The most outrageous (or ridiculous) conjectures in mathematics

The conjecture that an odd covering system exists. This was believed to be true by Paul Erdös (see Filaseta, M. and Ford, K. and Konyagin, S., On an irreducibility theorem of A. Schinzel associated with coverings of the integers. Illinois J. Math. (2000)) but recent results (see Hough, Robert D. and Nielsen, Pace P., Covering systems with restricted divisibility. Duke Math. J. (2019)) indicate that this is not the case.

The very old Catalan-Dickson conjecture, asserting that all aliquot sequences either terminate or are periodic. The reason It is infuriating, is that the so-called Lehmer five, 276, 552, 564, 660, and 966, are very likely counterexamples, as computed by Selfridge. Their aliquot sequences appear to go on forever without repetition. Who knows?

This one is due to Errett Bishop: "all meaningful mathematics is reducible to finite calculations with strings of $0$s and $1$s" (imho Bishop formulated this not as a conjecture but as an article of faith but that doesn't necessarily affect the truth or falsity thereof).

A reference for Bishop's claim is his article "Crisis in contemporary mathematics" (the link is to the mathscinet review of the article) which discusses the constructivist opposition to a principle called LPO ("limited principle of omniscience") related to the law of excluded middle. The LPO is discussed starting on page 511 of the article.

A weird number is a positive integer $n$ that is abundant and such that $n$ is not expressible as a sum of distinct divisors of $n$. The celebrated 20th century mathematician Paul Erdös offered a 10\$ prize for finding an odd weird number, and a 25\$ for a proof of non-existence. We may argue that Erdös thought an odd weird number should be more probable to exist. Nevertheless, recent work on finding such an odd weird number have been unsuccessful (https://arxiv.org/abs/2207.12906). So, at present the Erdos conjecture on the existence of odd weird numbers is likely to be false.

In 1956 Gleason (so 70 years ago) proved that every finite Fano projective plane is Desarguesian. A finite projective plane is known to be Desarguesian if and only if it is Pappian if and only if Moufang.

In general case we have the implications

Pappian $\Rightarrow$ Desarguesian $\Rightarrow$ Moufang,

none of which can be reversed (for infinite projective planes).

The following natural conjecture generalizing Gleason's theorem is generally believed to be false (but noone can prove or disprove it):

Conjecture. Every Fano projective plane is Moufang.

Let us recall that a projective plane is Fano if for every quadrangle $abcd$ its diagonal points $x\in\overline{ab}\cap\overline{cd}$, $y\in\overline{ac}\cap\overline{bd}$, $z\in\overline{ad}\cap\overline{bc}$ are collinear.

Historical Remark: The story with this Conjecture is very old and intriguing. It starts from the fact that Ruth Moufang herself thought that she proved this conjecture (in 1932) till Marshall Hall Jr (in 1943) has noticed a flaw in her arguments. One can read about this story in the footnote remark on page 176 of the book "Moufang Polygons" by Tits and Weiss. Fortunately, Google Books shows up this piece of the book which is copied-and-pasted below:

The Gaussian moat problem, posed by Basil Gordon in 1962: one can walk from origin to infinity using the Gaussian primes as stepping stones, and using steps of bounded size. Not likely to be true, since any route has to avoid many "moats" blocking the way, but as far we know, some route could be found.

Let $T(n^k)$ be the threshold of completeness of the $k$-th powers, the largest number which is not the sum of distinct $k$-th powers. Ërdos and Graham conjectured in 1980 that the sequence of these thresholds is not eventuallly monotonic, In other words, $T(n^k)>T(n^{k+1})$ happens infinitely often. But, it turns out that the sequence might be monotonic after all. In a paper by Kim (Kim, D. On the largest integer that is not a sum of distinct positive nth powers, Journal of Integer Sequences, Volume 20, Issue 7 (2017)), upper bounds for these thresholds are given, but these are very far from the lower bounds, meaning that even when using good candidates for $k$ that might break the monotonicity, such as $k=8$, the very next threshold might be closer to the upper bound that the lower bound. In fact computations by Michael J. Wiener in the paper "The Largest Integer Not the Sum of Distinct 8th Powers" have shown that $T(n^8)<T(n^9)$. This is an interesting case of a conjecture that was genuinely believed, despite plenty of computational evidence pointing in the opposite direction. But it could still be true. It is very outrageous, and It has some importance for the additive theory of $k$-th powers and complete sequences.

