# Examples of conjectures that were widely believed to be true but turned out to be false [duplicate]

Note: This is the same question, but it does not have enough answers and it is almost a year old. This question is similar, with many answers, but a conjecture is similar to, but not the same as a pattern. For example, one of the answers is:
"Choose n points around the circumference of a circle, and join every point to every other with a line segment. Assuming that no three of the line segments concur, how many regions does this divide the circle into?"
It is clear that this is not a conjecture.
Mathematical conjectures are proposed (usually) after being computationally tested for many cases. Because of this computational evidence, many are thought to be probably true, and many theorems are derived considering that they are true. But this not always the case. Some conjectures that were thought to be true (because of computational evidence) were false, for example that $$\pi(x)-\mathrm{li}(x)$$ is always positive (disproved by Littlewood). So my question is

What are some conjectures that were widely believed to be true but turned out to be false?

Like many big list question, please give only one example per answer.
Some include (these are from the answers, I wrote these because if this question gets many answers, than it would be inconvenient to scroll all the way down to see them):

• Squaring the circle
• Euler's sum of powers conjecture
• mathoverflow.net/questions/95865/… Nov 6, 2020 at 8:12
• The question is on other site, but will this question be closed?
– user841102
Nov 6, 2020 at 8:13
• No it doesn't have to be closed (at least I'm not voting to close it), but it already has answers on the other side. Nov 6, 2020 at 8:56
• More or less a duplicate of Examples of patterns that eventually fail. Nov 6, 2020 at 11:51
• If you are interested in just conjectures with empirical evidence, then the linked post should have most of them (duplicates do not need to be exactly the same question, as long as the answers address the question). If, however, you are asking for conjectures in general, then perhaps it is too broad or opinion-based. Nov 6, 2020 at 12:02

## 1 Answer

Euler's sum of powers conjecture (https://en.wikipedia.org/wiki/Euler%27s_sum_of_powers_conjecture) was an attempt to generalise the statement of Fermat's Last Theorem. No counterexamples were found until 1966.

From the article in Wikipedia:

$$27^5 + 84^5 + 110^5 + 133^5 = 144^5$$

was the first counterexample found by computer, by Lander and Parkin: the sum of $$4$$ fifth powers can be a fifth, whereas the conjecture says that you require at least $$5$$.

• Can you link an article (if any) of a disproof?
– user841102
Nov 6, 2020 at 8:01