12 isn't prime: it's composite.
So is 121.
So is 1211.
So is 12111.
So is 121111.
So is 1211111.
So is 12111111.
This seems to be a persistent pattern. Let's keep going. Seven 1’s, composite. Eight, still composite. Nine. Ten, eleven and twelve. We keep going. Everything up to twenty 1’s is composite. Up to thirty, still everything is composite. Forty. Fifty. Keep going. One hundred. They are all composite.
At this point it may seem reasonable to conjecture that these numbers are never prime. But this isn't true. The number with 138 digits, all 1’s except for the second digit which is 2, is prime.
To be clear, this isn't a particularly shocking example. It's not really that surprising. But it underscores the fact that some very simple patterns in numbers persist into pretty big territory, and then suddenly break down.
There appear to be two slightly different questions here. One is about statements which appear to be true, and are verifiably true for small numbers, but turn out to be false where the first counterexamples are massive. The other question is about such situations which have furthermore become actual conjectures, put forth and studied and perhaps widely believed in, until they were refuted.
In response to such questions, I used to cite a couple of nice examples given by Stark in his number theory textbook, by way of motivating his readers to see the need for proofs.
One example is Pólya’s conjecture, which is a famous instance of this phenomenon: it was a real conjecture suggested by a very prominent mathematician, supported by both numerical evidence and heuristic reasoning, which turned out to be false.
Another example given by Stark is a nice, smallish Diophantine equation which has a very large smallest solution. This isn't a case of a genuine conjecture, but it nicely demonstrates the necessity for proofs.
In that spirit, my new go-to example is definitely this gem. If you check numerically, you will quickly convince yourself that there aren't any natural numbers which satisfy
[math]\displaystyle \frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=4[/math].
You can search through millions, billions, trillions or quadrillions if you have the resources, and nothing works. It would seem very reasonable to expect that there aren't any solutions to this simple equation in natural numbers, just like many other similar equations. But that's false. There are infinitely many solutions, and the smallest one is ginormous.
This, to be clear, is not an instance of a “genuine conjecture”. I don't believe anyone actually suggested that this equation has no solutions: by the time it was stumbled upon, there was already enough theory to solve it. But it is a fantastic example of an equation which seems unsolvable unless you look very, very far.
I don't know if it was ever formally suggested that [math]\text{Li}(x)[/math], the logarithmic integral, is always greater than [math]\pi(x)[/math], the number of primes less than [math]x[/math]. The Prime Number Theorem says that the ratio between these two functions approaches [math]1[/math], but numerical evidence seems to strongly suggest that while this is of course true, it is consistently the logarithmic integral which has the upper hand. This is indeed true for values of [math]x[/math] up to billions, and likely much further.
But Littlewood proved that these two functions switch sides infinitely often. It's just that the first reversal occurs very, very far out. Littlewood’s student Skewes showed that it happens earlier than [math]e^{e^{e^{79}}}[/math], a number now known as Skewes’ number. Today it is known that it happens earlier, but still the first occurrence is gigantic.
Once again, I'm not sure if this was ever an “official” conjecture which got refuted, but it would have been a very reasonable conjecture to make even if it weren't.
A better example perhaps is Mertens’ conjecture from 1885, refuted exactly one hundred years later.
The Möbius function is an important function in number theory and combinatorics. It is defined as [math]\mu(n)=(-1)^k[/math] if [math]n[/math] has [math]k[/math] distinct prime factors, and [math]0[/math] otherwise (when [math]n[/math] has a repeated factor). The Mertens function is then
[math]M(n)=\sum_{k=1}^n \mu(k)[/math].
As early as 1885, Stieltjes conjectured that
[math]|M(n)|<\sqrt{n}[/math]
for all [math]n[/math], and this is supported by overwhelming numerical evidence. Moreover, it implies the Riemann Hypothesis – so it was definitely an interesting conjecture.
However, it's false. Odlyzko and te Riele proved in 1985 that [math]M(n)[/math] exceeds [math]\sqrt{n}[/math] by at least 6% infinitely many times. The first counterexample is probably around [math]10^{40}[/math] or so.
Finally, I'll mention a less elementary example, but one which (to me) was quite a shocker.
For any finite group [math]G[/math] we can form the group algebra [math]\mathbb{Z}G[/math], which is just the free abelian group generated by the elements of [math]G[/math], with multiplication of basis elements defined as in [math]G[/math] and then extended linearly. If you don't know what any of that means, you'll need to take my word for it: finite groups are among the most fundamental building blocks of math, and group algebras are a very natural, and extremely useful, construction involving them.
We may then wonder: the group determines its group algebra – is the opposite also true? Can you recover the group from the group algebra? In other words, if [math]\mathbb{Z}G \cong \mathbb{Z}H[/math], does it follow that [math]G \cong H[/math]?
This was not a named conjecture, but it was definitely a conjecture. It is mentioned, for example, in Isaacs’ “Character Theory of Finite Groups” from 1976 (p. 41 in my edition).
It turns out to be false, as Martin Hertweck showed in a 2001 paper[1] titled “A Counterexample to the Isomorphism Problem for Integral Group Rings”. His counterexample is a group of size [math]2^{21}97^{28}[/math], and people can be forgiven for taking long to find it.
Footnotes
[1] A Counterexample to the Isomorphism Problem for Integral Group Rings
