [29X[2XPowerModInt[102X( [3Xr[103X, [3Xe[103X, [3Xm[103X ) [32X function
  
  [33X[0;0Yreturns [22X[3Xr[103X^[3Xe[103X mod [3Xm[103X[122X for integers [3Xr[103X, [3Xe[103X and [3Xm[103X ([22X[3Xe[103X ≥ 0[122X).[133X
  
  [33X[0;0YNote  that  [2XPowerModInt[102X  can  reduce  intermediate  results  and  thus  will
  generally  be faster than using [3Xr[103X[10X^[110X[3Xe[103X[10X mod [110X[3Xm[103X, which would compute [22X[3Xr[103X^[3Xe[103X[122X first and
  reduces the result afterwards.[133X
  
  [33X[0;0Y[2XPowerModInt[102X is a method for the general operation [2XPowerMod[102X ([14X56.7-9[114X).[133X
  
  
  [1X14.4 [33X[0;0YPrime Integers and Factorization[133X[101X
  
  [1X14.4-1 Primes[101X
  
  [29X[2XPrimes[102X[32X global variable
  
  [33X[0;0Y[2XPrimes[102X is a strictly sorted list of the 168 primes less than 1000.[133X
  
  [33X[0;0YThis  is  used  in  [2XIsPrimeInt[102X  ([14X14.4-2[114X) and [2XFactorsInt[102X ([14X14.4-7[114X) to cast out
  small primes quickly.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPrimes[1];[127X[104X
    [4X[28X2[128X[104X
    [4X[25Xgap>[125X [27XPrimes[100];[127X[104X
    [4X[28X541[128X[104X
  [4X[32X[104X
  
  [1X14.4-2 IsPrimeInt[101X
  
  [29X[2XIsPrimeInt[102X( [3Xn[103X ) [32X function
  [29X[2XIsProbablyPrimeInt[102X( [3Xn[103X ) [32X function
  
  [33X[0;0Y[2XIsPrimeInt[102X returns [9Xfalse[109X if it can prove that the integer [3Xn[103X is composite and
  [9Xtrue[109X  otherwise.  By convention [10XIsPrimeInt(0) = IsPrimeInt(1) = false[110X and we
  define [10XIsPrimeInt(-[110X[3Xn[103X[10X) = IsPrimeInt([110X[3Xn[103X[10X)[110X.[133X
  
  [33X[0;0Y[2XIsPrimeInt[102X  will return [9Xtrue[109X for every prime [3Xn[103X. [2XIsPrimeInt[102X will return [9Xfalse[109X
  for  all  composite [3Xn[103X [22X< 10^18[122X and for all composite [3Xn[103X that have a factor [22Xp <
  1000[122X.  So  for integers [3Xn[103X [22X< 10^18[122X, [2XIsPrimeInt[102X is a proper primality test. It
  is conceivable that [2XIsPrimeInt[102X may return [9Xtrue[109X for some composite [3Xn[103X [22X> 10^18[122X,
  but no such [3Xn[103X is currently known. So for integers [3Xn[103X [22X> 10^18[122X, [2XIsPrimeInt[102X is a
  probable-primality  test.  [2XIsPrimeInt[102X will issue a warning when its argument
  is  probably  prime but not a proven prime. (The function [2XIsProbablyPrimeInt[102X
  will  do  a similar calculation but not issue a warning.) The warning can be
  switched  off  by  [10XSetInfoLevel(  InfoPrimeInt, 0 );[110X, the default level is [22X1[122X
  (also see [2XSetInfoLevel[102X ([14X7.4-3[114X) ).[133X
  
  [33X[0;0YIf  composites  that fool [2XIsPrimeInt[102X do exist, they would be extremely rare,
  and  finding  one  by pure chance might be less likely than finding a bug in
  [5XGAP[105X.  We  would  appreciate  being informed about any example of a composite
  number [3Xn[103X for which [2XIsPrimeInt[102X returns [9Xtrue[109X.[133X
  
  [33X[0;0Y[2XIsPrimeInt[102X  is  a deterministic algorithm, i.e., the computations involve no
  random  numbers,  and  repeated  calls  will  always return the same result.
  [2XIsPrimeInt[102X  first does trial divisions by the primes less than 1000. Then it
  tests  that  [3Xn[103X  is  a strong pseudoprime w.r.t. the base 2. Finally it tests
  whether [3Xn[103X is a Lucas pseudoprime w.r.t. the smallest quadratic nonresidue of
  [3Xn[103X.  A  better  description  can  be found in the comment in the library file
  [11Xprimality.gi[111X.[133X
  
  [33X[0;0YThe  time  taken  by  [2XIsPrimeInt[102X  is approximately proportional to the third
  power  of  the  number of digits of [3Xn[103X. Testing numbers with several hundreds
  digits is quite feasible.[133X
  
