Hi all, In a reply late last week concerning large values of x and x having many prime factors I made the comment that even pairs might be factored out by multiplying them and then finding a different way to factor the product. I as assuming that the product of two primes would not produce a product which in general could not be factored a different way. This seems to be in error. So far the vast majority of cases I have looked at have failed. This leaves the problem of large values of x which have many prime factors that appear either two or more times in n. At this point I don't see a way to use this relationship to further gain. Any ideas would be greatly appreciated. Side Note: I have lost the original post concerning this subject. I am still working on a paper concerning the behaviour of the mod function and would like to provide credit where credit is due. If you are the original author or can provide the e-mail address (please reply via e-mail) of the author it would be much appreciated. Take care.