Validity of number theories.

Question

Recently we talked about the Riemann hypothesis in class, and yesterday I stumbled across the Goldbach conjectures. I realized there are quite a few theories that assert a particular property to all numbers of a specific form (greater than some n), which have been confirmed for the first gazillion numbers. Yet that is not sufficient proof. (Of course, some of them have been proven.)

So out of interest: Are there any such theories that remain valid for a wide range of numbers and then suddenly break off?

By number theories, do you mean hypotheses or laws for natural numbers? — SBF, Jul 28 '14 at 11:43
Well, hypotheses, since laws should hold for all natural numbers per definition, right? — 355durch113, Jul 28 '14 at 11:47
Also, this thread has some examples of "almost true" statements in general, not only for number theory — Marcin Łoś, Jul 28 '14 at 11:51
See also http://mathoverflow.net/questions/15444/examples-of-eventual-counterexamples/ — Gerry Myerson, Jul 28 '14 at 12:36

score 2 · Answer 1 · answered Jul 28 '14 at 12:03

A few such stories:

1. Half the primes are of the form $4n+1$ and half of the form $4n+3$, but counting all the primes up to $x$, there seem to always be slightly more $4n+3$'s than $4n+1$'s. Does this continue forever?

2. There is no solution in small integers to $x^4 + y^4 + z^4 = w^4$. Is there a solution at all?

And last but by far the biggest:

3. The number $\pi(x)$ of primes less than $x$ seems to always be smaller than its estimate, the logarithmic integral $\operatorname{Li}(x)$. Is it generally true?

Validity of number theories.

1 Answers1