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October 28th, 2017, 08:41 PM  #1 
Member Joined: Apr 2017 From: Canada Posts: 32 Thanks: 2  Has anything been proven by this method?
Hi all, Are any of you aware of a mathematical conjecture that's been disproven not by FINDING a counterexample, but showing that one exists? For example, if I was trying to disprove Goldbach Strong Conjecture, I could try to find an even number which is not the sum of two primes. However, it's possible that this number is too massive to write down, comprehend, or even contain in our universe. So has a statement been proven/disproven by showing that a (counter)example exists, but not saying anything about its value? If so, could you link the paper? 
October 28th, 2017, 09:19 PM  #2 
Senior Member Joined: Aug 2012 Posts: 2,010 Thanks: 574 
Here's one that comes to mind. It takes a bit of explaining and I hope I can do justice to it. There's a thing called the logarithmic integral. It's defined for all positive real numbers $x \ne 1$ by $li(x) = \int_0^x \frac{dt}{\ln x}$. It turns out to be a good estimate of the number of primes less than $x$. Now there is a function called the prime counting function, $\pi(x)$, which gives the exact number of primes less than $x$. If you calculate out as many values of these functions for all the values of $x$ as you can, you find that $\pi(x) < li(x)$. Until 1914, people thought that $\pi(x)$ was always smaller than $li(x)$. In 1914, Littlewood (the guy played by Toby Jones in The Man Who Knew Infinity) proved that in fact these two functions alternate being larger or smaller than each other infinitely many times. Therefore there must be some smallest positive integer such that $\pi(x) > li(x)$. However Littlewood did not find out what that smallest integer is. In 1933, Littlewood's former student Skewes proved that if you assume the Riemann hypothesis, there is some integer $x$ satisfying $\pi(x) > li(x)$ for some value of $x < e^{e^{e^{79}}}$ which is around $10^{10^{10^{34}}}$. This is known as Skewes' number. We still don't know the exact value of the smallest such $x$, but Skewes' number is an upper bound on how large it could be. There's no hope of using a computer to figure out what the exact value is, these numbers are far too large. There have been some recent improvements in the upper bound using computers. There's an interesting survey of these results on the Wiki page for Skewes' number. To sum this up, there's some number that is the smallest number such that $\pi(x) > li(x)$. We have no idea what this number is, and it's out of reach of any computation that can be done with current technology. In fact, we currently do not know any specific number at all (the smallest one or any other one) that satisfies $\pi(x) > li(x)$, even though there must be infinitely many of them. Last edited by Maschke; October 28th, 2017 at 09:43 PM. 
October 29th, 2017, 07:34 PM  #3 
Senior Member Joined: Sep 2016 From: USA Posts: 444 Thanks: 254 Math Focus: Dynamical systems, analytic function theory, numerics 
For a long time it was believed that every continuous periodic function on $\mathbb{R}$ converged uniformly to its Fourier series. Not only is this not true, but it isn't even true that such a function must converge pointwise to its Fourier series. The first proof of this fact showed that for any $x \in \mathbb{R}$, the set of continuous functions converging pointwise to its Fourier series at $x$ is a meager set (Baire category 1). This means that despite being unable to find a specific function which didn't converge pointwise everywhere, this result proves that "most" continuous functions fail to converge pointwise to their own Fourier series at every point in their domain. Of course now there are explicit examples. 

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