Numbers

java.lang.Object
- org.mathIT.numbers.Numbers

```
public class Numbers
extends Object
```
This class provides basic number theoretic functions.

Version:

1.1

Author:

Andreas de Vries

Field Summary

Fields
Modifier and Type	Field and Description
`static long[][]`	`binomial` Array containing the binomial coefficients (n over k) = `binomial`[n][k] for 0 ≤ n, k ≤ 66.
`static double`	`gamma` Euler-Mascheroni constant γ = 0.577215664901532860605.
`static double`	`GAMMA` Euler-Mascheroni constant γ = 0.577215664901532860605.
`static double`	`RADIANS` The constant 2π/360, the ratio of 1 radians per degree.
`static double`	`SQRT_1_2` Square root of 1/2.
`static double`	`SQRT2` constant representing the square root of 2.
`static double`	`SQRT3` constant representing the square root of 3.
`static char`	`TB0` Symbol in balanced ternary system for 0.
`static char`	`TBM` Symbol in balanced ternary system for -1.
`static char`	`TBP` Symbol in balanced ternary system for +1.
`static double`	`ZETA_11` The value of the Riemann Zeta function ζ(11).
`static double`	`ZETA_13` The value of the Riemann Zeta function ζ(13).
`static double`	`ZETA_15` The value of the Riemann Zeta function ζ(15).
`static double`	`ZETA_17` The value of the Riemann Zeta function ζ(17).
`static double`	`ZETA_19` The value of the Riemann Zeta function ζ(19).
`static double`	`ZETA_21` The value of the Riemann Zeta function ζ(21).
`static double`	`ZETA_23` The value of the Riemann Zeta function ζ(23).
`static double`	`ZETA_25` The value of the Riemann Zeta function ζ(25).
`static double`	`ZETA_3` The value of the Riemann Zeta function ζ(3).
`static double`	`ZETA_5` The value of the Riemann Zeta function ζ(5).
`static double`	`ZETA_7` The value of the Riemann Zeta function ζ(7).
`static double`	`ZETA_9` The value of the Riemann Zeta function ζ(9).