  [33X[0;0Y[2XIsPrimeInt[102X is a method for the general operation [2XIsPrime[102X ([14X56.5-8[114X).[133X
  
  [33X[0;0YRemark:  In  future  versions  of  [5XGAP[105X  we  hope to change the definition of
  [2XIsPrimeInt[102X  to  return  [9Xtrue[109X  only  for  proven primes (currently, we lack a
  sufficiently   good   primality  proving  function).  In  applications,  use
  explicitly [2XIsPrimeInt[102X or [2XIsProbablyPrimeInt[102X with this change in mind.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsPrimeInt( 2^31 - 1 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsPrimeInt( 10^42 + 1 );[127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X14.4-3 PrimalityProof[101X
  
  [29X[2XPrimalityProof[102X( [3Xn[103X ) [32X function
  
  [33X[0;0YConstruct  a  machine  verifiable  proof  of  the primality of (the probable
  prime)  [3Xn[103X,  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,  [10X[  "F",  divisor,  base  ][110X,  indicates  a  successful  Fermat
  pseudoprimality test, where [3Xn[103X is a strong pseudoprime at [9Xbase[109X with order not
  divisible by [22X([3Xn[103X-1)/divisor[122X. The second kind, [10X[ "L", divisor, discriminant, P
  ][110X indicates a successful Lucas pseudoprimality test, for a quadratic form of
  given [9Xdiscriminant[109X and middle term [9XP[109X with an extra check at [22X([3Xn[103X+1)/divisor[122X.[133X
  
  [1X14.4-4 IsPrimePowerInt[101X
  
  [29X[2XIsPrimePowerInt[102X( [3Xn[103X ) [32X function
  
  [33X[0;0Y[2XIsPrimePowerInt[102X  returns  [9Xtrue[109X  if  the integer [3Xn[103X is a prime power and [9Xfalse[109X
  otherwise.[133X
  
  [33X[0;0YAn  integer  [22Xn[122X  is  a  [13Xprime  power[113X if there exists a prime [22Xp[122X and a positive
  integer  [22Xi[122X  such  that [22Xp^i = n[122X. If [22Xn[122X is negative the condition is that there
  must  exist a negative prime [22Xp[122X and an odd positive integer [22Xi[122X such that [22Xp^i =
  n[122X. The integers 1 and -1 are not prime powers.[133X
  
  [33X[0;0YNote   that   [2XIsPrimePowerInt[102X   uses   [2XSmallestRootInt[102X   ([14X14.2-10[114X)   and   a
  probable-primality test (see [2XIsPrimeInt[102X ([14X14.4-2[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XIsPrimePowerInt( 31^5 );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsPrimePowerInt( 2^31-1 );  # 2^31-1 is actually a prime[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XIsPrimePowerInt( 2^63-1 );[127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XFiltered( [-10..10], IsPrimePowerInt );[127X[104X
    [4X[28X[ -8, -7, -5, -3, -2, 2, 3, 4, 5, 7, 8, 9 ][128X[104X
  [4X[32X[104X
  
  [1X14.4-5 NextPrimeInt[101X
  
  [29X[2XNextPrimeInt[102X( [3Xn[103X ) [32X function
  
  [33X[0;0Y[2XNextPrimeInt[102X  returns  the  smallest prime which is strictly larger than the
  integer [3Xn[103X.[133X
  
  [33X[0;0YNote  that  [2XNextPrimeInt[102X  uses  a  probable-primality  test  (see [2XIsPrimeInt[102X
  ([14X14.4-2[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNextPrimeInt( 541 ); NextPrimeInt( -1 );[127X[104X
    [4X[28X547[128X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  [1X14.4-6 PrevPrimeInt[101X
  
  [29X[2XPrevPrimeInt[102X( [3Xn[103X ) [32X function
  
  [33X[0;0Y[2XPrevPrimeInt[102X  returns  the  largest prime which is strictly smaller than the
  integer [3Xn[103X.[133X
  
  [33X[0;0YNote  that  [2XPrevPrimeInt[102X  uses  a  probable-primality  test  (see [2XIsPrimeInt[102X
  ([14X14.4-2[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPrevPrimeInt( 541 ); PrevPrimeInt( 1 );[127X[104X
    [4X[28X523[128X[104X
    [4X[28X-2[128X[104X
  [4X[32X[104X
  
  [1X14.4-7 FactorsInt[101X
  
  [29X[2XFactorsInt[102X( [3Xn[103X ) [32X function
  [29X[2XFactorsInt[102X( [3Xn:[103X [3XRhoTrials[103X [3X:=[103X [3Xtrials[103X ) [32X function
  
