Re: [math-fun] Integers with many 0s in base 2 and base 3 (fwd)
This question remindes me of the news item that's been posted on the homepage of Eric Weisstein's MathWorld for a number of months: << A paper submitted by P. Mihailescu on April 18 [2002] allegedly proves Catalan's 158-year-old conjecture.
Am I right that this is the conjecture that the only positive integral powers of 2 and 3 that differ by 1 are 3 - 2 = 1 and 9 - 8 = 1 ??? Does anyone know whether this proof has been accepted by experts? If so, can someone please give a sketch of the proof? Thanks, Dan
Actually this fact is much easier to prove (once you correct it by adding 4 - 3 = 1). First you show that 3^(2^(n-1)) = d2^n + 1 where d is odd, from which it follows that 3 is a primitive root (mod 2^n). If 3^a = 2^b +/- 1 then 3^a == +/-1 (mod 2^b) => a >= 2^(b-2) => 3^(2^(b-2)) <= 3^a <= 2^b + 1, and this last inequality can only be satisfied for b <= 3. J.P. On Sun, 22 Dec 2002 asimovd@aol.com wrote:
This question remindes me of the news item that's been posted on the homepage of Eric Weisstein's MathWorld for a number of months:
<< A paper submitted by P. Mihailescu on April 18 [2002] allegedly proves Catalan's 158-year-old conjecture.
Am I right that this is the conjecture that the only positive integral powers of 2 and 3 that differ by 1 are 3 - 2 = 1 and 9 - 8 = 1 ???
Does anyone know whether this proof has been accepted by experts? If so, can someone please give a sketch of the proof?
Thanks,
Dan
On Sun, 22 Dec 2002 asimovd@aol.com wrote:
This question remindes me of the news item that's been posted on the homepage of Eric Weisstein's MathWorld for a number of months:
<< A paper submitted by P. Mihailescu on April 18 [2002] allegedly proves Catalan's 158-year-old conjecture.
Am I right that this is the conjecture that the only positive integral powers of 2 and 3 that differ by 1 are 3 - 2 = 1 and 9 - 8 = 1 ???
I think the conjecture states that the only solution to x^a - y^b =1, where x,y,a,b are any integers > 1, is the solution 3^2 - 2^3 = 1.
Does anyone know whether this proof has been accepted by experts? If so, can someone please give a sketch of the proof?
The home page of Yuri Bilu http://www.math.u-bordeaux.fr/~yuri/ contains a Seminaire Bourbaki article which is an exposition of Mihailescu's claimed proof. This article thanks Hendrik Lenstra and Yann Bugeaud and says they read the manuscript carefully. This would all seem to suggest the proof is valid. I can't sketch the argument. After a quick glance: We may assume a=p and b=q are prime. Part of the argument is to show p^{q-1} = 1 mod q^2 and q^{p-1} = 1 mod p^2 (p and q are called double Wieferich pairs - only 6 such pairs are known). Another part of the argument uses results of Baker on logarithmic forms. Also, the proof seems to depend on a computer computation. Gary McGuire
participants (3)
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asimovd＠aol.com -
Gary McGuire -
JP Grossman