Method Summary

All Methods Static Methods Concrete Methods
Modifier and Type	Method and Description
`static String`	`add3(String m, String n)` Returns the sum of m and n in the ternary system.
`static char`	`bbp(long position)` Returns the hexadecimal digit of π at the given position, according to the Bailey-Borwein-Plouffe algorithm.
`static Rational`	`bernoulli(int n)` Returns the Bernoulli number B_n of the nonnegative integer n.
`static Rational[]`	`bernoulliNumbers(int n)` Returns an array of n+1 rational numbers with entry number j representing the Bernoulli number B_j, j = 0, 1, ..., n.
`static long[]`	`bestRationalApproximation(double x, int limit)` Returns the best rational approximation of a real number x, that is, the integers p, q such that x ≈ p/q.
`static double`	`beta(double z, double w)` Returns the beta function value B(z,w) = ∫₀¹ t^z-1 (1-t)^w-1 dt.
`static double`	`binomial(long n, long k)` Returns the binomial coefficient (n choose k) as a floating point number.
`static long`	`binToDec(String bin)` Returns a binary string as an integer.
`static double`	`binToDouble(String bin)` Returns a binary string as a decimal floating-point number.
`static String`	`binToHex(String bin)` Returns a binary string as a hexadecimal string.
`static long[]`	`continuedFraction(double x, int limit)` Returns an array containing coefficients of the continued fraction of the number x, with at most `limit` coefficients.
`static String`	`decToBin(double z)` Returns z as a binary string with at most 70 digits right of the binary point.
`static String`	`decToBin(double z, int limit)` Returns z as a binary string with at most `limit` positions right of the binary point.
`static String`	`decToBin(long n)` Returns n as a binary string.
`static String`	`decToBin(long n, int minimumLength)` Returns n as a binary string of the specified minimum length.
`static String`	`decToBinByte(long n)` Returns n as a binary string in a 1-byte form.
`static String`	`decToHex(double z)` Returns z as a hexadecimal string with at most 20 digits right of the hexadecimal point.
`static String`	`decToHex(double z, int limit)` Returns z as a hexadecimal string with at most `limit` positions right of the hexadecimal point.
`static String`	`decToHex(long n)` Returns n as a hexadecimal string.
`static String`	`decToHex2Byte(long n)` Returns n as a hexadecimal string in 2-byte form.
`static String`	`decToTern(double z)` Returns z as a ternary string with at most 40 digits right of the ternary point.
`static String`	`decToTern(double z, int limit)` Returns z as a ternary string with at most `limit` positions right of the ternary point.
`static String`	`decToTern(long n)` Returns n as a ternary string.
`static String`	`decToTern(long n, int minimumLength)` Returns n as a ternary string of the specified minimum length.
`static String`	`decToTernB(double z)` Returns z as a balanced ternary string with at most 40 digits right of the ternary point.
`static String`	`decToTernB(double z, int limit)` Returns z as a balanced ternary string with at most `limit` positions right of the ternary point.
`static String`	`decToTernB(long n)` Returns n as a balanced ternary string.
`static String`	`decToTernB(long n, int minimumLength)` Returns n as a balanced ternary string of the specified minimum length.
`static long`	`div3b(long n)` Returns the integer division n/3 in the balanced ternary system.
`static long[]`	`euclid(long m, long n)` Returns an array of three integers x[0], x[1], x[2] as given by the extended Euclidian algorithm for positive integers m and n.
`static BigInteger`	`exactBinomial(int n, int k)` Returns the exact binomial coefficient (n choose k) as an integer.
`static BigInteger`	`factorial(int n)` Returns the exact value of n!.
`static double`	`gamma(double x)` The Euler Gamma function
`static long`	`gcd(long m, long n)` Returns the greatest common divisor of the integers m and n.
`static String`	`grayCode(long x)` Returns the Gray code of an integer.
`static String`	`grayCode(long x, int minimumLength)` Returns the Gray code of an integer x, with a given minimum length.
`static String`	`grayCodeToBinary(String grayCode)` Returns binary representation of the integer in which is represented by a Gray code string.
`static String`	`grayCodeToBinary(String grayCode, int minimumLength)` Returns binary representation of the integer in which is represented by a Gray code string.
`static long`	`grayCodeToLong(String grayCode)` Returns the integer represented by a Gray code string.
`static String`	`hexToBin(String hex)` Returns a hexadecimal string as a binary string.
`static long`	`hexToDec(String hex)` Returns a hexadecimal string as a decimal integer.
`static double`	`hexToDouble(String hex)` Returns a hexadecimal string as a decimal floating-point number.
`static double`	`hypotenuse(double a, double b)` Returns sqrt(a^2 + b^2) without under/overflow.
`static double`	`incBeta(double a, double b, double x)` The incomplete beta function for 0 <= x <= 1.
`static double`	`incGamma(double a, double x)` incomplete Gammma function.
`static boolean`	`isPower(long n)` Tests whether there exist integers m and k such that n = m^k.
`static boolean`	`isPrime(long n)` Tests deterministically whether the given integer is prime.
`static boolean`	`isStrongProbablePrime(int n, int a)` Returns true if n is a strong probable prime to base a, and false if n is not prime.
`static int`	`largestPrimeFactor(int n)` Determines the largest prime factor of a number, using naive trial division, appropriate only for comparatively small integers.
`static long`	`lcm(long m, long n)` Returns the least common multiple of m and n.
`static int`	`leastPrimeFactor(int n)` Determines the least prime factor of a number, using naive trial division, appropriate only for comparably small integers.
`static double`	`lnFactorial(long n)` Returns ln (n!).
`static double`	`lnGamma(double x)` Logarithm of the Euler Gamma function.
`static double`	`mod(double n, double m)` Returns the value of n mod m, even for negative values.
`static long`	`mod(long n, long m)` Returns the value of n mod m, even for negative values.
`static char`	`mod3b(long n)` Returns the symbol in the balanced ternary system representing the value n mod 3.
`static double`	`modPow(double x, long e, double n)` Returns the value of (x^e) mod n.
`static int`	`modPow(long x, long e, int m)` Returns the value of x^e mod n.
`static long`	`modPow(long x, long e, long n)` Returns the value of x^e mod n.
`static BigInteger[]`	`numberOfPartitions(int n)` Returns an array of the numbers of partitions up to the number n.
`static long`	`ord(long m, long n)` Returns the order ord(m,n) of m modulo n.
`static ArrayList<LinkedList<Integer>>`	`partitions(int n)` Returns a list of all partitions of a small integer n.
`static double`	`pow(double x, long e)` Returns the value of x^e.
`static long`	`pow(long x, long e)` Returns the value of x^e.
`static long`	`pow2(int n)` Returns the value of 2ⁿ.
`static double`	`root(int n, double z)` Returns the nth root of z, with an accuracy of 1e-12 z.
`static long[]`	`solveLinearDiophantine(long a, long b, long c)` Returns the array {d,x,y} such that the linear Diophantine equation ax + by = c is solved and d = gcd(a,b).
`static long`	`sqr(long x)` Returns the value of x².
`static int`	`sum(Collection<Integer> list)` Returns the sum of the integers of the specified list.
`static String`	`ternaryComplement(String tern)` Returns the ternary complement of the specified number in ternary representation.
`static long`	`ternBToDec(String tern)` Returns a balanced ternary string as an integer.
`static double`	`ternBToDouble(String tern)` Returns a balanced ternary string as a decimal floating-point number.
`static long`	`ternToDec(String tern)` Returns a ternary string as an integer.
`static double`	`ternToDouble(String tern)` Returns a ternary string as a decimal floating-point number.
`static String`	`ternToTernB(String tern)` Returns a ternary string as an balanced ternary string.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail
- gamma
```
public static final double gamma
```
  Euler-Mascheroni constant γ = 0.577215664901532860605. According to general mathematical conventions, it is written in small letters.
  
  See Also:
  
  GAMMA, BigNumbers.GAMMA, Constant Field Values
- GAMMA
```
public static final double GAMMA
```
  Euler-Mascheroni constant γ = 0.577215664901532860605.
  
  See Also:
  
  BigNumbers.GAMMA, Constant Field Values
- SQRT2
```
public static final double SQRT2
```
  constant representing the square root of 2.
  
  See Also:
  
  BigNumbers.SQRT_TWO, Constant Field Values
- SQRT3
```
public static final double SQRT3
```
  constant representing the square root of 3.
  
