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Math::Big - routines (cos,sin,primes,hailstone,euler,fibbonaci etc) with big numbers


    use Math::Big qw/primes fibonacci hailstone factors wheel
      cos sin tan euler bernoulli arctan arcsin pi/;

    @primes     = primes(100);          # first 100 primes
    $count      = primes(100);          # number of primes <= 100
    @fib        = fibonacci (100);      # first 100 fibonacci numbers
    $fib_1000   = fibonacci (1000);     # 1000th fibonacci number
    $hailstone  = hailstone (1000);     # length of sequence
    @hailstone  = hailstone (127);      # the entire sequence

    $factorial  = factorial(1000);      # factorial 1000!

    $e = euler(1,64);                   # e to 64 digits

    $b3 = bernoulli(3);

    $cos        = cos(0.5,128);         # cosinus to 128 digits
    $sin        = sin(0.5,128);         # sinus to 128 digits
    $cosh       = cosh(0.5,128);        # cosinus hyperbolicus to 128 digits
    $sinh       = sinh(0.5,128);        # sinus hyperbolicus to 128 digits
    $tan        = tan(0.5,128);         # tangens to 128 digits
    $arctan     = arctan(0.5,64);       # arcus tangens to 64 digits
    $arcsin     = arcsin(0.5,32);       # arcus sinus to 32 digits
    $arcsinh    = arcsin(0.5,18);       # arcus sinus hyperbolicus to 18 digits

    $pi         = pi(1024);             # first 1024 digits
    $log        = log(64,2);            # $log==6, because 2**6==64
    $log        = log(100,10);          # $log==2, because 10**2==100
    $log        = log(100);             # base defaults to 10: $log==2


perl5.006002, Exporter, Math::BigInt, Math::BigFloat


Exports nothing on default, but can export primes(), fibonacci(), hailstone(), bernoulli, euler, sin, cos, tan, cosh, sinh, arctan, arcsin, arcsinh, pi, log and factorial.


This module contains some routines that may come in handy when you want to do some math with really, really big (or small) numbers. These are primarily examples.


        @primes = primes($n);
        $primes = primes($n);

Calculates all the primes below N and returns them as array. In scalar context returns the prime count of N (the number of primes less than or equal to N).

This uses an optimized version of the Sieve of Eratosthenes, which takes half of the time and half of the space, but is still O(N).

        @fib = fibonacci($n);
        $fib = fibonacci($n);

Calculates the first N fibonacci numbers and returns them as array. In scalar context returns the Nth number of the Fibonacci series.

The scalar context version uses an ultra-fast conquer-divide style algorithm to calculate the result and is many times faster than the straightforward way of calculating the linear sum.

        @hail = hailstone($n);          # sequence
        $hail = hailstone($n);          # length of sequence

Calculates the Hailstone sequence for the number N. This sequence is defined as follows:

        while (N != 0)
          if (N is even)
            N is N /2
            N = N * 3 +1

It is not yet proven whether for every N the sequence reaches 1, but it apparently does so. The number of steps is somewhat chaotically.

        ($n,$a) = base($number,$base);

Reduces a number to $base to the $nth power plus $a. Example:

        use Math::BigInt :constant;
        use Math::Big qw/base/;

        print base ( 2 ** 150 + 42,2);

This will print 150 and 42.

        $string = to_base($number,$base);

        $string = to_base($number,$base, $alphabet);

Returns a string of $number in base $base. The alphabet is optional if $base is less or equal than 36. $alphabet is a string.


        print to_base(15,2);            # 1111
        print to_base(15,16);           # F
        print to_base(31,16);           # 1F
        $n = factorial($number);

Calculate n! for n = 0>.

Uses internally Math::BigInt's bfac() method.

        $b = bernoulli($n);
        ($c,$d) = bernoulli($n);        # $b = $c/$d

Calculate the Nth number in the Bernoulli series. Only the first 40 are defined for now.

        $e = euler($x,$d);

Calculate Euler's constant to the power of $x (usual 1), to $d digits. Defaults to 1 and 42 digits.

        $sin = sin($x,$d);

Calculate sinus of $x, to $d digits.

        $cos = cos($x,$d);

Calculate cosinus of $x, to $d digits.

        $tan = tan($x,$d);

Calculate tangens of $x, to $d digits.

        $arctan = arctan($x,$d);

Calculate arcus tangens of $x, to $d digits.

        $arctanh = arctanh($x,$d);

Calculate arcus tangens hyperbolicus of $x, to $d digits.

        $arcsin = arcsin($x,$d);

Calculate arcus sinus of $x, to $d digits.

        $arcsinh = arcsinh($x,$d);

Calculate arcus sinus hyperbolicus of $x, to $d digits.

        $cosh = cosh($x,$d);

Calculate cosinus hyperbolicus of $x, to $d digits.

        $sinh = sinh($x,$d);

Calculate sinus hyperbolicus of $<$x>, to $d digits.

        $pi = pi($N);

The number PI to $N digits after the dot.

        $log = log($number,$base,$A);

Calculates the logarithmn of $number to base $base, with $A digits accuracy and returns a new number as the result (leaving $number alone).

Math::BigInt objects are promoted to Math::BigFloat objects, meaning you will never get a truncated integer result like when using Math::BigInt-blog()>.


  • Primes and the Fibonacci series use an array of size N and will not be able to calculate big sequences due to memory constraints.

    The exception is fibonacci in scalar context, this is able to calculate arbitrarily big numbers in O(N) time:

            use Math::Big;
            use Math::BigInt qw/:constant/;
            $fib = Math::Big::fibonacci( 2 ** 320 );
  • The Bernoulli numbers are not yet calculated, but looked up in a table, which has only 40 elements. So bernoulli($x) with $x > 42 will fail.

    If you know of an algorithmn to calculate them, please drop me a note.


Please report any bugs or feature requests to bug-math-big at, or through the web interface at (requires login). We will be notified, and then you'll automatically be notified of progress on your bug as I make changes.


You can find documentation for this module with the perldoc command.

    perldoc Math::Big

You can also look for information at:


This program is free software; you may redistribute it and/or modify it under the same terms as Perl itself.


  • Tels 2001-2007.

  • Peter John Acklam <> 2016-.