Problems & Puzzles: Conjectures

Conjecture 12. Are Mersenne and Fermat numbers square free ?

A "squarefree number" is a composite number which has no repeated factor.

Well, nobody knows if the Mersenne numbers (2p -1, p=prime) or the Fermat numbers are squarefree or not…

A close problem related with this one is the following : From the Fermat (little) theorem we know that if p is prime then p divides 2p -2. But how many primes p<106 are such that p2 divides 2p -2. The answer is p= 1093 and p=3511, only.

Now (following a challenge of Selfridge) would you like to compute fifty more primes p such that p2 divides 2p -2 ?…

(Ref. 2, pp. 8-9)

Solution

Mr. Jacques Basaldúa, from Spain, sent (16/12/2001) the following message:

"All Fermat and Mersenne numbers are squarefree. My proof of this conjecture is detailed at: http://www.dybot.com/numbers/sqfree.htm

Very shortly: If ß(n) is the order of the subgroup of the powers of 2 modulo any odd integer n: ß(n) = LCM(a^(i-1)·ß(a), b^(j-1)·ß(b), c^(k-1)·ß(c), ... ) where a^i·b^j·c^k· ... is the prime decomposition of n. Also:ß(Fk) = 2^(k+1) From which follows: ß(Fk) has no odd prime factor. ß(any non squarefree number) has at least one odd prime factor. Non squarefree Fermat numbers do not exist. The article contains much more. Please those interested in the subject read my proof and try to find any possible flaws."

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But, this proof has a flaw, as Mr. Pavlos N saw (message 4609):

--- In primenumbers@y..., Pavlos N <pavlos199@y...> wrote:

> Can anyone comment or find a (possible) flaw in :

Yes, there is a flaw. I can't pinpoint the exact error in reasoning (basically, I don't want to spend the time to find it :-), but the equation which is "proven" in section 4.2 is not true. There is an easily computable counterexample.

The author states that: ß(p^n) = p^(n-1)·ß(p) (for any prime p) This is not true if p = 1093 & n = 2:

ß(p^n) == 364

p^(n-1)·ß(p) == 1093*364

I believe that in general, the postulate in section 4.2 is not true for any Wieferich prime.

Somebody apparently tried to prove the Mersennes squarefree using this technique in '96 and came up against this same problem:

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