  See Also:
  
  Constant Field Values
- SQRT_1_2
```
public static final double SQRT_1_2
```
  Square root of 1/2.
  
  See Also:
  
  BigNumbers.SQRT_ONE_HALF, Constant Field Values
- RADIANS
```
public static final double RADIANS
```
  The constant 2π/360, the ratio of 1 radians per degree.
  
  See Also:
  
  Math.PI, BigNumbers.RADIANS, Constant Field Values
- ZETA_3
```
public static final double ZETA_3
```
  The value of the Riemann Zeta function ζ(3). See M. Abramowitz and I. A. Stegun: Handbook of Mathematical Functions, p. 811.
  
  See Also:
  
  BigNumbers.ZETA_3, Constant Field Values
- ZETA_5
```
public static final double ZETA_5
```
  The value of the Riemann Zeta function ζ(5). See M. Abramowitz and I. A. Stegun: Handbook of Mathematical Functions, p. 811.
  
  See Also:
  
  BigNumbers.ZETA_5, Constant Field Values
- ZETA_7
```
public static final double ZETA_7
```
  The value of the Riemann Zeta function ζ(7). See M. Abramowitz and I. A. Stegun: Handbook of Mathematical Functions, p. 811.
  
  See Also:
  
  BigNumbers.ZETA_7, Constant Field Values
- ZETA_9
```
public static final double ZETA_9
```
  The value of the Riemann Zeta function ζ(9). See M. Abramowitz and I. A. Stegun: Handbook of Mathematical Functions, p. 811.
  
  See Also:
  
  BigNumbers.ZETA_9, Constant Field Values
- ZETA_11
```
public static final double ZETA_11
```
  The value of the Riemann Zeta function ζ(11). See M. Abramowitz and I. A. Stegun: Handbook of Mathematical Functions, p. 811.
  
  See Also:
  
  BigNumbers.ZETA_11, Constant Field Values
- ZETA_13
```
public static final double ZETA_13
```
  The value of the Riemann Zeta function ζ(13). See M. Abramowitz and I. A. Stegun: Handbook of Mathematical Functions, p. 811.
  
  See Also:
  
  BigNumbers.ZETA_13, Constant Field Values
- ZETA_15
```
public static final double ZETA_15
```
  The value of the Riemann Zeta function ζ(15). See M. Abramowitz and I. A. Stegun: Handbook of Mathematical Functions, p. 811.
  
  See Also:
  
  BigNumbers.ZETA_15, Constant Field Values
- ZETA_17
```
public static final double ZETA_17
```
  The value of the Riemann Zeta function ζ(17). See M. Abramowitz and I. A. Stegun: Handbook of Mathematical Functions, p. 811.
  
  See Also:
  
  BigNumbers.ZETA_17, Constant Field Values
- ZETA_19
```
public static final double ZETA_19
```
  The value of the Riemann Zeta function ζ(19). See M. Abramowitz and I. A. Stegun: Handbook of Mathematical Functions, p. 811.
  
  See Also:
  
  BigNumbers.ZETA_19, Constant Field Values
- ZETA_21
```
public static final double ZETA_21
```
  The value of the Riemann Zeta function ζ(21). See M. Abramowitz and I. A. Stegun: Handbook of Mathematical Functions, p. 811.
  
  See Also:
  
  BigNumbers.ZETA_21, Constant Field Values
- ZETA_23
```
public static final double ZETA_23
```
  The value of the Riemann Zeta function ζ(23). See M. Abramowitz and I. A. Stegun: Handbook of Mathematical Functions, p. 811.
  
  See Also:
  
  BigNumbers.ZETA_23, Constant Field Values
- ZETA_25
```
public static final double ZETA_25
```
  The value of the Riemann Zeta function ζ(25). See M. Abramowitz and I. A. Stegun: Handbook of Mathematical Functions, p. 811.
  
  See Also:
  
  BigNumbers.ZETA_25, Constant Field Values
- binomial
```
public static final long[][] binomial
```
  Array containing the binomial coefficients (n over k) = binomial[n][k] for 0 ≤ n, k ≤ 66. Here binomial[n][k] = 0 if n < k.
- TBM
```
public static final char TBM
```
  Symbol in balanced ternary system for -1.
  
  See Also:
  
  Constant Field Values
- TB0
```
public static final char TB0
```
  Symbol in balanced ternary system for 0.
  
  See Also:
  
  Constant Field Values
- TBP
```
public static final char TBP
```
  Symbol in balanced ternary system for +1.
  
  See Also:
  
  Constant Field Values

Method Detail

beta
```
public static double beta(double z,
                          double w)
```
Returns the beta function value B(z,w) = ∫₀¹ t^z-1 (1-t)^w-1 dt. We have B(z,w) = B(w,z) = Γ(z) Γ(w) / Γ(z+w).

