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An optimal strategy for choosing these polynomials is not known; one simple method is to pick a degree d for a polynomial, consider the expansion of n in base m (allowing digits between −m and m) for a number of different m of order n 1/d, and pick f(x) as the polynomial with the smallest coefficients and g(x) as x − m.
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.
If the original polynomial is the product of factors at least two of which are of degree 2 or higher, this technique only provides a partial factorization; otherwise the factorization is complete. In particular, if there is exactly one non-linear factor, it will be the polynomial left after all linear factors have been factorized out.
In other words, if R is a UFD with quotient field K, and if an element k in K is a root of a monic polynomial with coefficients in R, then k is an element of R. Let S be a multiplicatively closed subset of a UFD A. Then the localization S −1 A is a UFD. A partial converse to this also holds; see below.
Polynomial factoring algorithms use basic polynomial operations such as products, divisions, gcd, powers of one polynomial modulo another, etc. A multiplication of two polynomials of degree at most n can be done in O(n 2) operations in F q using "classical" arithmetic, or in O(nlog(n) log(log(n)) ) operations in F q using "fast" arithmetic.
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In practice, one wouldn't bother with that last row until b is an integer. But observe that if N had a subroot factor above a − b = 47830.1 {\displaystyle a-b=47830.1} , Fermat's method would have found it already.
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