  [33X[0;0Y[2XFactorsInt[102X returns a list of factors of a given integer [3Xn[103X such that [10XProduct(
  FactorsInt(  [3Xn[103X[10X  )  ) = [3Xn[103X[10X[110X. If [22X|n| ≤ 1[122X the list [10X[[3Xn[103X[10X][110X 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 [3Xn[103X.[133X
  
  [33X[0;0YSee  [2XPrimeDivisors[102X  ([14X14.4-8[114X) for a function that returns a set of (probable)
  primes dividing [3Xn[103X.[133X
  
  [33X[0;0YNote   that   [2XFactorsInt[102X  uses  a  probable-primality  test  (see [2XIsPrimeInt[102X
  ([14X14.4-2[114X)).  Thus  [2XFactorsInt[102X  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 [10XSetInfoLevel( InfoPrimeInt, 0 );[110X
  (also see [2XSetInfoLevel[102X ([14X7.4-3[114X)).[133X
  
  [33X[0;0YThe  time  taken  by  [2XFactorsInt[102X is approximately proportional to the square
  root  of  the  second  largest prime factor of [3Xn[103X, which is the last one that
  [2XFactorsInt[102X 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  [3Xn[103X. [2XFactorsInt[102X is guaranteed to find all factors less than [22X10^6[122X and
  will  find  most  factors  less  than  [22X10^10[122X. If [3Xn[103X contains multiple factors
  larger than that [2XFactorsInt[102X may not be able to factor [3Xn[103X and will then signal
  an error.[133X
  
  [33X[0;0Y[2XFactorsInt[102X is used in a method for the general operation [2XFactors[102X ([14X56.5-9[114X).[133X
  
  [33X[0;0YIn  the  second  form, [2XFactorsInt[102X calls [10XFactorsRho[110X with a limit of [3Xtrials[103X 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 [3Xtrials[103X, it
  is  usually more advisable to install the [5XFactInt[105X package and then simply to
  use  the  operation  [2XFactors[102X  ([14X56.5-9[114X). The factorization of the 8-th Fermat
  number by Pollard's Rho below takes already a while.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFactorsInt( -Factorial(6) );[127X[104X
    [4X[28X[ -2, 2, 2, 2, 3, 3, 5 ][128X[104X
    [4X[25Xgap>[125X [27XSet( FactorsInt( Factorial(13)/11 ) );[127X[104X
    [4X[28X[ 2, 3, 5, 7, 13 ][128X[104X
    [4X[25Xgap>[125X [27XFactorsInt( 2^63 - 1 );[127X[104X
    [4X[28X[ 7, 7, 73, 127, 337, 92737, 649657 ][128X[104X
    [4X[25Xgap>[125X [27XFactorsInt( 10^42 + 1 );[127X[104X
    [4X[28X[ 29, 101, 281, 9901, 226549, 121499449, 4458192223320340849 ][128X[104X
    [4X[25Xgap>[125X [27XFactorsInt(2^256+1:RhoTrials:=100000000);[127X[104X
    [4X[28X[ 1238926361552897, [128X[104X
    [4X[28X  93461639715357977769163558199606896584051237541638188580280321 ][128X[104X
  [4X[32X[104X
  
  [1X14.4-8 PrimeDivisors[101X
  
  [29X[2XPrimeDivisors[102X( [3Xn[103X ) [32X attribute
  
  [33X[0;0Y[2XPrimeDivisors[102X  returns  for  a  non-zero  integer  [3Xn[103X  a  set of its positive
  (probable)  primes  divisors.  In  rare  cases  the  result  could contain a
  composite    number    which    passed    certain   primality   tests,   see
  [2XIsProbablyPrimeInt[102X ([14X14.4-2[114X) and [2XFactorsInt[102X ([14X14.4-7[114X) for more details.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPrimeDivisors(-12);[127X[104X
    [4X[28X[ 2, 3 ][128X[104X
    [4X[25Xgap>[125X [27XPrimeDivisors(1);[127X[104X
    [4X[28X[  ][128X[104X
  [4X[32X[104X
  
  [1X14.4-9 PartialFactorization[101X
  
  [29X[2XPartialFactorization[102X( [3Xn[103X[, [3Xeffort[103X] ) [32X operation
  
  [33X[0;0Y[2XPartialFactorization[102X  returns  a  partial factorization of the integer [3Xn[103X. No
  assertions  are  made  about  the  primality of the factors, except of those
  mentioned below.[133X
  
  [33X[0;0YThe argument [3Xeffort[103X, if given, specifies how intensively the function should
  try to determine factors of [3Xn[103X. The default is [3Xeffort[103X = 5.[133X
  
  [30X    [33X[0;6YIf  [3Xeffort[103X = 0,  trial  division  by  the  primes  below  100 is done.
        Returned factors below [22X10^4[122X are guaranteed to be prime.[133X
  