Parameters:

z - first argument

w - second argument

Returns:

value of the beta function B(z,w)

See Also:

gamma(double)

incBeta

public static double incBeta(double a,
                             double b,
                             double x)

The incomplete beta function for 0 <= x <= 1. For a>0, b>0 it is defined by


                      1
       I_x (a,b) = --------  int_0^x  t^{a-1} (1-t)^{b-1} dt.
                    B(a,b)

It can be well approximated by the continued fraction


                    x^a (1-x)^b   {  1                }
       I_x (a,b) = -------------  { ----------------- }
                      a B(a,b)    {        d_1        }
                                  {  1 + ------------ }
                                  {          d_2      }
                                  {      1 + -------- }
                                  {              d_3  }
                                  {          1 + ---- }
                                  {               ... }

with the coefficients


                     (a+m)(a+b+m)                  m(b-mm)
       d_{2m+1} = - -------------- x,  d_{2m} = --------------  x.
                    (a+2m)(a+2m-1)              (a+2m-1)(a+2m)

The approximation converges rapidly for


            a+1
       x > -----.
           a+b+1

If this condition is not satisfied, then just 1-x does, and we can apply the symmetry relation


       I_x (a,b) = 1 - I_{1-x} (b,a).

Parameters:: a - first parameter of the beta function, a > 0; b - second parameter of the beta function, b > 0; x - index
Returns:: the incomplete beta function value
See Also:: beta(double,double)

gamma
```
public static double gamma(double x)
```
The Euler Gamma function
Γ(x) = ∫₀^∞ t^x-1 e^-t dt.

Parameters:

x - the argument

Returns:

the value Γ(x)

See Also:

beta(double, double)

lnGamma
```
public static double lnGamma(double x)
```
Logarithm of the Euler Gamma function. It is computed according to the Lanczos approximation. See S. Brandt: Datenanalyse. Spektrum Akademischer Verlag, Heidelberg Berlin, §D.1

Parameters:

x - the argument

Returns:

the logarithm ln(Gamma(x)) of the Gamma function

See Also:

gamma(double)

incGamma
```
public static double incGamma(double a,
                              double x)
```
incomplete Gammma function.

Parameters:

a - the argument, a > 0

x - the argument

Returns:

the incGamma(a,x)

See Also:

gamma(double)

exactBinomial
```
public static BigInteger exactBinomial(int n,
                                       int k)
```
Returns the exact binomial coefficient (n choose k) as an integer.

Parameters:

n - an integer

k - an integer

Returns:

(n over k)

binomial
```
public static double binomial(long n,
                              long k)
```
Returns the binomial coefficient (n choose k) as a floating point number.

Parameters:

n - an integer

k - an integer

Returns:

(n over k)

lnFactorial
```
public static double lnFactorial(long n)
```
Returns ln (n!).

Parameters:

n - an integer

Returns:

ln (n!)

factorial
```
public static BigInteger factorial(int n)
```
Returns the exact value of n!. Argument values n %gt;> 10000 lead to unreasonable running times.

Parameters:

n - the value to be computed

Returns:

the value of n!

Throws:

IllegalArgumentException - if n < 0

mod
```
public static long mod(long n,
                       long m)
```
Returns the value of n mod m, even for negative values. Here the usual periodic definition is used, i.e.,
n mod m = n - ⎣n/m⎦ for m ≠ 0, and n mod 0 = n.
For instance, 5 mod 3 = 2, but -5 mod 3 = 1, 5 mod (-3) = -1, and -5 mod (-3) = -2. See R.L. Graham, D.E. Knuth, O. Patashnik: Concrete Mathematics. 2nd Edition. Addison-Wesley, Upper Saddle River, NJ 1994, §3.4 (p.82)

Parameters:

n - the value to be raised to the power

m - the modulus

Returns:

the value n mod m

mod
```
public static double mod(double n,
                         double m)
```
Returns the value of n mod m, even for negative values. Here the usual periodic definition is used, i.e., 5 mod 3 = 2, but -5 mod 3 = 1, 5 mod (-3) = -1, and -5 mod (-3) = -2. See R.L. Graham, D.E. Knuth, O. Patashnik: Concrete Mathematics. 2nd Edition. Addison-Wesley, Upper Saddle River, NJ 1994, §3.4 (p.82)

Parameters:

n - the value to be computed

m - the modulus

Returns:

the value n mod m

pow
```
public static long pow(long x,
                       long e)
```
Returns the value of x^e.

Parameters:

x - the value to be raised to the power

e - the exponent

Returns:

the value of x^e

Throws:

IllegalArgumentException - if e < 0

pow
```
public static double pow(double x,
                         long e)
```
Returns the value of x^e.

Parameters:

x - the value to be raised to the power

e - the exponent

Returns:

the value of x^e

pow2
```
public static long pow2(int n)
```
Returns the value of 2ⁿ.

Parameters:

n - the exponent

Returns:

the value of 2ⁿ

modPow
```
public static int modPow(long x,
                         long e,
                         int m)
```
Returns the value of x^e mod n.

Parameters:

x - the value to be raised to the power

e - the exponent

m - the modulus

Returns:

the value of x^e mod n

Throws:

ArithmeticException - if e < 0 and gcd(x, n) > 1

modPow
```
public static long modPow(long x,
                          long e,
                          long n)
```
Returns the value of x^e mod n.

Parameters:

x - the value to be raised to the power

e - the exponent

n - the modulus

Returns:

the value of x^e mod n

Throws:

ArithmeticException - if e < 0 and gcd(x, n) > 1

modPow
```
public static double modPow(double x,
                            long e,
                            double n)
```
Returns the value of (x^e) mod n.

