"Sander Hoogendoorn" <[EMAIL PROTECTED]> asks
> Can sombody please tell me when a number is completely factored?
The prime factors of 2^29 - 1 are 233, 1103, and 2089.
The equation
2^29 - 1 = 233 * 1103 * 2089
gives the _factorization_ or _complete factorization_ of 2^29 - 1.
Whereas
2^29 - 1 = 233 * 2304167 (where 2304167 = 1103 * 2089)
2^29 - 1 = 1103 * 486737 (where 486737 = 233 * 2089)
2^29 - 1 = 2089 * 256999 (where 256999 = 233 * 1103)
are _partial factorizations_ of 2^29 - 1. A complete factorization
factors a number into prime factors. A partial factorization
allows composite cofactors (which are smaller than the original number).
If someone finds the factor 233 of 2^29 - 1, he/she can compute
the cofactor 2304167, and subject the cofactor to a probable prime test.
If the cofactor is definitely composite, we have
only a partial factorization, and more work is needed.
If instead the cofactor passes the probable prime test,
we try to prove it prime so we can be sure the factorization is complete.
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