[QUOTE=LaurV;375800]This equals (2^2^251)(2^2^25+1).
Since when F25 is full factored? :shock:[/QUOTE]Okay, maybe my mistake. I thought that 2[sup]ab[/sup]1 could be factored by (2[sup]a[/sup]1) x (1+2[sup]a[/sup]+2[sup]2a[/sup]+2[sup]3a[/sup]+...+2[sup](b1)a[/sup]) Edit: So, nevermind I guess it is still easy to construct one with 2[sup]pq[/sup]1? 
[QUOTE=Gordon;375796]Then there's that pesky thing called a dictionary :smile:
probable 1. likely to occur or prove true 2. having more evidence for than against, or evidence that inclines the mind to belief but leaves some room for doubt. 3. affording ground for belief. So by definition, it is not [U][B]completely[/B][/U] factored.[/QUOTE] Then there's that other pesky thing called a definition: A probable prime is defined as a number that has passed a probable prime test. 
[QUOTE=BudgieJane;375824]Then there's that other pesky thing called a definition: A probable prime is defined as a number that has passed a probable prime test.[/QUOTE]
Yep, probable, see my previous dictionary definition. This number is [U][B]100% NOT competely factored[/B][/U], you can dress it up however you want, but the facts don't lie. 
Well, if you want me to be completely pedantic, if you know that the Mersenne number shown is not 100% completely factored, that means that the 173528digit pseudoprime is composite. What is your proof?

[QUOTE=alpertron;375847]Well, if you want me to be completely pedantic, if you know that the Mersenne number shown is not 100% completely factored, that means that the 173528digit pseudoprime is composite. What is your proof?[/QUOTE]
That's just it you see, I don't need proof of anything, I'm not the one making the unverifiable claim, remember [U]your[/U] claim was "...completely factorized" So for that to be true there can be no PRP rubbish spouted, show all the factors else [U]you are making a false and misleading claim[/U]. Amazing how many people on here struggle with simple English and seem to believe that Probable == Definitively It doesn't... :mooc: 
Congratz Dario, nice found. Your prp is now listet as #6 of Henri Lifchitz's page of Mersenne cofactors:
[URL]http://www.primenumbers.net/prptop/searchform.php?form=%282^n1%29%2F%3F&action=Search[/URL] @Gordon: maybe an expression like '(prp)completet' is more accurate, but everyone who is interestet in factoring such large Mnumbers knows, that a complete factorization of a Mnumber with an exponent ~500k will have one or more prp's involved. Simply compare the size of the largest proven prime cofactor of a Mnumber (I don't know the size, I think somewhere 10002000 bits) with the size of this Mnumber. Do you expectet 500 factors with average size of ~1000 bits and all factors are proven as prime (with primo or so)? And the complete thread is talking about "looking for prp's". See f.e.: [URL]http://www.mersenne.ca/prp.php?show=1&min_exponent=100000&max_exponent=1000000[/URL] Dario's prp is already included :wink: mfg Matthias 
[QUOTE=MatWurS530113;375868]
[snip] that a complete factorization of a Mnumber with an exponent ~500k will have one or more prp's involved. Simply compare the size of the largest proven prime cofactor of a Mnumber (I don't know the size, I think somewhere 10002000 bits) with the size of this Mnumber. Do you expectet 500 factors with average size of ~1000 bits and all factors are proven as prime [snip] [/QUOTE] Precisely the point, you believe with some degree of confidence that it is prime, it may well be, we'll probably never know in our lifetimes. All you do know with 100% certainty is that it is [B][U]NOT FACTORED COMPLETELY[/U][/B] I am reminded of this famous quote “The less people know, the more stubbornly they know it.” Osho 
[QUOTE=Gordon;375882]All you do know with 100% certainty is that it is [B][U]NOT FACTORED COMPLETELY[/U][/B][/QUOTE]
Consider the two statements: A: We do NOT know with 100% certainty that it is factored completely. B: We do know with 100% certainty that it is NOT factored completely. Do you believe these two to be identical? 
[QUOTE=Gordon;375882]All you do know with 100% certainty is that it is [B][U]NOT FACTORED COMPLETELY[/U][/B][/QUOTE]Actually we don't know that. It might be completely factored, or it might not. We have a high confidence that it is completely factored. But we can't say for certainty that it is not.

I possibly more or less but not definitely rejected the idea that there is in no way any amount of uncertainty that I undeniably do or do not know that it is completely factored.

[QUOTE=retina;375894]Actually we don't know that. It might be completely factored, or it might not. We have a high confidence that it is completely factored. But we can't say for certainty that it is not.[/QUOTE]
You are bandying words and undefined terminology. Start by defining "completely factored". My definition would be: A number is completely factored when it is represented as the product of primes. Since the number in question has not been represented as the product of primes, then it most definitely has NOT been completely factored. 
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