We say a formula $ f (x) $, for $ x in Bbb {Z} $ a given compound, *factors $ x $* Yes $ gcd (f (x), x) notin {1, x } $.

Suppose that $ gcd (f (x), x) = d $and take any rational formula, including radicals, to $ f (x) $and you will find that yes $ f $ you cannot write in less operations, then there is a constant $ c $ what factors $ x $ or $ gcd (x, c) mid x $ and it is not one of the trivial divisors.

Intuitively, this says that an optimal formula for a divisor does not make sense because to arrive at a formula it is necessary to have "seen" a divisor. If you are not only doing a linear search through the integers (as in naive factorization), then your factoring method is calculating some formula for a divisor. I find it interesting

By *rational, including radicals* formula I mean any formula in the variable $ x $ come to take $ sqrt (a) {x} $ where $ a in Bbb {N} $, adding or subtracting, or multiplying or dividing a finite, constant number of times.

So for example yes $ x = $ 33. We have $ f (x) = dfrac {x – 1} {2} – 1 = $ 15 however, further reducing the number of operations we have $ f (x) = dfrac {x – 3} {2} $ then there is a constant term $ geq 3 $ In the optimal expression.

by $ x = 4 $. You really can't get to $ 2, 6, 8, $ or $ dots $ no first meeting 2.

Try this for different numbers.

How can I do a better job formalizing this concept in the context of integer factorization algorithms?