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gap> START_TEST("integers.tst");
#
gap> Basis(Integers);
CanonicalBasis( Integers )
gap> CanonicalBasis(Integers);
CanonicalBasis( Integers )
gap> Coefficients(Basis(Integers), 5);
[ 5 ]
gap> Coefficients(Basis(Integers), 5/2);
fail
#
gap> BestQuoInt(5, 3);
2
gap> BestQuoInt(-5, 3);
-2
gap> BestQuoInt(-5, -3);
2
gap> BestQuoInt(5, -3);
-2
#
gap> QuoInt(5, 3);
1
gap> QuoInt(-5, 3);
-1
gap> QuoInt(-5, -3);
1
gap> QuoInt(5, -3);
-1
#
gap> RoundCyc(3);
3
gap> RoundCycDown(3);
3
#
gap> PrimeDivisors(0);
Error, <n> must be non zero
gap> List([1..10], PrimeDivisors);
[ [ ], [ 2 ], [ 3 ], [ 2 ], [ 5 ], [ 2, 3 ], [ 7 ], [ 2 ], [ 3 ], [ 2, 5 ] ]
gap> last = List([1..10], n->PrimeDivisors(-n));
true
#
gap> n:=(2^31-1)*(2^61-1);; # product of two "small" primes
gap> PartialFactorization(n);
[ 2147483647, 2305843009213693951 ]
gap> FactorsInt(n);
[ 2147483647, 2305843009213693951 ]
gap> n:=2^155-19;; # not a prime; GAP fails to fully factorize it, though FactInt finds all 4 factors
gap> PartialFactorization(n);
[ 167, 273484587823896504154881143846609846492502347 ]
gap> n:=(2^2203-1)*(2^2281-1);; # product of two "large" primes
gap> PartialFactorization(n) = [ n ];
true
gap> n:=2^255-19;; # this is a "large" prime for which GAP only knows it is probably prime
gap> PartialFactorization(n) = [ n ];
true
gap> FactorsInt(n) = [ n ];
#I FactorsInt: used the following factor(s) which are probably primes:
#I 57896044618658097711785492504343953926634992332820282019728792003956564819949
true
#
gap> StringPP(-3);
"-3"
gap> StringPP(0);
"0"
gap> StringPP(-10);
"-2*5"
gap> StringPP(10);
"2*5"
gap> StringPP(10000);
"2^4*5^4"
#
gap> Filtered([-4..20], IsPrimePowerInt);
[ -3, -2, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19 ]
gap> IsPrimePowerInt(1009^1009);
true
gap> IsPrimePowerInt(1009^1009*1013);
false
#
gap> LogInt(0, 2);
Error, <n> must be a positive integer
gap> LogInt(1, 1);
Error, <base> must be an integer greater than 1
gap> ForAll([2,8,16,10,10000, 2^64], b->
> List([ 1, b-1, b, b+1, b^2-1, b^2, b^2+1 ], n->LogInt(n,b))
> = [ 0, 0, 1, 1, 1, 2, 2 ]);
true
#
gap> List([-8..8], NextPrimeInt);
[ -7, -5, -5, -3, -3, -2, 2, 2, 2, 2, 3, 5, 5, 7, 7, 11, 11 ]
gap> List([-8..8], PrevPrimeInt);
[ -11, -11, -7, -7, -5, -5, -3, -2, -2, -2, -2, 2, 3, 3, 5, 5, 7 ]
#
gap> PrimePowersInt(180);
[ 2, 2, 3, 2, 5, 1 ]
gap> PrimePowersInt(1);
[ ]
gap> PrimePowersInt(2);
[ 2, 1 ]
gap> PrimePowersInt(0);
Error, <n> must be non zero
#
gap> EuclideanDegree(Integers, -5);
5
gap> EuclideanDegree(Integers, 0);
0
gap> EuclideanDegree(Integers, 5);
5
#
gap> EuclideanQuotient(5, 3);
1
gap> EuclideanQuotient(-5, 3);
-1
gap> EuclideanQuotient(-5, -3);
1
gap> EuclideanQuotient(5, -3);
-1
#
gap> EuclideanRemainder(5, 3);
2
gap> EuclideanRemainder(-5, 3);
-2
gap> EuclideanRemainder(-5, -3);
-2
gap> EuclideanRemainder(5, -3);
2
#
gap> iter := Iterator(Integers);
<iterator of Integers at 0>
gap> List([1..10], i -> NextIterator(iter));
[ 0, 1, -1, 2, -2, 3, -3, 4, -4, 5 ]
gap> it2 := ShallowCopy(iter);
<iterator of Integers at 5>
gap> NextIterator(iter);
-5
gap> it2;
<iterator of Integers at 5>
gap> iter;
<iterator of Integers at -5>
#
gap> iter := Iterator(PositiveIntegers);
<iterator>
gap> List([1..10], i -> NextIterator(iter));
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ]
gap> it2 := ShallowCopy(iter);
<iterator>
gap> NextIterator(iter);
11
gap> NextIterator(it2);
11
#
gap> List([-1,0,1,5/2], i -> i in Integers);
[ true, true, true, false ]
gap> List([-1,0,1,5/2], i -> i in PositiveIntegers);
[ false, false, true, false ]
gap> List([-1,0,1,5/2], i -> i in NonnegativeIntegers);
[ false, true, true, false ]
#
gap> Iterator(5);
Error, You cannot loop over the integer 5 did you mean the range [1..5]
gap> for x in 5 do od;
Error, You cannot loop over the integer 5 did you mean the range [1..5]
#
gap> CoefficientsQadic(0,3);
[ ]
gap> CoefficientsQadic(2,3);
[ 2 ]
gap> CoefficientsQadic(3^5*2^7,3);
[ 0, 0, 0, 0, 0, 2, 0, 2, 1, 1 ]
gap> CoefficientsQadic(3^5*2^7,2);
[ 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1 ]
#
gap> STOP_TEST("integers.tst");
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