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Old 2020-11-13, 03:48   #3
VBCurtis
 
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"Curtis"
Feb 2005
Riverside, CA

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I think you are asking if we could rule out primality of a Mersenne candidate by showing its corresponding perfect-number-candidate isn't perfect. But, to do that you'd have to completely factor that number.
Factoring numbers is many orders of magnitude harder than testing for primality.
Numbers with special form (where the form helps us factor faster) can be factored up to about 300 digits. Beyond that size, you have to hope all the factors are small- like under 70 digits. And, you only know you're done factoring when the factors you've broken the candidate into are tested to be prime- which means a primality test on th Mersenne candidate is a component of your hope of finding a perfect number directly!
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