Parameters:

x - the value to be raised to the power

e - the exponent

n - the modulus

Returns:

the value of x^e mod n

hypotenuse
```
public static double hypotenuse(double a,
                                double b)
```
Returns sqrt(a^2 + b^2) without under/overflow. This method can be conveniently used to compute iteratively the norm norm of a vector x = (x₀, ..., x_n-1) by
```
    double norm = 0;
    for (int i = 0; i < x.length; i++) {
       norm = Numbers.hypotenuse(norm, x[i]);
    }
  
```
Parameters:

a - the first triangle leg

b - the first triangle leg

Returns:

sqrt(a^2 + b^2)

root
```
public static double root(int n,
                          double z)
```
Returns the nth root of z, with an accuracy of 1e-12 z. The root is understood as the principal root r with the unique real number with the same sign as z such that rⁿ = z. If n = 0, the values 0, 1, or Double.POSITIVE_INFINITY is returned depending on whether |z| < 0, |z| = 1, or z > 1; if n = 0 and z < -1, then the value is not defined and an exception is thrown. If n < 0, then root(-n, z) is returned.

Parameters:

n - the radical

z - the radicand

Returns:

the principal nth root r of z such that rⁿ = z

Throws:

IllegalArgumentException - if (a) n = 0 and z < -1, or (b) z < 0 and n is even

sqr
```
public static long sqr(long x)
```
Returns the value of x².

Parameters:

x - the value to be squared

Returns:

the square of the input x, i.e., x²

gcd
```
public static long gcd(long m,
                       long n)
```
Returns the greatest common divisor of the integers m and n.

Parameters:

m - the first integer

n - the second integer

Returns:

the greatest commom divisor of m and n

euclid
```
public static long[] euclid(long m,
                            long n)
```
Returns an array of three integers x[0], x[1], x[2] as given by the extended Euclidian algorithm for positive integers m and n. The three integers satisfy the equations
x[0] = gcd(m, n) = x[1] m + x[2] n.
This methods implements a recursive version of the extended Euclidian algorithm.

Parameters:

m - the first integer, must be positive

n - the second integer, must be positive

Returns:

an array of three integers x[0], x[1], x[2] such that x[0] = gcd(m, n) = x[1] m + x[2] n

See Also:

gcd(long,long), BigNumbers.euclid(java.math.BigInteger,java.math.BigInteger)

lcm
```
public static long lcm(long m,
                       long n)
```
Returns the least common multiple of m and n. It is computed by lcm(m, n) = mn / gcd(m, n).

Parameters:

m - the first integer

n - the second integer

Returns:

the least common multiple of m and n

See Also:

gcd(long,long)

isPrime
```
public static boolean isPrime(long n)
```
Tests deterministically whether the given integer is prime. This algorithm is a naive trial division primality test, appropriate only for comparably small integers.

Parameters:

n - the integer to test

Returns:

true if and only if n i prime.

solveLinearDiophantine
```
public static long[] solveLinearDiophantine(long a,
                                            long b,
                                            long c)
```
Returns the array {d,x,y} such that the linear Diophantine equation
ax + by = c
is solved and d = gcd(a,b). If the equation is not solvable, then {0,0,0} is returned. Note that the equation is solvable if and only if gcd(a,b) divides c. Moreover, from a given solution {x₀, y₀} a general solution can be computed by
x = x₀ + bt/d, y = y₀ - at/d
for t ∈ ℤ. Cf. R. Schulze.Pillot: Elementare Algebra und Zahlentheorie. Springer, Berlin Heidelberg 2007, Satz 3.12.

Parameters:

a - first coefficient of the linear Diophantine equation

b - second coefficient of the linear Diophantine equation

c - third coefficient of the linear Diophantine equation

Returns:

the array {d,x,y} such that the linear Diophantine equation ax + by = c is solved and d = gcd(a,b), or {0,0,0} if the equation is not solbvable

isStrongProbablePrime
```
public static boolean isStrongProbablePrime(int n,
                                            int a)
```
Returns true if n is a strong probable prime to base a, and false if n is not prime. This algorithm is the core of the Miller-Rabin primality test, cf. R. Crandall & C. Pomerance: Prime Numbers. A Computational Perspective. 2^nd edition. Springer, New York 2005, &sec;3.5. The number n must be an odd number > 3, and 1 < a < n-1.

Parameters:

n - the number to be tested on strong probable primality

a - the base of the strong probable primality test

Returns:

true if n is a strong probable prime to base a, and false if n is not prime

Throws:

IllegalArgumentException - if n ≤ 3, or a ≤ 1, or a ≥ n - 1

See Also:

BigNumbers.isStrongProbablePrime(BigInteger,BigInteger)

leastPrimeFactor
```
public static int leastPrimeFactor(int n)
```
Determines the least prime factor of a number, using naive trial division, appropriate only for comparably small integers.

Parameters:

n - an integer

Returns:

the least prime factor of n

largestPrimeFactor
```
public static int largestPrimeFactor(int n)
```
Determines the largest prime factor of a number, using naive trial division, appropriate only for comparatively small integers.

Parameters:

n - an integer

Returns:

the largest prime factor of n

ord
```
public static long ord(long m,
                       long n)
```
Returns the order ord(m,n) of m modulo n. More precisely, we have
ord(m,n) = min_i { i > 0 : mⁱ = 1 mod n },
The order is found by Floyd's cycle finding algorithm.

