Instructions for use of this factorizer:

Return to the Math APIs page.

• This page allows for five different types of calculations, each related to the factorization of integers. The first four depend upon the following pairs of circumstances:
  • real integers vs complex integers" (aka Gaussian integers)
  • single integer (factorization thereof) vs list of integers (greatest common divisor - GCD - thereof)
  The fifth type is the conversion of a decimal (either finite or repeating) to a fraction.
• URL:
  • After ...herokuapp.com above you should type either /complex/ if you want to consider Gaussian integers or a single slash if you want to consider real numbers.
  • After this you should type json/ if you want the output in json or nothing if you want the output in html.
  • After this you should type your input, as described next.
• Input(s):
  • You are discouraged from using spaces, because %20 will replace each space after you hit return, thereby making the address uglier.
  • Do not use E-notation.
  • All real numbers must be less than 2⁶³ in magnitude (approximately "9" followed by 18 zeros). Exception: the limit on the number of digits in your Gaussian integer equals half of this.
  • Complex numbers should be of the form m + ni, where m and n represent integers as described in the previous bullet. You may use standard shortcuts such as 3 + i instead of 3 + 1i, or 2i instead of 0 + 2i.
  • Use commas to separate numbers in a list, ie when determining GCD.
  • In order to represent a decimal that has both a terminating part and a repeating part, separate the two parts by either repeat, r, R, or , (ie a comma).
• Factorization theorems and definitions:
  • real factorization:
    Any nonzero (real) integer may be expressed uniquely as a product of ±1 and zero or more prime numbers (e.g., 2, 3, 5, 7, 11, etc), each raised to a power of 1 or more.
  • complex factorization:
    Any nonzero Gaussian integer may be expressed as a product of either ±1 or ±i, and zero or more Gaussian primes, each raised to a power of 1 or more. This expression is unique if the Gaussian primes are restricted to the first quadrant of the complex plane, including the positive real axis but not including the positive imaginary axis.
  • Whether real or complex, any nonzero integer other than ±1 or ±i is categorized as one of the following:
    • A prime number contains exactly one prime factor in its factorization.
    • A composite number contains two or more prime factors in its factorization. Although this definition usually is used for positive integers, it extends naturally to Gaussian integers.
  • Just as an individual integer may be classified as either prime or composite, so may a list of integers be classified by its GCD: the list is relatively prime (or "coprime" or "mutually prime") if its GCD equals 1, or relatively composite (or "mutually composite") if its GCD exceeds 1.
• Output:
  
  • The factors property of the json output will be an array of 2-component arrays, the first element of each signifying a different prime number, and the other element equalling the power to which the particular prime is raised when forming the factorization. For complex integers, the value of the exponent property equals the power to which i should be raised when multiplying the prime factors.
  • The GCD is given in factored form, for the html output.
  • The json return for a conversion from decimal to fraction will have three properties, the value for each of which will be an integer, and the names for which are integer_part, numerator, and denominator.
• Examples:
  

  type	input	result	json output
  real, single	9134567890123456789	prime
  	999999	33·7·11·13·37	{"factors" : [[3, 3], [7, 1], [11, 1], [13, 1], [37, 1]]}
  complex, single	2	-i(1 + i)2	{"exponent": 3, "factors": [["1+i", 2]]}
  	3	prime
  	1 + 3i	(1 + i)(2 + i)	{"exponent": 0, "factors": [["1+i": "1"], ["2+i": "1"]]}
  real, list	9,12	3
  	9,10	relatively prime
  	8,10,12	2
  complex, list	2i, -1 + 3i, 1 + 3i	1 + i
  decimal-to-fraction	3.4	3 2⁄5	{"integer_part": 3, "numerator": 2, "denominator": 5}
  	3.4r56	3 226⁄495	{"integer_part": 3, "numerator": 226, "denominator": 495}

creator:  Peter Knipp