Problems & Puzzles: Conjectures

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

Well, nobody knows if the Mersenne numbers (2^p -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 2^p -2. But how many primes p<10⁶ are such that p² divides 2^p -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 2^p -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."

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

> http://www.dybot.com/numbers/sqfree.htm

http://www2.netdoor.com/~acurry/mersenne/archive2/0037.html

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