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Belgian mathematician Catalan in $1876$ made next conjecture: If we consider the following sequence of Mersenne prime numbers: $2^2-1=3 , 2^3-1=7 , 2^7-1=127 , 2^{127}-1$ then $$2^{2^{127}-1}-1$$ is also prime number. The last term has more than $10^{38}$digits and cannot be tested at present, so I would like to know is there any theoretical indication that Catalan's conjecture could be true ?


At London Curt Noll's prime page I have found statement that this number has no prime divisors below $5*10^{51}.$

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Note that this is not the same conjecture of Catalan proven by Preda Mihăilescu in 2002. – Dan Brumleve Oct 11 '11 at 10:28
I've never seen any argument in its favor. – Gerry Myerson Oct 11 '11 at 11:25
@Dan: Hopefully the new title is less confusing on that front. – Chris Eagle Oct 11 '11 at 11:32
@Gerry,There are well known Cunningham this sequence might be chain of some specific length size... – pedja Oct 11 '11 at 11:40

I don't know of any theoretical reason to think it would be true. I would conjecture it to be false.

Standard heuristics suggest that the first unknown example would be prime with probability $$e^\gamma\cdot2^{-120}\approx1.34\cdot10^{-34}\%$$ which is small.

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