  [30X    [33X[0;6YIf  [3Xeffort[103X = 1,  trial  division  by  the  primes  below 1000 is done.
        Returned factors below [22X10^6[122X are guaranteed to be prime.[133X
  
  [30X    [33X[0;6YIf [3Xeffort[103X = 2, additionally trial division by the numbers in the lists
        [10XPrimes2[110X  and [10XProbablePrimes2[110X is done, and perfect powers are detected.
        Returned factors below [22X10^6[122X are guaranteed to be prime.[133X
  
  [30X    [33X[0;6YIf  [3Xeffort[103X = 3, additionally [10XFactorsRho[110X (Pollard's Rho) with [10XRhoTrials[110X
        = 256 is used.[133X
  
  [30X    [33X[0;6YIf [3Xeffort[103X = 4, as above, but [10XRhoTrials[110X = 2048.[133X
  
  [30X    [33X[0;6YIf  [3Xeffort[103X = 5, as above, but [10XRhoTrials[110X = 8192. Returned factors below
        [22X10^12[122X are guaranteed to be prime, and all prime factors below [22X10^6[122X are
        guaranteed to be found.[133X
  
  [30X    [33X[0;6YIf  [3Xeffort[103X = 6  and  the package [5XFactInt[105X is loaded, in addition to the
        above quite a number of special cases are handled.[133X
  
  [30X    [33X[0;6YIf  [3Xeffort[103X = 7 and the package [5XFactInt[105X 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.[133X
  
  [33X[0;0YIncreasing  the  value  of  the argument [3Xeffort[103X by one usually results in an
  increase of the runtime requirements by a factor of (very roughly!) 3 to 10.
  (Also see [2XCheapFactorsInt[102X ([14XEDIM: CheapFactorsInt[114X)).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XList([0..5],i->PartialFactorization(97^35-1,i)); [127X[104X
    [4X[28X[ [ 2, 2, 2, 2, 2, 3, 11, 31, 43, [128X[104X
    [4X[28X      2446338959059521520901826365168917110105972824229555319002965029 ], [128X[104X
    [4X[28X  [ 2, 2, 2, 2, 2, 3, 11, 31, 43, 967, [128X[104X
    [4X[28X      2529823122088440042297648774735177983563570655873376751812787 ],[128X[104X
    [4X[28X  [ 2, 2, 2, 2, 2, 3, 11, 31, 43, 967, [128X[104X
    [4X[28X      2529823122088440042297648774735177983563570655873376751812787 ],[128X[104X
    [4X[28X  [ 2, 2, 2, 2, 2, 3, 11, 31, 43, 967, 39761, 262321, [128X[104X
    [4X[28X      242549173950325921859769421435653153445616962914227 ], [128X[104X
    [4X[28X  [ 2, 2, 2, 2, 2, 3, 11, 31, 43, 967, 39761, 262321, 687121, [128X[104X
    [4X[28X      352993394104278463123335513593170858474150787 ], [128X[104X
    [4X[28X  [ 2, 2, 2, 2, 2, 3, 11, 31, 43, 967, 39761, 262321, 687121, [128X[104X
    [4X[28X      20241187, 504769301, 34549173843451574629911361501 ] ][128X[104X
  [4X[32X[104X
  
  [1X14.4-10 PrintFactorsInt[101X
  
  [29X[2XPrintFactorsInt[102X( [3Xn[103X ) [32X function
  
  [33X[0;0Yprints the prime factorization of the integer [3Xn[103X in human-readable form.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPrintFactorsInt( Factorial( 7 ) ); Print( "\n" );[127X[104X
    [4X[28X2^4*3^2*5*7[128X[104X
  [4X[32X[104X
  
  [1X14.4-11 PrimePowersInt[101X
  
  [29X[2XPrimePowersInt[102X( [3Xn[103X ) [32X function
  
  [33X[0;0Yreturns  the prime factorization of the integer [3Xn[103X as a list [22X[ p_1, e_1, ...,
  p_k, e_k ][122X with [3Xn[103X = [22Xp_1^{e_1} ⋅ p_2^{e_2} ⋅ ... ⋅ p_k^{e_k}[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XPrimePowersInt( Factorial( 7 ) );[127X[104X
    [4X[28X[ 2, 4, 3, 2, 5, 1, 7, 1 ][128X[104X
  [4X[32X[104X
  
  [1X14.4-12 DivisorsInt[101X
  
  [29X[2XDivisorsInt[102X( [3Xn[103X ) [32X function
  
  [33X[0;0Y[2XDivisorsInt[102X  returns  a  list  of all divisors of the integer [3Xn[103X. The list is
  sorted,  so  that  it  starts  with  1  and  ends  with  [3Xn[103X.  We  define that
  [10XDivisorsInt( -[3Xn[103X[10X ) = DivisorsInt( [3Xn[103X[10X )[110X.[133X
  
