Numbers / Cryptography...?

Question

-24060626 = 2^2 * 'a small number e.g. 5' * 'a large number e.g. 39628'.
Does anyone know how to do this?
Thanks!

Answer 1

First, 2² is not a factor of that number. 2 is, but 24060626/2 = 12030313, which is not divisible by 2.

The full factorization is 24060626 = 2*1637*7349. And there's no simple way to do this other than guessing until you find a factor. There are complicated ways to do this which are much, much faster than this method (e.g. elliptic curve factorization, quadratic sieve, general number field sieve), but with the exception of Shor's algorithm, all of them require more than polynomial time, so it is quite easy to find numbers sufficiently large that they cannot be factored in any reasonable amount of time using any of the classical algorithms. Of course, Shor's algorithm is an exception to this, but it only works on a quantum computer, which you probably don't have. This fact is required for the security of the RSA algorithm, which is one of the most widely-used public-key algorithms on the internet. So for the moment, it's probably a good thing that nobody knows how to do this.

Answer 2

do you mean something like this (very simple example):
917497 = 131071 * 7

131071 is a prime number and so is 7. the product of multiplying them creates a number with only two factors (besides itself and one).

if the first prime is huge, i mean HUGE and the second is reasonably large then the product would be a very large number with only two factors. factoring large numbers is very time expensive, taking hundreds of years or some such. anyway, you use the HUGE prime number to encrypt data and send the product of the primes with the data. you can easily decrypt the data if you have the smaller prime number or you can try to factor the product. modern encryption schemes use this method.

Answer 3

Not solvable.