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Old 2021-09-27, 14:26   #1
Dr Sardonicus
 
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Feb 2017
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Default Proper factorization known, no prime factors known

This is merely recreational mathematics. If this more properly fits in an area other than "Puzzles," feel free to move it there.

My "jumping off place" is the fact that (currently, AFAIK) the character (prime or composite) of F33 is unknown.

The Prime factors k*2n + 1 of Fermat numbers Fm and complete factoring status page lists Fm for the values m = 33, 34, 35, 41, 44, 45, 46, 47, 49, 50, 51, . . . as "Character unknown."

So it occurred to me that there are numbers with known proper factorizations, for which the "minimal" proper factors (those which are not the product of smaller known proper factors) are all of unknown character. A trivial example would be N = F33*F33.

Of course, one can cobble together other such trivial examples by tracking down numbers of unknown character and multiplying them together.

It occurred to me that one could probably construct "more natural" examples using algebraic factorizations. As a potential example, I give the number N = googolplex + 1 = 1010^100 + 1. We have

N\;=\;\(10^{2^{100}}\;+\;1\)\cdot\Phi_{5}\(10^{2^{100}}\)\cdot\Phi_{25}\(10^{2^{100}}\)\cdot\Phi_{125}\(10^{2^{100}}\)\cdot...\cdot\Phi_{5^{100}}\(10^{2^{100}}\)

According to Prime factors of generalized Fermat numbers Fm(10) and complete factoring status, F100(10) appears to be of unknown character [though two prime factors are known for F99(10)], so the first factor at least is of unknown character. I'm guessing nobody has tried to find factors of the other algebraic factors, but I don't know. I am not suggesting that anybody waste their time looking for them.

I imagine much smaller examples are to be found; a possible candidate is 23*2^33 + 1. But for all I know, someone may have found a divisor of the larger algebraic factor.

I don't know whether there's a number of unknown character smaller than F33. Anybody?

Last fiddled with by Dr Sardonicus on 2021-09-27 at 14:31 Reason: insert missing multiplication symbol
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