From: Jan Kristian Haugland Subject: Re: Goldbach Conjecture Date: Sun, 17 Sep 2000 20:53:41 +0200 Newsgroups: sci.math Summary: [missing] Fred Galvin wrote: > On 17 Sep 2000, Daniel McLaury wrote: > > > I don't think that Euler would have been fooled by something > > as simple as x^2 + x + 41. After all, it is obvious that f(41) > > is 41*43. He may have given this as an example of why not to > > rely on anecdotal evidence. One of my favorites is the polynomial > > (x-1)(x-2)(x-3)*...(x-n), multiplied out so that its form is not > > so obvious. Then the prof/teacher says, f(1) = 0, f(2) = 0, ... > > so therefore f(x) must always equal 0. Half the students are > > convinced, and then the teacher takes f(n+1) = n!, which is > > _very_ far from being 0. > > A silly example: the "identity" (1+2+3+...+n)! = 1!3!5!...(2n-1)! > which is true for n = 0, 1, 2, 3, and 4, but false for all larger > values of n. Not in the same league with Euler's x^2+x+41 of course. > > -- > "Any clod can have the facts, but having opinions is an art."--McCabe How about n! = k^3 - k where k = 2^(n - 3) + 1 for n = 3, 4, 5, 6? ;-) ============================================================================== From: "denis-feldmann" Subject: Re: Goldbach Conjecture Date: Sun, 17 Sep 2000 21:52:06 +0200 Newsgroups: sci.math Summary: [missing] Jan Kristian Haugland a écrit dans le message : 39C51335.726798CB@studNOSPAM.hia.no... [quote of previous message deleted --djr] Or number of regions from n points on a circle = 2^(n-1) for n=1,2,3,4,5 ? ============================================================================== From: Erick Wong Subject: Re: Goldbach Conjecture Date: Sun, 17 Sep 2000 05:22:32 -0700 Newsgroups: sci.math Summary: [missing] daniel_mcl@hotmail.com (Daniel McLaury) wrote: > An example of the "law of small numbers" in real maths > is given by the conjecture that Li(x) is always less than Pi(x). > This is true for values well over a googol, but eventually it > becomes false for all values over a certain number. Slight correction: Littlewood actually showed that Li(x) - Pi(x) alternates in sign infinitely often, and gave an upper bound for the position of the first sign change. Another example is Borsuk's conjecture on decomposing a bounded set in d dimensions into d+1 sets of strictly smaller diameter. It was shown to be false in dimension 1325 (and it's now known to be false in at least dimensions 561 and up). -- Erick