Open problems in mathematics

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In 1900 only two unsolved conjectures remained in mathematics, and David Hilbert said in a famous speech at Göttingen University, "Turn the lights off when you're done. I'm going to open a Bierkeller and get pissed every night from now on." How wrong he was!

Number theory

Ulanova's conjecture

“If n is a number, n + 1 is also a number.”

~ Igor Ulanova on Ulanova's conjecture

This is easy to state, so that even non-mathematicians can understand it (if they can be seriously said to understand anything, the simps), but surprisingly difficult to prove. An essentially equivalent formulation was known to the ancients. Euclid pointed out in a footnote in his treatise On Big Numbers that "it is tempting to speculate that this exceedingly large number, greater than the number of grains of sand on a beach, could be exceeded by adding a smaller number to it." Bhaskara drew a diagram of a large number of dots next to another dot, separated by a slightly greater distance, and a later hand added in the margin the Sanskrit word for "perhaps". Leonhard Euler knew it to be true for all numbers up to a million. Experimental results on modern computers have extended that bound a thousand-fold, and a distributed computing project is being crowdfunded to double the current bound.

The real difficulty is not ultimately one of computer power, but one of logic. It was not appreciated until Galina Ulanova's rigorous formulation of the conjecture in 1941 that even for low bounding numbers there is a potentially infinite search space, which makes the problem one of non-polynomial complexity. In addition to confirming that numbers such as 317 have a successor cardinal such as 318 (a merely computational difficulty), it is necessary to show that if Queen Victoria or a raven or the present King of France is not a number, then adding 1 to the raven also does not produce a number. Annual meetings of the International Logicians' Union have descended into fist-fights over this problem.

For the interested, raven + 1 = Queen Victoria. For obvious reasons this was not proved until 1837. Addition is not commutative on either corvids or royalty, so 1 + raven equals a mirror image Prince Albert, and to get the original parity Prince Albert you need to subtract 1 and add 1 again. The intermediate stage yields an upside-down jackdaw with hemophilia.

Bakst–Fokine conjecture

“For any number n, there is some number N such that N is greater than n.”

~ Ryszard Bakst on Bakst–Fokine conjecture

Deceptively similar to Ulanova's conjecture, but it only implies it under the assumption of a version of the Axiom of Choice more extreme than most mathematicians are prepared to accept.

A result by Tom Mix circulated on the Internet in March 2016 led to a flurry of interest both among experts and in the wider non-numerate population. Mix appeared to have proved that for any number n that could be specified in a finite number of elementary operations, there was a bounding pair (u, v) such that u was less than n if and only if v was greater than n. Minor flaws were quickly found in the proof by his horse, but these are not considered fatal. A proof for the case n = 2 is one of those still lacking, but most mathematicians believe a general proof for even numbers may be achievable.

The problem is made more difficult by debates over how to pronounce N and n to distinguish them. The Big-Enian school holds that big N should be said in a louder voice and this is sufficient, if not strictly necessary.

Ippolitov-Ivanov heuristic

“For any given number n (less than some arbitrary infinite cardinal κ), if you can count up to it, there is some bastard in your immediate circle of acquaintance who has already counted further.”

~ Igoretta Ivanov on Ippolitov-Ivanov heuristic

Another result that was known informally to the ancients (Euclid, according to tradition, lost a tooth brawling with the guy), but which has resisted rigorous proof. Henson showed in 1992 that if an infinite number of people all know more than some other person, there is no upper bound on the amount of knowledge a given person can have on a given subject. As a corollary, it is possible for one person to know more than another. The difficulty lies in showing that you are never that person.

Wesendonk–Lieder–Erlkönig theorem

“1 and 2 are both numbers. Furthermore, if either one of them is not a number, the other one is not either.”

~ Anonymous on Wesendonk–Lieder–Erlkönig theorem

This was actually proved in 2006 by a team of computer scientists led by Kim Wilde and Gene Wilder, but in a 2009 review by WMADB (the World Mathematical Anti-Doping Busybodies), the proof was disallowed on account of steroid use by one of the team. (It was never revealed who, but Elliot Page suspiciously resigned from the group in 2011 to pursue a successful solo career as a lesbian hottie.)

Riemann hypothesis

“Some crap about some fancy pretend numbers no-one believes in anyway.”

~ Bernhard Riemann on Riemann hypothesis

Listed for descheduling at the 2018 Congress for Mathematical Neatness in Biarritz. With the phasing out of complex numbers in 2015, and a proposal before the CMN to abolish most rational numbers by 2030, there is no longer any need for theorems involving silly stuff.

Algebra

Artan–Tiggywinkle stroke of genius

“Any group that is also a field can be extended to a bigger group that is a bigger field.”

~ Mrs Tiggywinkle on Tiggywinkle's lemma

Has passed all tests so far, with groups of 2, 3, 4, and 10 elements, so a proposal has been made to declare it "true by effective exhaustion", a conveniently powerful tool adopted by international agreement in 1998. Known to Cauchy in the weaker form that any empty group was extensible to a field that might or might not be empty. Proved by Dedekind for the cases of empty groups of order 0, 1, and 7. Disproved by Gospodin-Bogomirsky in 1974 for an anti-symmetric group of order 712, but the guy was a tosser so his writings are ignored by most of us.

Graph theory

Ofra Harnoy's embarrassing blunder

“Any graph containing a subgraph contains at least one subgraph.”

~ Captain Obvious on Ofra Harnoy's embarrassing blunder

An interesting example of a topological statement that was widely regarded as trivially false until someone set out to prove it was non-trivially false. Harnoy spent eight years of her life trying to disprove it once and for all; and later gave up to become an amateur nude cellist. Her cleaning lady, about to throw her notes in the bin, spotted the elementary error:

−1 × −1 = −1

a statement Harnoy later admitted she had always implicitly assumed but never checked. Mathematical habits die hard, and in her later career she once played the Bach Toccata and Fugue BWV 565 under the impression it was the Elgar Cello Concerto. Bernstein gave up after a few minutes and went for a smoke.

Spessivtseva's painting algorithm

“Given any connected Suslin tree of adjacent adjunct matrices, such that any (n, m)-cluster of nodes has the Vishnevskaya metric on pairwise disjoint amicable sets, any finite Lie group of randomly selected quasi-compact Grothendieck representatives touches only a Ramsey limit cardinal of witnesses.”

~ Most scholars in the field on Spessivtseva's painting algorithm

Considered obvious by most scholars in the field, but fine details of the proof elude capture and remain hard to spell. The case n = 0, m = 0 is trivial.

Philosophy of mathematics

Perelman's conjecture

“If someone is willing to pay you a million dollars to solve a problem, they've probably got an agenda and you don't want to be part of it.”

~ Oscar Wilde on Perelman's conjecture

Considered obvious by most working mathematicians, but hotly disputed by logicians. Intuitionists, philosophers of science, and other jerkwads who should be first against the wall when the mathematicians are called to take over but are too busy contemplating circles in the sand and consequently miss the opportunity.

Goldbach's little conjecture

“There are prime numbers in different colours.”

~ Christian Goldbach on Goldbach's little conjecture

Difficult to solve because it is not clear how to define what constitutes a different colour. Can be generalized to theorems in algebra, analysis, and so on, about differently coloured vector spaces, open coverings of the plane, standard deviations, etc. Over drinks after the conference, most mathematicians are prepared to hazard a guess that Goldbach was past it by this time.