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List of conjectures
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==Disproved (no longer conjectures)== The conjectures in following list were not necessarily generally accepted as true before being disproved. * [[Atiyah conjecture]] (not a conjecture to start with) * [[Borsuk's conjecture]] * [[Chinese hypothesis]] (not a conjecture to start with) * [[Doomsday conjecture]] * [[Euler's sum of powers conjecture]] * [[Ganea conjecture]] * [[Generalized Smith conjecture]] * [[Hauptvermutung]] * [[Hedetniemi's conjecture]], counterexample announced 2019<ref>{{cite web |last1=Kalai |first1=Gil |title=A sensation in the morning news – Yaroslav Shitov: Counterexamples to Hedetniemi's conjecture. |url=https://gilkalai.wordpress.com/2019/05/10/sansation-in-the-morning-news-yaroslav-shitov-counterexamples-to-hedetniemis-conjecture/ |website=Combinatorics and more |language=en |date=10 May 2019}}</ref> * [[Hirsch conjecture]] (disproved in 2010) * [[Intersection graph conjecture]] * [[Kelvin's conjecture]] * [[Kouchnirenko's conjecture]] * [[Mertens conjecture]] * [[Pólya conjecture]], 1919 (1958) * [[Ragsdale conjecture]] * [[Schoenflies conjecture]] (disproved 1910)<ref>{{SpringerEOM|title=Schoenflies conjecture|id=Schoenflies_conjecture}}</ref> * [[Tait's conjecture]] * [[Von Neumann conjecture]] * [[Weyl–Berry conjecture]] * [[Williamson conjecture]] In [[mathematics]], ideas are supposedly not accepted as fact until they have been rigorously proved. However, there have been some ideas that were fairly accepted in the past but which were subsequently shown to be false. The following list is meant to serve as a repository for compiling a list of such ideas. *The idea of the [[Pythagoreanism|Pythagoreans]] that all numbers can be expressed as a ratio of two [[integer|whole number]]s. This was disproved by one of [[Pythagoras]]' own disciples, [[Hippasus]], who showed that the square root of two is what we today call an [[irrational number]]. One story claims that he was thrown off the ship in which he and some other Pythagoreans were sailing because his discovery was too heretical.<ref>{{cite book | last = Farlow | first = Stanley J. | title = Paradoxes in Mathematics | year = 2014 | url = https://books.google.com/books?id=d4YUAwAAQBAJ&pg=PA57 | publisher = [[Courier Corporation]] | page = 57 | isbn = 978-0-486-49716-7 }}</ref> *[[Euclid]]'s [[parallel postulate]] stated that if two lines cross a third in a [[plane (mathematics)|plane]] in such a way that the sum of the "interior angles" is not 180° then the two lines meet. Furthermore, he implicitly assumed that two separate intersecting lines meet at only one point. These assumptions were believed to be true for more than 2000 years, but in light of [[General Relativity]] at least the second can no longer be considered true. In fact the very notion of a straight line in four-dimensional curved [[space-time]] has to be redefined, which one can do as a [[geodesic]]. (But the notion of a plane does not carry over.) It is now recognized that [[Euclidean geometry]] can be studied as a mathematical abstraction, but that the [[universe]] is [[non-Euclidean]]. *[[Fermat]] conjectured that all numbers of the form <math>2^{2^m}+1</math> (known as [[Fermat number]]s) were prime. However, this conjecture was disproved by [[Euler]], who found that <math>2^{2^5}+1=4,294,967,297 = 641 \times 6,700,417.</math><ref>{{cite book | last1 = Krizek | first1 = Michal | last2 = Luca | first2 = Florian | last3 = Somer | first3 = Lawrence | title = 17 Lectures on Fermat Numbers: From Number Theory to Geometry | year = 2001 | publisher = [[Springer (publisher)|Springer]] | url = https://books.google.com/books?id=6JCBqZ0CMqgC&pg=PA1 | page = 1 | isbn = 0-387-95332-9 | doi = 10.1007/978-0-387-21850-2 }}</ref> *The idea that [[transcendental number]]s were unusual. Disproved by [[Georg Cantor]] who [[Cantor's first uncountability proof|showed]] that there are so many transcendental numbers that it is impossible to make a one-to-one mapping between them and the [[algebraic number]]s. In other words, the [[cardinality]] of the set of transcendentals (denoted <math>\beth_1</math>) is greater than that of the set of algebraic numbers (<math>\aleph_0</math>).<ref>{{cite book | last = McQuarrie | first = Donald Allan | title = Mathematical Methods for Scientists and Engineers | year = 2003 | url = https://books.google.com/books?id=FmAAwE8MSwoC&pg=PA711 | page = 711 | publisher = University Science Books | isbn = 978-1-891389-24-5 }}</ref> *[[Bernhard Riemann]], at the end of his famous 1859 paper "[[On the Number of Primes Less Than a Given Magnitude]]", stated (based on his results) that the [[logarithmic integral function|logarithmic integral]] gives a somewhat too high estimate of the [[prime-counting function]]. The evidence also seemed to indicate this. However, in 1914 [[J. E. Littlewood]] proved that this was not always the case, and in fact it is now known that the first ''x'' for which <math>\pi(x) > \mathrm{li}(x)</math> occurs somewhere before 10<sup>317</sup>. See [[Skewes' number]] for more detail. *Naïvely it might be expected that a continuous function must have a derivative or else that the set of points where it is not differentiable should be "small" in some sense. This was disproved in 1872 by [[Karl Weierstrass]], and in fact examples had been found earlier of functions that were nowhere differentiable (see [[Weierstrass function]]). According to Weierstrass in his paper, earlier mathematicians including [[Gauss]] had often assumed that such functions did not exist. *It was conjectured in 1919 by [[George Pólya]], based on the evidence, that most numbers less than any particular limit have an odd number of [[prime factor]]s. However, this [[Pólya conjecture]] was disproved in 1958. It turns out that for some values of the limit (such as values a bit more than 906 million),<ref>{{cite journal | last = Lehman | first = R. S. | year =1960 | title = On Liouville's function | journal = [[Mathematics of Computation]] | doi = 10.1090/S0025-5718-1960-0120198-5 | doi-access=free | mr = 0120198 | jstor = 2003890 | volume = 14 | issue = 72 | pages = 311–320 }}</ref><ref>{{cite journal |last=Tanaka |first=M. |date=1980 |title=A Numerical Investigation on Cumulative Sum of the Liouville Function |journal=[[Tokyo Journal of Mathematics]] |doi=10.3836/tjm/1270216093 |mr=0584557 |volume=3 |issue=1 |pages=187–189 |doi-access=free}}</ref> most numbers less than the limit have an even number of prime factors.
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