mth(3)

NAME
mth - standard math module

INTRODUCTION
This chapter covers the mathematical facilities available in the AFNIX standard math module. All AFNIX math service objects are located in the afnix-mth module. This module must be loaded prior any operation. Multiple calls to the module initialization routine are harmless. The interpreter method module loads a specific module by name. When the module has been loaded, the object are available in the afnix:mth nameset.

interp:library "afnix-mth"

Random number services
The math module provides various functions that generate random numbers in different formats.
Function | Description
get-random-integer | return a random integer number
get-random-real | return a random real number between 0.0 and 1.0
get-random-relatif | return a random relatif number
get-random-prime | return a random probable prime relatif number

The numbers are generated with the help of the system random generator. Such generator is machine dependant and results can vary from one machine to another.

Primality testing services
The math module provides various predicates that test a number for a primality condition. Most of these predicates are intricate and are normally not used except the prime-probable-p predicate.
Predicate | Description
fermat-p | Fermat test predicate
miller-rabin-p | Miller-Rabin test predicate
prime-probable-p | general purpose prime probable test
get-random-prime | return a random probable prime relatif number

The fermat-p and miller-rabin-p predicates return true if the primality condition is verified. These predicate operate with a base number. The prime number to test is the second argument.

Fermat primality testing
The fermat-p predicate is a simple primality test based on the "little Fermat theorem". A base number greater than 1 and less than the number to test must be given to run the test.

afnix:mth:fermat-p 2 7

In the preceeding example, the number 7 is tested, and the fermat-p predicate returns true. If a number is prime, it is guaranted to pass the test. The oppositte is not true. For example, 561 is a composite number, but the Fermat test will succeed with the base 2. Numbers that successfully pass the Fermat test but which are composite are called Carmichael numbers. For those numbers, a better test needs to be employed, such like the Miller-Rabin test.

Miller-Rabin primality testing
The miller-rabin-p predicate is a complex primality test that is more efficient in detecting prime number at the cost of a longer computation. A base number greater than 1 and less than the number to test must be given to run the test.

afnix:mth:miller-rabin-p 2 561

In the preceeding example, the number 561, which is a Carmichael number, is tested, and the miller-rabin-p predicate returns false. The probability that a number is prime depends on the number of times the test is ran. Numerous studies have been made to determine the optimal number of passes that are needed to declare that a number is prime with a good probability. The prime-probable-p predicate takes care to run the optimal number of passes.

General primality testing
The prime-probable-p predicate is a complex primality test that incorporates various primality tests. To make the story short, the prime candidate is first tested with a series of small prime numbers. Then a fast Fermat test is executed. Finally, a series of Miller-Rabin tests are executed. Unlike the other primality tests, this predicate operates with a number only and optionally, the number of test passes. This predicate is the recommended test for the folks who want to test their numbers.

afnix:mth:prime-probable-p 17863

STANDARD MATH REFERENCE
This appendix is a reference of the AFNIX standard math module.
Symbol | Description
afnix-mth | module
afnix:mth | nameset

Functions

    get-random-integer -> Integer (none|Integer)
    The get-random-integer function returns a random integer number. Without argument, the integer range is machine dependent. With one integer argument, the resulting integer number is less than the specified maximum bound.

    get-random-real -> Real (none|Boolean)
    The get-random-real function returns a random real number between 0.0 and 1.0. In the first form, without argument, the random number is between 0.0 and 1.0 with 1.0 included. In the second form, the boolean flag controls whether or not the 1.0 is included in the result. If the argument is false, the 1.0 value is guaranted to be excluded from the result. If the argument is true, the 1.0 is a possible random real value. Calling this function with the argument set to true is equivalent to the first form without argument.

    get-random-relatif -> Relatif (Integer|Integer Boolean)
    The get-random-relatif function returns a n bits random positive relatif number. In the first form, the argument is the number of bits. In the second form, the first argument is the number of bits and the second argument, when true produce an odd number, or an even number when false.

    get-random-prime -> Relatif (Integer)
    The get-random-prime function returns a n bits random positive relatif probable prime number. The argument is the number of bits. The prime number is generated by using the Miller-Rabin primality test. As such, the returned number is declared probable prime. The more bits neede, the longer it takes to generate such number.

    fermat-p -> Boolean (Integer|Relatif Integer|Relatif)
    The fermat-p predicate returns true if the little fermat theorem is validated. The first argument is the base number and the second argument is the prime number to validate.

    miller-rabin-p -> Boolean (Integer|Relatif Integer|Relatif)
    The miller-rabin-p predicate returns true if the Miller-Rabin test is validated. The first argument is the base number and the second argument is the prime number to validate.

    prime-probable-p -> Boolean (Integer|Relatif [Integer])
    The prime-probable-p predicate returns true if the argument is a probable prime. In the first form, only an integer or relatif number is required. In the second form, the number of iterations is specified as the second argument. By default, the number of iterations is specified to 56.