Parameters:

m - the first integer

n - the second integer

Returns:

the order of m mod n

isPower
```
public static boolean isPower(long n)
```
Tests whether there exist integers m and k such that n = m^k.

Parameters:

n - the number to be checked

Returns:

true if and only if there exist integers m and k such that n = m^k

continuedFraction
```
public static long[] continuedFraction(double x,
                                       int limit)
```
Returns an array containing coefficients of the continued fraction of the number x, with at most limit coefficients. Note that by the finite precision of x the higher continuous fraction coefficients get more and more imprecise. For instance, for the Euler number the coefficients are correct up to the limit 20.

Parameters:

x - the number to be expanded as a continuous fraction

limit - the maximum number of continuous fraction coefficients to be computed

Returns:

an array of length limit containing the continuous fraction coefficients

bestRationalApproximation
```
public static long[] bestRationalApproximation(double x,
                                               int limit)
```
Returns the best rational approximation of a real number x, that is, the integers p, q such that x ≈ p/q. The algorithm computes the continued fraction coefficients corresponding to x, where the number of coefficients is bounded by limit. Usually, the value of limit should be about 20.

Parameters:

x - the number to be approximated

limit - the maximum number of continued fraction coefficients being considered

Returns:

a two-element array y where y[0] = p and y[1] = q such that x ≈ p/q

See Also:

continuedFraction(double,int)

bernoulli

public static Rational bernoulli(int n)

Returns the Bernoulli number B_n of the nonnegative integer n. B_n is recursively defined as

B_n

= -

n+1

n-1

j = 0

(

n+1

)

B_j

and B₀ = 1. Note that in case of repeatedly required Bernoulli numbers the method bernoulliNumbers(int) may be used more efficiently, giving the whole list of Bernoulli numbers up to B_n.

Parameters:: n - a nonnegative integer
Returns:: a rational number representing B_n
Throws:: IllegalArgumentException - if n<0
See Also:: bernoulliNumbers(int)

bernoulliNumbers
```
public static Rational[] bernoulliNumbers(int n)
```
Returns an array of n+1 rational numbers with entry number j representing the Bernoulli number B_j, j = 0, 1, ..., n. Since the Bernoulli numbers are defined recursively, this method allows efficient implementations for algorithms which repeatedly use Bernoulli numbers.

Parameters:

n - a nonnegative integer

Returns:

a rational number representing B_n

Throws:

IllegalArgumentException - if n<0

See Also:

bernoulli(int)

numberOfPartitions
```
public static BigInteger[] numberOfPartitions(int n)
```
Returns an array of the numbers of partitions up to the number n. A partition of a positive integer n is a set of tuples consisting of positive summands yielding a total of exactly n. For instance, there are three partitions of n = 3, namely {(1,1,1), (1,2), (3)} since 1 + 1 + 1 = 1 + 2 = 3.
For details see R.L. Graham, D.E. Knuth, O. Patashnik: Concrete Mathematics. 2nd Edition. Addison-Wesley, Upper Saddle River, NJ 1994, §7.1 (p. 330)

Parameters:

n - a positive integer

Returns:

an array p of the numbers of partitions where p[i] is the number of partitions of i ≤ n.

See Also:

partitions(int)

partitions
```
public static ArrayList<LinkedList<Integer>> partitions(int n)
```
Returns a list of all partitions of a small integer n. A partition of a positive integer is a set of positive integers {n₁, ..., n_k} whose sum equals n, i.e., n₁ + ... + n_k = n. For instance, there are three partitions of n = 3, namely {{1,1,1}, {1,2}, {3}} since 1 + 1 + 1 = 1 + 2 = 3.
The running time behavior of this algorithm is very bad, the required time complexity is estimated as O((n!)²). The computation of the partitions for a small number n < 50 is done in less than 30 seconds, but for instance n = 60 requires about 4 minutes on a 2 GHz dual core processor system. If you are interested only in the number of partitions, you may consider numberOfPartitions(int).

For details see R.L. Graham, D.E. Knuth, O. Patashnik: Concrete Mathematics. 2nd Edition. Addison-Wesley, Upper Saddle River, NJ 1994, §7.1 (p. 330)

Parameters:

n - a positive integer

Returns:

list of all partitions of the specified integer n.

See Also:

numberOfPartitions(int)

sum
```
public static int sum(Collection<Integer> list)
```
Returns the sum of the integers of the specified list.

Parameters:

list - a list of integers

Returns:

the sum of the integers contained in the list

bbp
```
public static char bbp(long position)
```
Returns the hexadecimal digit of π at the given position, according to the Bailey-Borwein-Plouffe algorithm. The digit is computed in n+1 iterations each of which requiring O(log n) elementary arithmetic operations, yielding a time complexity of O(n log n) for the entire algorithm.

Parameters:

position - the position of the searched hexadecimal digit

Returns:

the hexadecimal digit of π at the given position

decToBin
```
public static String decToBin(long n)
```
Returns n as a binary string.

Parameters:

n - the decimal value to be represented in binary form

Returns:

string representing the binary representation of n

decToBin
```
public static String decToBin(long n,
                              int minimumLength)
```
Returns n as a binary string of the specified minimum length.

Parameters:

n - the decimal value to be represented in binary form

minimumLength - the minimum length of the returned binary string

Returns:

string representing the binary representation of n

decToBinByte
```
public static String decToBinByte(long n)
```
Returns n as a binary string in a 1-byte form.

