GAP (Utils) - Chapter 4: Number-theoretic functions

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4 Number-theoretic functions
4.1 Functions for integers
4.1-1 AllSmoothIntegers
4.1-2 AllProducts
4.1-3 RestrictedPartitionsWithoutRepetitions
4.1-4 NextProbablyPrimeInt
4.1-5 PrimeNumbersIterator

### 4 Number-theoretic functions

#### 4.1 Functions for integers

##### 4.1-1 AllSmoothIntegers

`‣ AllSmoothIntegers`( maxp, maxn ) | ( function )

`‣ AllSmoothIntegers`( L, maxp ) | ( function )

This function has been transferred from package RCWA.

The function `AllSmoothIntegers(maxp,maxn)` returns the list of all positive integers less than or equal to maxn whose prime factors are all in the list \(L = \{p ~|~ p \leqslant maxp, p~\mbox{prime} \}\).

In the alternative form, when \(L\) is a list of primes, the function returns the list of all positive integers whose prime factors lie in \(L\).

    gap> AllSmoothIntegers( 3, 1000 );
    [ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 54, 64, 72, 81, 96,
      108, 128, 144, 162, 192, 216, 243, 256, 288, 324, 384, 432, 486, 512, 576,
      648, 729, 768, 864, 972 ]
    gap> AllSmoothIntegers( [5,11,17], 1000 );
    [ 1, 5, 11, 17, 25, 55, 85, 121, 125, 187, 275, 289, 425, 605, 625, 935 ]
    gap> Length( last );
    16
    gap> List( [3..20], n -> Length( AllSmoothIntegers( [5,11,17], 10^n ) ) );
    [ 16, 29, 50, 78, 114, 155, 212, 282, 359, 452, 565, 691, 831, 992, 1173,
      1374, 1595, 1843 ]

##### 4.1-2 AllProducts

`‣ AllProducts`( L, k ) | ( function )

This function has been transferred from package RCWA.

The command `AllProducts(L,k)` returns the list of all products of k entries of the list L. Note that every ordering of the entries is used so that, in the commuting case, there are bound to be repetitions.

    gap> AllProducts([1..4],3);
    [ 1, 2, 3, 4, 2, 4, 6, 8, 3, 6, 9, 12, 4, 8, 12, 16, 2, 4, 6, 8, 4, 8, 12,
      16, 6, 12, 18, 24, 8, 16, 24, 32, 3, 6, 9, 12, 6, 12, 18, 24, 9, 18, 27,
      36, 12, 24, 36, 48, 4, 8, 12, 16, 8, 16, 24, 32, 12, 24, 36, 48, 16, 32,
      48, 64 ]
    gap> Set(last);
    [ 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 27, 32, 36, 48, 64 ]
    gap> AllProducts( [(1,2,3),(2,3,4)], 2 );
    [ (2,4,3), (1,2)(3,4), (1,3)(2,4), (1,3,2) ]

##### 4.1-3 RestrictedPartitionsWithoutRepetitions

`‣ RestrictedPartitionsWithoutRepetitions`( n, S ) | ( function )

This function has been transferred from package RCWA.

For a positive integer n and a set of positive integers S, this function returns the list of partitions of n into distinct elements of S. Unlike `RestrictedPartitions`, no repetitions are allowed.

    gap> RestrictedPartitions( 20, [4..10] );
    [ [ 4, 4, 4, 4, 4 ], [ 5, 5, 5, 5 ], [ 6, 5, 5, 4 ], [ 6, 6, 4, 4 ],
      [ 7, 5, 4, 4 ], [ 7, 7, 6 ], [ 8, 4, 4, 4 ], [ 8, 6, 6 ], [ 8, 7, 5 ],
      [ 8, 8, 4 ], [ 9, 6, 5 ], [ 9, 7, 4 ], [ 10, 5, 5 ], [ 10, 6, 4 ],
      [ 10, 10 ] ]
    gap> RestrictedPartitionsWithoutRepetitions( 20, [4..10] );
    [ [ 10, 6, 4 ], [ 9, 7, 4 ], [ 9, 6, 5 ], [ 8, 7, 5 ] ]
    gap> RestrictedPartitionsWithoutRepetitions( 10^2, List([1..10], n->n^2 ) );
    [ [ 100 ], [ 64, 36 ], [ 49, 25, 16, 9, 1 ] ]

##### 4.1-4 NextProbablyPrimeInt

`‣ NextProbablyPrimeInt`( n ) | ( function )

This function has been transferred from package RCWA.

The function `NextProbablyPrimeInt(n)` does the same as `NextPrimeInt(n)` except that for reasons of performance it tests numbers only for `IsProbablyPrimeInt(n)` instead of `IsPrimeInt(n)`. For large n, this function is much faster than `NextPrimeInt(n)`

    gap> n := 2^251;
    3618502788666131106986593281521497120414687020801267626233049500247285301248
    gap> NextProbablyPrimeInt( n );
    3618502788666131106986593281521497120414687020801267626233049500247285301313
    gap> time;
    1
    gap> NextPrimeInt( n );
    3618502788666131106986593281521497120414687020801267626233049500247285301313
    gap> time;
    213

##### 4.1-5 PrimeNumbersIterator

`‣ PrimeNumbersIterator`( [chunksize] ) | ( function )

This function has been transferred from package RCWA.

This function returns an iterator which runs over the prime numbers n ascending order; it takes an optional argument `chunksize` which specifies the length of the interval which is sieved in one go (the default is \(10^7\)), and which can be used to balance runtime vs. memory consumption. It is assumed that `chunksize` is larger than any gap between two consecutive primes within the range one intends to run the iterator over.

    gap> iter := PrimeNumbersIterator();;
    gap> for i in [1..100] do  p := NextIterator(iter);  od;
    gap> p;
    541
    gap> sum := 0;;
    gap> ## "prime number race" 1 vs. 3 mod 4
    gap> for p in PrimeNumbersIterator() do
    >       if p <> 2 then sum := sum + E(4)^(p-1); fi;
    >       if sum > 0 then break; fi;
    >    od;
    gap> p;
    26861

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