  [33X[0;0YSince  the  set of divisors of 0 is infinite calling [10XDivisorsInt( 0 )[110X causes
  an error.[133X
  
  [33X[0;0Y[2XDivisorsInt[102X  may call [2XFactorsInt[102X ([14X14.4-7[114X) to obtain the prime factors. [2XSigma[102X
  ([14X15.5-1[114X)  and  [2XTau[102X  ([14X15.5-2[114X)  compute  the  sum  and  the number of positive
  divisors, respectively.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDivisorsInt( 1 ); DivisorsInt( 20 ); DivisorsInt( 541 );[127X[104X
    [4X[28X[ 1 ][128X[104X
    [4X[28X[ 1, 2, 4, 5, 10, 20 ][128X[104X
    [4X[28X[ 1, 541 ][128X[104X
  [4X[32X[104X
  
  
  [1X14.5 [33X[0;0YResidue Class Rings[133X[101X
  
  [33X[0;0Y[2XZmodnZ[102X  ([14X14.5-2[114X)  returns  a  residue  class ring of [2XIntegers[102X ([14X14[114X) modulo an
  ideal.  These  residue  class rings are rings, thus all operations for rings
  (see Chapter [14X56[114X) apply. See also Chapters [14X59[114X and [14X15[114X.[133X
  
  [1X14.5-1 \mod[101X
  
  [29X[2X\mod[102X( [3Xr/s[103X, [3Xn[103X ) [32X operation
  
  [33X[0;0YIf  [3Xr[103X, [3Xs[103X and [3Xn[103X are integers, [10X[3Xr[103X[10X / [3Xs[103X[10X[110X as a reduced fraction is [10Xp/q[110X, where [10Xq[110X and
  [3Xn[103X  are  coprime,  then [10X[3Xr[103X[10X / [3Xs[103X[10X mod [3Xn[103X[10X[110X is defined to be the product of [10Xp[110X and the
  inverse of [10Xq[110X modulo [3Xn[103X. See Section [14X4.13[114X for more details and definitions.[133X
  
  [33X[0;0YWith the above definition, [10X4 / 6 mod 32[110X equals [10X2 / 3 mod 32[110X and hence exists
  (and is equal to 22), despite the fact that 6 has no inverse modulo 32.[133X
  
  [1X14.5-2 ZmodnZ[101X
  
  [29X[2XZmodnZ[102X( [3Xn[103X ) [32X function
  [29X[2XZmodpZ[102X( [3Xp[103X ) [32X function
  [29X[2XZmodpZNC[102X( [3Xp[103X ) [32X function
  
  [33X[0;0Y[2XZmodnZ[102X returns a ring [22XR[122X isomorphic to the residue class ring of the integers
  modulo  the  positive  integer  [3Xn[103X.  The element corresponding to the residue
  class  of  the integer [22Xi[122X in this ring can be obtained by [10Xi * One( R )[110X, and a
  representative  of  the residue class corresponding to the element [22Xx ∈ R[122X can
  be computed by [10XInt[110X[22X( x )[122X.[133X
  
  [33X[0;0Y[10XZmodnZ( [3Xn[103X[10X )[110X is equal to [10XIntegers mod [3Xn[103X[10X[110X.[133X
  
  [33X[0;0Y[2XZmodpZ[102X  does the same if the argument [3Xp[103X is a prime integer, additionally the
  result is a field. [2XZmodpZNC[102X omits the check whether [3Xp[103X is a prime.[133X
  
  [33X[0;0YEach  ring  returned  by  these  functions  contains the whole family of its
  elements  if  [3Xn[103X  is  not  a prime, and is embedded into the family of finite
  field elements of characteristic [3Xn[103X if [3Xn[103X is a prime.[133X
  
  [1X14.5-3 ZmodnZObj[101X
  
  [29X[2XZmodnZObj[102X( [3XFam[103X, [3Xr[103X ) [32X operation
  [29X[2XZmodnZObj[102X( [3Xr[103X, [3Xn[103X ) [32X operation
  
  [33X[0;0YIf  the  first argument is a residue class family [3XFam[103X then [2XZmodnZObj[102X returns
  the element in [3XFam[103X whose coset is represented by the integer [3Xr[103X.[133X
  