Parameters:

n - the decimal number to be represented in a 1 byte binary form.

Returns:

string representing the binary representation of n

binToDec
```
public static long binToDec(String bin)
```
Returns a binary string as an integer.

Parameters:

bin - the binary string to be represented in decimal form

Returns:

the long integer representing the binary string

Throws:

NumberFormatException - if the string is not binary

decToBin
```
public static String decToBin(double z)
```
Returns z as a binary string with at most 70 digits right of the binary point.

Parameters:

z - the decimal value to be represented in binary form.

Returns:

the binary representation of z

decToBin
```
public static String decToBin(double z,
                              int limit)
```
Returns z as a binary string with at most limit positions right of the binary point.

Parameters:

z - the decimal value to be represented in binary form.

limit - the maximum position after the binary point.

Returns:

the binary representation of z

binToDouble
```
public static double binToDouble(String bin)
```
Returns a binary string as a decimal floating-point number.

Parameters:

bin - the binary string to be represented in decimal form.

Returns:

the double number representing the binary string

Throws:

NumberFormatException - if the string is not binary

binToHex
```
public static String binToHex(String bin)
```
Returns a binary string as a hexadecimal string.

Parameters:

bin - the binary string to be represented in hexadecimal form

Returns:

the hexadecimal string representing the binary string

Throws:

NumberFormatException - if the string is not binary

decToTern
```
public static String decToTern(long n)
```
Returns n as a ternary string.

Parameters:

n - the decimal value to be represented in ternary form

Returns:

string representing the ternary representation of n

decToTern
```
public static String decToTern(long n,
                               int minimumLength)
```
Returns n as a ternary string of the specified minimum length.

Parameters:

n - the decimal value to be represented in ternary form

minimumLength - the minimum length of the returned ternary string

Returns:

string representing the ternary representation of n

ternToDec
```
public static long ternToDec(String tern)
```
Returns a ternary string as an integer.

Parameters:

tern - the ternary string to be represented in decimal form

Returns:

the long integer representing the ternary string

Throws:

NumberFormatException - if the string is not ternary

decToTern
```
public static String decToTern(double z)
```
Returns z as a ternary string with at most 40 digits right of the ternary point.

Parameters:

z - the decimal value to be represented in ternary form.

Returns:

the ternary representation of z

decToTern
```
public static String decToTern(double z,
                               int limit)
```
Returns z as a ternary string with at most limit positions right of the ternary point.

Parameters:

z - the decimal value to be represented in ternary form.

limit - the maximum position after the ternary point.

Returns:

the ternary representation of z

ternToDouble
```
public static double ternToDouble(String tern)
```
Returns a ternary string as a decimal floating-point number.

Parameters:

tern - the ternary string to be represented in decimal form.

Returns:

the double number representing the ternary string

Throws:

NumberFormatException - if the string is not ternary

ternaryComplement
```
public static String ternaryComplement(String tern)
```
Returns the ternary complement of the specified number in ternary representation. Here every digit '0' is changed to '2' and vice versa.

Parameters:

tern - number in ternary representation

Returns:

ternary complement of the specified number tern

add3
```
public static String add3(String m,
                          String n)
```
Returns the sum of m and n in the ternary system. It is defined by [n/3] except for the case n % 2 where it gives [n/3] + 1.

Parameters:

m - number in ternary representation

n - number in ternary representation

Returns:

the sum m + n in the ternary representation

div3b
```
public static long div3b(long n)
```
Returns the integer division n/3 in the balanced ternary system. It is defined by [n/3] except for the case n % 2 where it gives [n/3] + 1.

Parameters:

n - the decimal value to be divised

Returns:

the integer division n/3 in the balanced ternary system

mod3b
```
public static char mod3b(long n)
```
Returns the symbol in the balanced ternary system representing the value n mod 3.

Parameters:

n - the decimal value to be divised

Returns:

the symbol in the balanced ternary system representing the value n mod 3.

decToTernB
```
public static String decToTernB(long n)
```
Returns n as a balanced ternary string.

Parameters:

n - the decimal value to be represented in ternary form

Returns:

string representing the balanced ternary representation of n

decToTernB
```
public static String decToTernB(long n,
                                int minimumLength)
```
Returns n as a balanced ternary string of the specified minimum length.

Parameters:

n - the decimal value to be represented in balanced ternary form

minimumLength - the minimum length of the returned ternary string

Returns:

string representing the balanced ternary representation of n

ternBToDec
```
public static long ternBToDec(String tern)
```
Returns a balanced ternary string as an integer.

Parameters:

tern - the ternary string to be represented in decimal form

Returns:

the long integer representing the ternary string

Throws:

NumberFormatException - if the string is not ternary

decToTernB
```
public static String decToTernB(double z)
```
Returns z as a balanced ternary string with at most 40 digits right of the ternary point.

Parameters:

z - the decimal value to be represented in balanced ternary form.

Returns:

the balanced ternary representation of z

decToTernB
```
public static String decToTernB(double z,
                                int limit)
```
Returns z as a balanced ternary string with at most limit positions right of the ternary point.

Parameters:

z - the decimal value to be represented in balanced ternary form.

limit - the maximum position after the ternary point.

