Search results
Results from the WOW.Com Content Network
RSA-120 1993-06-09 ppmpqs ... and 14.2 hours to find the factors using the square root) ... (this pair has a yield of relations approximately 13.5 times that of a ...
To factorize a small integer n using mental or pen-and-paper arithmetic, the simplest method is trial division: checking if the number is divisible by prime numbers 2, 3, 5, and so on, up to the square root of n. For larger numbers, especially when using a computer, various more sophisticated factorization algorithms are more efficient.
The square root of a positive integer is the product of the roots of its prime factors, because the square root of a product is the product of the square roots of the factors. Since p 2 k = p k , {\textstyle {\sqrt {p^{2k}}}=p^{k},} only roots of those primes having an odd power in the factorization are necessary.
As 120 is a factorial and one less than a square (! =), it—with 11—is one of the few Brown number pairs. 120 appears in Pierre de Fermat's modified Diophantine problem as the largest known integer of the sequence 1, 3, 8, 120. Fermat wanted to find another positive integer that, when multiplied by any of the other numbers in the sequence ...
A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using ordinal numbers, as in fourth root, twentieth root, etc. The computation of an n th root is a root extraction. For example, 3 is a square root of 9, since 3 2 = 9, and −3 is also a square root of 9, since (−3) 2 = 9.
The first: 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600 (sequence A000142 in the OEIS). 0! = 1 is sometimes included. A k-smooth number (for a natural number k) has its prime factors ≤ k (so it is also j-smooth for any j > k). m is smoother than n if the largest prime factor of m is below the largest of n.
The polynomial x 2 + cx + d, where a + b = c and ab = d, can be factorized into (x + a)(x + b).. In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind.
Yves Gallot's proth.exe has been used to find factors of large Fermat numbers. Édouard Lucas, improving Euler's above-mentioned result, proved in 1878 that every factor of the Fermat number , with n at least 2, is of the form + + (see Proth number), where k is a positive integer. By itself, this makes it easy to prove the primality of the ...