  [33X[0;0YIf  the  two  arguments  are  an  integer  [3Xr[103X  and  a positive integer [3Xn[103X then
  [2XZmodnZObj[102X  returns  the  element  in [10XZmodnZ( [3Xn[103X[10X )[110X (see [2XZmodnZ[102X ([14X14.5-2[114X)) whose
  coset is represented by the integer [3Xr[103X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xr:= ZmodnZ(15);[127X[104X
    [4X[28X(Integers mod 15)[128X[104X
    [4X[25Xgap>[125X [27Xfam:=ElementsFamily(FamilyObj(r));;[127X[104X
    [4X[25Xgap>[125X [27Xa:= ZmodnZObj(fam,9);[127X[104X
    [4X[28XZmodnZObj( 9, 15 )[128X[104X
    [4X[25Xgap>[125X [27Xa+a;[127X[104X
    [4X[28XZmodnZObj( 3, 15 )[128X[104X
    [4X[25Xgap>[125X [27XInt(a+a);[127X[104X
    [4X[28X3[128X[104X
  [4X[32X[104X
  
  [1X14.5-4 IsZmodnZObj[101X
  
  [29X[2XIsZmodnZObj[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsZmodnZObjNonprime[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsZmodpZObj[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsZmodpZObjSmall[102X( [3Xobj[103X ) [32X Category
  [29X[2XIsZmodpZObjLarge[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YThe elements in the rings [22XZ / n Z[122X are in the category [2XIsZmodnZObj[102X. If [22Xn[122X is a
  prime  then  the elements are of course also in the category [2XIsFFE[102X ([14X59.1-1[114X),
  otherwise they are in [2XIsZmodnZObjNonprime[102X. [2XIsZmodpZObj[102X is an abbreviation of
  [10XIsZmodnZObj   and   IsFFE[110X.   This   category   is   the  disjoint  union  of
  [2XIsZmodpZObjSmall[102X  and  [2XIsZmodpZObjLarge[102X,  the former containing all elements
  with [22Xn[122X at most [10XMAXSIZE_GF_INTERNAL[110X.[133X
  
  [33X[0;0YThe reasons to distinguish the prime case from the nonprime case are[133X
  
  [30X    [33X[0;6Ythat  objects  in  [2XIsZmodnZObjNonprime[102X have an external representation
        (namely the residue in the range [22X[ 0, 1, ..., n-1 ][122X),[133X
  
  [30X    [33X[0;6Ythat  the  comparison  of elements can be defined as comparison of the
        residues, and[133X
  
  [30X    [33X[0;6Ythat  the  elements  lie in a family of type [10XIsZmodnZObjNonprimeFamily[110X
        (note that for prime [22Xn[122X, the family must be an [10XIsFFEFamily[110X).[133X
  
  [33X[0;0YThe reasons to distinguish the small and the large case are that for small [22Xn[122X
  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 [22Xn[122X.[133X
  
  [33X[0;0YNote  that we [13Xcannot[113X claim that every finite field element of degree 1 is in
  [2XIsZmodnZObj[102X,  since finite field elements in internal representation may not
  know that they lie in the prime field.[133X
  
  
  [1X14.6 [33X[0;0YCheck Digits[133X[101X
  
  [1X14.6-1 CheckDigitISBN[101X
  
  [29X[2XCheckDigitISBN[102X( [3Xn[103X ) [32X function
  [29X[2XCheckDigitISBN13[102X( [3Xn[103X ) [32X function
  [29X[2XCheckDigitPostalMoneyOrder[102X( [3Xn[103X ) [32X function
  [29X[2XCheckDigitUPC[102X( [3Xn[103X ) [32X function
  
  [33X[0;0YThese  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 [10Xtrue[110X or [10Xfalse[110X), 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 [2XCheckDigitISBN[102X (since arithmetic is module 11, a
  digit  11  is  represented  by  an  X);  the  newer  13-digit  ISBN-13 using
  [2XCheckDigitISBN13[102X;  the  numbers  of  11-digit  US  postal money orders using
  [2XCheckDigitPostalMoneyOrder[102X; and the 12-digit UPC bar code found on groceries
  using [2XCheckDigitUPC[102X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCheckDigitISBN("052166103");[127X[104X
    [4X[28XCheck Digit is 'X'[128X[104X
    [4X[28X'X'[128X[104X
    [4X[25Xgap>[125X [27XCheckDigitISBN("052166103X");[127X[104X
    [4X[28XChecksum test satisfied[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XCheckDigitISBN(0,5,2,1,6,6,1,0,3,1);[127X[104X
    [4X[28XChecksum test failed[128X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27XCheckDigitISBN(0,5,2,1,6,6,1,0,3,'X'); # note single quotes![127X[104X
    [4X[28XChecksum test satisfied[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XCheckDigitISBN13("9781420094527");[127X[104X
    [4X[28XChecksum test satisfied[128X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27XCheckDigitUPC("07164183001");[127X[104X
    [4X[28XCheck Digit is 1[128X[104X
    [4X[28X1[128X[104X
    [4X[25Xgap>[125X [27XCheckDigitPostalMoneyOrder(16786457155);[127X[104X
    [4X[28XChecksum test satisfied[128X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X14.6-2 CheckDigitTestFunction[101X
  