There is an outrageous conjecture which I abbreviate to:

Algebraic Integrability Conjecture: on a complex variety $V$, a foliation $\mathcal{F}\subset T_{V/\mathbb{C}}$ is given by some rational fibration $f:V \dashrightarrow W$, i.e., $\mathcal{F}=T_f$, (algebraically integrable) if and only if the reduction modulo $p$ of the foliation is $p$-closed for almost all primes.

Being $p$-closed means that if $D\in \mathcal{F}$, then its $p$-times composition, which is also a derivation (!) if $p=0$, is also there, i.e., $D^p\in \mathcal{F}$. It is a positive characteristic feature.

This conjecture is outrageous because it mixes complex analysis with the modulo $p$, and it has strong implications for monodromy theory: it would characterize the systems with finite ones (something topological, analytical) by their arithmetic properties.

Some history. There is something called the Poincaré--Painlevé problem (1890s): for a vector field with rational coefficients on a plane $D=f\partial_x +g\partial_y$, where $f,g\in \mathbb{C}(x,y)$. It is hard to compute explicit solutions to equations like $D(F)=0$; thus, Poincaré asked whether we could bound the genus of any $D$-invariant curve in terms of the degrees of $ f$ and $ g$: we cannot. One is required to add more global invariants to do anything about it.

From a different angle, Grothendieck (1960s), while working on crystalline cohomology, suggested that bundles with a connection which, modulo almost all primes, is $p$-flat, should be algebraic fibrations. We know some cases by Katz, Bost, and others. Katz was the first to write about it explicitly and prove the first cases; thus, we call it the Grothendieck--Katz conjecture.

However, bundles with $p$-flat connections are just a special case of foliations that are $p$-closed, thus in a never-published and not anymore publicly available preprint, Ekedahl, Shephard-Baron, and Taylor (1990s) proposed ``Conjecture F'', which is what I call the algebraic integrability conjecture, because calling it:

Poincaré--Painlevé--Grothendieck--Katz--Ekedahl--Shephard-Baron--Taylor Conjecture/Problem

is ridiculous. Moreover, some people like to joke about the authors' opinions, whether the conjecture is true or not, since each supposedly had a different one. True, false, ?. This is just a repeat of jokes; I do not claim that this is factual.

Furthermore, as a recent survey Algebraic solutions of linear differential equations: an arithmetic approach explains, this conjecture would also tell us a lot about hypergeometric series, and any other series, about when they are rational functions and when not, simply by computing some easy modulo $p$ data.

Naively, one might think of just computing the solution and checking the equations to verify its existence: both conditions are kinds of equations to be satisfied, there is inclusion, but why these big sets of equations are equal, who knows? There are infinitely many of them, and even finitely many of them grow too fast with bigger degrees of coefficients and solutions we allow...

Finally, this is a conjecture about geometric objects, but I cannot imagine how something so general could be proved geometrically, because the number of invariants and their relations grow rapidly with dimension and vary with rank, the variety it is on. Therefore, a proof would have to give us a way to go through all the geometries of varieties and, at the same time, ignore them all...

This is one of my favourites. Let $\sigma^k{(n)}$ be the $k$-fold iteration of the sum of divisors function on $n$. Then for any $m,n$ greater than one, there are $i,j$ such that ${\sigma^i}(m)={\sigma^j}(n)$. In other words, there is essentialy only one sequence of this type. It was asked by van Wijngaarden, back in the 50s. It is obviously very important to the undertanding of the behaviour of arithmetical functions. You can trivially check that the sequences generated by $2,3$ and $4$ satisfy this, but already in the smallest non trivial case $m=5$, $n=2$, the generated sequences appear to be disjoint. In fact, Graeme L. Cohen and Herman J. J. te Riele (see their paper "Iterating the Sum-of-Divisors Function") computed the sequences starting with $2$ and $5$ to values up to $200$ digits without finding any common element. But the conjecture could very well be true: no one seems to know for certain, which is very annoying.