Returns:

the balanced ternary representation of z

ternBToDouble
```
public static double ternBToDouble(String tern)
```
Returns a balanced ternary string as a decimal floating-point number.

Parameters:

tern - the ternary string to be represented in decimal form.

Returns:

the BigDecimal number representing the balanced ternary string

Throws:

NumberFormatException - if the string is not balanced ternary

ternToTernB
```
public static String ternToTernB(String tern)
```
Returns a ternary string as an balanced ternary string.

Parameters:

tern - the ternary string to be represented in balanced ternary form

Returns:

the balanced ternary string representing the ternary string

Throws:

NumberFormatException - if the string is not ternary

decToHex
```
public static String decToHex(long n)
```
Returns n as a hexadecimal string.

Parameters:

n - the decimal value to be represented in hexadecimal form.

Returns:

string representing the hexadecimal representation of n

decToHex
```
public static String decToHex(double z)
```
Returns z as a hexadecimal string with at most 20 digits right of the hexadecimal point.

Parameters:

z - the decimal value to be represented in hexadecimal form.

Returns:

the hexadecimal representation of z

decToHex
```
public static String decToHex(double z,
                              int limit)
```
Returns z as a hexadecimal string with at most limit positions right of the hexadecimal point.

Parameters:

z - the decimal value to be represented in hexadecimal form.

limit - the maximum position after the hexadecimal point.

Returns:

the hexadecimal representation of z

decToHex2Byte
```
public static String decToHex2Byte(long n)
```
Returns n as a hexadecimal string in 2-byte form.

Parameters:

n - the decimal number to be represented in a 2 byte hexadecimal form.

Returns:

string representing the hexadecimal representation of n

hexToBin
```
public static String hexToBin(String hex)
```
Returns a hexadecimal string as a binary string.

Parameters:

hex - the hexadecimal string to be represented in decimal form.

Returns:

the binary string representing the hexadecimal string

Throws:

NumberFormatException - if the string is not hexadecimal

hexToDec
```
public static long hexToDec(String hex)
```
Returns a hexadecimal string as a decimal integer.

Parameters:

hex - the hexadecimal string to be represented in decimal form.

Returns:

the long integer representing the hexadecimal string

Throws:

NumberFormatException - if the string is not hexadecimal

hexToDouble
```
public static double hexToDouble(String hex)
```
Returns a hexadecimal string as a decimal floating-point number.

Parameters:

hex - the hexadecimal string to be represented in decimal form.

Returns:

the long integer representing the hexadecimal string

Throws:

NumberFormatException - if the string is not hexadecimal

grayCode
```
public static String grayCode(long x,
                              int minimumLength)
```
Returns the Gray code of an integer x, with a given minimum length. If the minimum length is greater than the bit length of the integer x, the Gray code string is padded with leading zeros.

Parameters:

x - an integer

minimumLength - the minimum length of the returned Gray code string

Returns:

the Gray code of x as a string

grayCode
```
public static String grayCode(long x)
```
Returns the Gray code of an integer.

Parameters:

x - an integer

Returns:

the Gray code of x as a string

grayCodeToLong
```
public static long grayCodeToLong(String grayCode)
```
Returns the integer represented by a Gray code string.

Parameters:

grayCode - a Gray code string

Returns:

the integer represented by the Gray code string

Throws:

NumberFormatException - if the Gray code string does not contain a parsable long

grayCodeToBinary
```
public static String grayCodeToBinary(String grayCode,
                                      int minimumLength)
```
Returns binary representation of the integer in which is represented by a Gray code string.

Parameters:

grayCode - a Gray code string

minimumLength - the minimum length of the returned Gray code string

Returns:

the binary representation of the integer represented by the Gray code string

Throws:

NumberFormatException - if the string does not represent a Gray code

See Also:

grayCodeToBinary(String)

grayCodeToBinary
```
public static String grayCodeToBinary(String grayCode)
```
Returns binary representation of the integer in which is represented by a Gray code string.

Parameters:

grayCode - a Gray code string

Returns:

the binary representation of the integer represented by the Gray code string

Throws:

NumberFormatException - if the string does not represent a Gray code

See Also:

grayCodeToBinary(String, int)

Class Numbers

Field Summary

Method Summary

Methods inherited from class java.lang.Object

Field Detail

gamma

GAMMA

SQRT2

SQRT3

SQRT_1_2

RADIANS

ZETA_3

ZETA_5

ZETA_7

ZETA_9

ZETA_11

ZETA_13

ZETA_15

ZETA_17

ZETA_19

ZETA_21

ZETA_23

ZETA_25

binomial

TBM

TB0

TBP

Method Detail

beta

incBeta

gamma

lnGamma

incGamma

exactBinomial

binomial

lnFactorial

factorial

mod

mod

pow

pow

pow2

modPow

modPow

modPow

hypotenuse

root

sqr

gcd

euclid

lcm

isPrime

solveLinearDiophantine

isStrongProbablePrime

leastPrimeFactor

largestPrimeFactor

ord

isPower

continuedFraction

bestRationalApproximation

bernoulli

bernoulliNumbers

numberOfPartitions

partitions

sum

bbp

decToBin

decToBin

decToBinByte

binToDec

decToBin

decToBin

binToDouble

binToHex

decToTern

decToTern

ternToDec

decToTern

decToTern

ternToDouble