  [29X[2XCheckDigitTestFunction[102X( [3Xl[103X, [3Xm[103X, [3Xf[103X ) [32X function
  
  [33X[0;0YThis  function  creates  check  digit  test functions such as [2XCheckDigitISBN[102X
  ([14X14.6-1[114X)  for  check  digit schemes that use the inner products with a fixed
  vector  modulo  a  number.  The  scheme creates will use strings of [3Xl[103X digits
  (including  the  check  digits),  the  check consists of taking the standard
  product of the vector of digits with the fixed vector [3Xf[103X modulo [3Xm[103X; the result
  needs  to  be  0.  The function returns a function that then can be used for
  testing or determining check digits.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xisbntest:=CheckDigitTestFunction(10,11,[1,2,3,4,5,6,7,8,9,-1]); [127X[104X
    [4X[28Xfunction( arg ) ... end[128X[104X
    [4X[25Xgap>[125X [27Xisbntest("038794680");[127X[104X
    [4X[28XCheck Digit is 2[128X[104X
    [4X[28X2[128X[104X
  [4X[32X[104X
  
  
  [1X14.7 [33X[0;0YRandom Sources[133X[101X
  
  [33X[0;0Y[5XGAP[105X  provides  [2XRandom[102X ([14X30.7-1[114X) methods for many collections of objects. On a
  lower  level  these methods use [13Xrandom sources[113X which provide random integers
  and random choices from lists.[133X
  
  [1X14.7-1 IsRandomSource[101X
  
  [29X[2XIsRandomSource[102X( [3Xobj[103X ) [32X Category
  
  [33X[0;0YThis is the category of random source objects which are defined to have, for
  an  object  [3Xrs[103X  in  this  category,  methods  available  for  the  following
  operations  which  are  explained  in  more detail below: [10XRandom( [3Xrs[103X[10X, [3Xlist[103X[10X )[110X
  giving  a  random element of a list, [10XRandom( [3Xrs[103X[10X, [3Xlow[103X[10X, [3Xhigh[103X[10X )[110X giving a random
  integer  between [3Xlow[103X and [3Xhigh[103X (inclusive), [2XInit[102X ([14X14.7-3[114X), [2XState[102X ([14X14.7-3[114X) and
  [2XReset[102X ([14X14.7-3[114X).[133X
  
  [33X[0;0YUse [2XRandomSource[102X ([14X14.7-5[114X) to construct new random sources.[133X
  
  [33X[0;0YOne  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.[133X
  
  [33X[0;0YRandom source objects lie in the family [10XRandomSourcesFamily[110X.[133X
  
  [1X14.7-2 Random[101X
  
  [29X[2XRandom[102X( [3Xrs[103X, [3Xlist[103X ) [32X operation
  [29X[2XRandom[102X( [3Xrs[103X, [3Xlow[103X, [3Xhigh[103X ) [32X operation
  
  [33X[0;0YThis operation returns a random element from list [3Xlist[103X, or an integer in the
  range  from  the  given (possibly large) integers [3Xlow[103X to [3Xhigh[103X, respectively.
  The  choice should only depend on the random source [3Xrs[103X and have no effect on
  other random sources.[133X
  
  [1X14.7-3 State[101X
  
  [29X[2XState[102X( [3Xrs[103X ) [32X operation
  [29X[2XReset[102X( [3Xrs[103X[, [3Xseed[103X] ) [32X operation
  [29X[2XInit[102X( [3Xprers[103X[, [3Xseed[103X] ) [32X operation
  
  [33X[0;0YThese  are the basic operations for which random sources (see [2XIsRandomSource[102X
  ([14X14.7-1[114X)) must have methods.[133X
  
  [33X[0;0Y[2XState[102X  should  return  a data structure which allows to recover 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 [2XState[102X should return [9Xfail[109X.[133X
  
  [33X[0;0Y[10XReset(  [3Xrs[103X[10X, [3Xseed[103X[10X )[110X resets the random source [3Xrs[103X to a state described by [3Xseed[103X,
  if  the  random  source  can be reset (otherwise it should do nothing). Here
  [3Xseed[103X  can  be  an output of [2XState[102X and then should reset to that state. Also,
  the  methods should always allow integers as [3Xseed[103X. Without the [3Xseed[103X argument
  the default [22X[3Xseed[103X = 1[122X is used.[133X
  
  [33X[0;0Y[2XInit[102X  is  the  constructor  of  a  random source, it gets an empty component
  object  [3Xprers[103X  which  has  already  the  correct type and should fill in the
  actual  data  which are needed. Optionally, it should allow one to specify a
  [3Xseed[103X for the initial state, as explained for [2XReset[102X.[133X
  
