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A Gaussian integer is either the zero, one of the four units (±1, ±i), a Gaussian prime or composite. The article is a table of Gaussian Integers x + iy followed either by an explicit factorization or followed by the label (p) if the integer is a Gaussian prime.
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10 100. Heuristically, its complexity for factoring an integer n (consisting of ⌊log 2 n ⌋ + 1 bits) is of the form
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second-fastest method known (after the general number field sieve).It is still the fastest for integers under 100 decimal digits or so, and is considerably simpler than the number field sieve.
The following tables list the computational complexity of various algorithms for common mathematical operations. Here, complexity refers to the time complexity of performing computations on a multitape Turing machine. [1] See big O notation for an explanation of the notation used.
In mathematics, integer factorization is the decomposition of a positive integer into a product of integers. Every positive integer greater than 1 is either the product of two or more integer factors greater than 1, in which case it is a composite number, or it is not, in which case it is a prime number.
The conjugate of a Gaussian prime is also a Gaussian prime (this implies that Gaussian primes are symmetric about the real and imaginary axes). A positive integer is a Gaussian prime if and only if it is a prime number that is congruent to 3 modulo 4 (that is, it may be written 4 n + 3 , with n a nonnegative integer) (sequence A002145 in the ...
"The table might have been further reduced to the integers in the first octant of the complex plane using the symmetry y + ix =i (x − iy)." That symmetry maps between the 1st and 4th quadrants. It rotates by +/- 90 degrees so maps between the 1st and 7th octant and between the 2nd and 8th octant. e.g. 2+i == i(1-2i) is a first octant term ...
Integer factorization is the process of determining which prime numbers divide a given positive integer.Doing this quickly has applications in cryptography.The difficulty depends on both the size and form of the number and its prime factors; it is currently very difficult to factorize large semiprimes (and, indeed, most numbers that have no small factors).