  [33X[0;0YMost   methods   for   [2XRandom[102X   ([14X30.7-1[114X)   in   the   [5XGAP[105X  library  use  the
  [2XGlobalMersenneTwister[102X  ([14X14.7-4[114X)  as  random  source.  It can be reset into a
  known state as in the following example.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xseed := State(GlobalMersenneTwister);;[127X[104X
    [4X[25Xgap>[125X [27XList([1..10],i->Random(Integers));[127X[104X
    [4X[28X[ -1, -3, -2, 1, -2, -1, 0, 1, 0, 1 ][128X[104X
    [4X[25Xgap>[125X [27XList([1..10],i->Random(Integers));[127X[104X
    [4X[28X[ -1, 0, 2, 0, 4, -1, -3, 1, -4, -1 ][128X[104X
    [4X[25Xgap>[125X [27XReset(GlobalMersenneTwister, seed);;[127X[104X
    [4X[25Xgap>[125X [27XList([1..10],i->Random(Integers));[127X[104X
    [4X[28X[ -1, -3, -2, 1, -2, -1, 0, 1, 0, 1 ][128X[104X
  [4X[32X[104X
  
  [1X14.7-4 IsMersenneTwister[101X
  
  [29X[2XIsMersenneTwister[102X( [3Xrs[103X ) [32X Category
  [29X[2XIsGAPRandomSource[102X( [3Xrs[103X ) [32X Category
  [29X[2XIsGlobalRandomSource[102X( [3Xrs[103X ) [32X Category
  [29X[2XGlobalMersenneTwister[102X[32X global variable
  [29X[2XGlobalRandomSource[102X[32X global variable
  
  [33X[0;0YCurrently,   the  [5XGAP[105X  library  provides  three  types  of  random  sources,
  distinguished by the three listed categories.[133X
  
  [33X[0;0Y[2XIsMersenneTwister[102X 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 [22X2^19937-1[122X and the numbers have a [22X623[122X-dimensional equidistribution.
  For  more  details  and  the origin of the code used in the [5XGAP[105X kernel, see:
  [7Xhttp://www.math.sci.hiroshima-u.ac.jp/~m-mat/MT/emt.html[107X.[133X
  
  [33X[0;0YUse  the  Mersenne  twister  if  possible, in particular for generating many
  large random integers.[133X
  
  [33X[0;0YThere  is also a predefined global random source [2XGlobalMersenneTwister[102X which
  is used by most of the library methods for [2XRandom[102X ([14X30.7-1[114X).[133X
  
  [33X[0;0Y[2XIsGAPRandomSource[102X  uses  the  same number generator as [2XIsGlobalRandomSource[102X,
  but  you  can  create  several  of these random sources which generate their
  random numbers independently of all other random sources.[133X
  
  [33X[0;0Y[2XIsGlobalRandomSource[102X  gives  access to the [13Xclassical[113X global random generator
  which  was  used by [5XGAP[105X 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 [2XGlobalRandomSource[102X. This uses
  the  additive  random  number  generator  described  in [Knu98] (Algorithm A
  in 3.2.2 with lag [22X30[122X).[133X
  
  [1X14.7-5 RandomSource[101X
  
  [29X[2XRandomSource[102X( [3Xcat[103X[, [3Xseed[103X] ) [32X operation
  
  [33X[0;0YThis  operation is used to create new random sources. The first argument [3Xcat[103X
  is  the  category  describing  the type of the random generator, an optional
  [3Xseed[103X  which can be an integer or a type specific data structure can be given
  to specify the initial state.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xrs1 := RandomSource(IsMersenneTwister);[127X[104X
    [4X[28X<RandomSource in IsMersenneTwister>[128X[104X
    [4X[25Xgap>[125X [27Xstate1 := State(rs1);;[127X[104X
    [4X[25Xgap>[125X [27Xl1 := List([1..10000], i-> Random(rs1, [1..6]));;  [127X[104X
    [4X[25Xgap>[125X [27Xrs2 := RandomSource(IsMersenneTwister);;[127X[104X
    [4X[25Xgap>[125X [27Xl2 := List([1..10000], i-> Random(rs2, [1..6]));;[127X[104X
    [4X[25Xgap>[125X [27Xl1 = l2;[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27Xl1 = List([1..10000], i-> Random(rs1, [1..6])); [127X[104X
    [4X[28Xfalse[128X[104X
    [4X[25Xgap>[125X [27Xn := Random(rs1, 1, 2^220);[127X[104X
    [4X[28X1598617776705343302477918831699169150767442847525442557699717518961[128X[104X
  [4X[32X[104X
  
