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What are examples of math hoaxes/interesting jokes published on April Fool's day?

For a start P=NP.

Added 2019-04-01 Anything new in 2019?

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    $\begingroup$ mathforum.org/kb/thread.jspa?threadID=437522 . Also Zeilberger has, of course, many other ones. $\endgroup$ – Vesselin Dimitrov Apr 1 '16 at 7:36
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    $\begingroup$ John Baez has been at it overnight: plus.google.com/117663015413546257905/posts/EnuZypVRAj5 By contrast with some examples, it is pretty low-key. $\endgroup$ – potentially dense Apr 1 '16 at 8:34
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    $\begingroup$ I'm voting to close this question as off-topic because it seems to have run its course. $\endgroup$ – Suvrit Apr 2 '16 at 14:13
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    $\begingroup$ @Suvrit Hoaxes last much longer than the single april first. $\endgroup$ – joro Apr 2 '16 at 14:29
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    $\begingroup$ Joro: Permission for minor edits is implicitly expected on MathOverflow, and is part of the design of the site. Please do not admonish users for doing so. $\endgroup$ – S. Carnahan Apr 11 '16 at 15:43

23 Answers 23

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I enjoyed the hexasphere by A. V. Akopyan, J. Crowder, H. Edelsbrunner, R. Guseinov
from last year:

http://pub.ist.ac.at/~edels/hexasphere/

enter image description here

In the link, the sphere is animated, so you can look at it from all sides.

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    $\begingroup$ I found a pentagon! $\endgroup$ – M.G. Apr 1 '16 at 7:52
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    $\begingroup$ 2 July, There are two of them. Try to find the second one! $\endgroup$ – Arseniy Akopyan Apr 4 '16 at 14:53
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    $\begingroup$ I've partially deobfuscated the source here: gist.github.com/sleepygarden/1c2496991d9b8534f0cf948bacd22f05 $\endgroup$ – mcornell Apr 6 '16 at 19:20
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    $\begingroup$ I tried very hard but couldn't find any pentagon. I'm starting to think the claim that there are pentagons is another april fools joke! $\endgroup$ – Alnitak Apr 10 '16 at 0:56
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    $\begingroup$ @Alnitak: If you have a powerful enough imagination you will notice some slight discrepancy between the displayed model and your mental model. (In other words it is a fake rendering, much like moillusions.com/wp-content/uploads/2013/05/choc.gif.) $\endgroup$ – user21820 Apr 18 '16 at 13:37
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In 2009, it was discovered that the numerical value of $\pi$ has changed over time. This is a truly interdisciplinary work connecting the study of ancient cultures with string theory, cosmology and bicycle tires. Let me quote from the introduction:

Physicists have long speculated that the fundamental constants might not, in fact, be constant, but instead might vary with time. Dirac was the first to suggest this possibility, and time variation of the fundamental constants has been investigated numerous times since then. Among the various possibilities, the fine structure constant and the gravitational constant have received the greatest attention, but work has also been done, for example, on constants related to the weak and strong interactions, the electron-proton mass ratio, and several others.

It is well-known that only time variation of dimensionless fundamental constants has any physical meaning. Here we consider the time variation of a dimensionless constant not previously discussed in the literature: $\pi$. It is impossible to overstate the significance of this constant. Indeed, nearly every paper in astrophysics makes use of it. [..]

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    $\begingroup$ "More speculatively, one might consider the possibility that the values of the integers could vary with time, a result suggested by several early Fortran simulations." (p. 2) $\endgroup$ – r.e.s. Apr 2 '16 at 14:15
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    $\begingroup$ The whole paper is beautiful. No sentence too much. The "leakage" paragraph and Section IV are my favorites. $\endgroup$ – darij grinberg Apr 2 '16 at 20:57
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    $\begingroup$ It is also conjectured that the constant 1 is subject to infinitesimal quantum fluctuations and assumes at times the value of 0.999... . Unfortunately no experimental setup allows to observe the phenomenon in its entirety. $\endgroup$ – Florian F Apr 3 '16 at 21:14
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    $\begingroup$ @Florian, this is related to recent observations of particles travelling at speeds higher than the speed of light, related to the upward infinitesimal fluctuations of the constant $1$ assuming the value of $1.000\ldots 1$. $\endgroup$ – Mikhail Katz Apr 12 '16 at 9:54
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In 1975 Martin Gardner produced a map with 110 regions which he claimed required five colours:

Martin Gardner April Fool '5-colour' map

http://mathworld.wolfram.com/Four-ColorTheorem.html

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    $\begingroup$ At that time I was an Assistant Professor. My mother read the Gardner column in Scientific American and sent me a letter asking whether I knew that the Four Color Conjecture had been disproved. $\endgroup$ – Richard Stanley Apr 2 '16 at 13:58
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    $\begingroup$ The puzzle in your answer is different from the one on the page you linked to. So I solved it: i.imgur.com/jTbwfnm.png $\endgroup$ – UTF-8 Apr 3 '16 at 0:34
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    $\begingroup$ @UTF-8 I spotted two touching vertical red pieces on the middle of the RHS. $\endgroup$ – ᴇʟᴇvᴀтᴇ Apr 3 '16 at 18:50
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    $\begingroup$ Is the punchline that the top half and the bottom half seem to be moving relative to each other in an optical illusion? $\endgroup$ – Andrew Grimm Apr 6 '16 at 11:25
  • $\begingroup$ @aetheria: but it's fine: they are not touching, there is a black color in between... oh, wait... that makes it 5 colors (... or 2 ? if we take this line into account, only 1 other color is required ^^) $\endgroup$ – Olivier Dulac Apr 25 '16 at 16:21
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Someone (widely believed to be Henri Darmon) circulated the following email on April Fools' Day, 1994:

There has been a really amazing development today on Fermat's Last Theorem. Noam Elkies has announced a counterexample, so that FLT is not true after all! His spoke about this at the Institute today. The solution to Fermat that he constructs involves an incredibly large prime exponent (larger that 10^20), but it is constructive. The main idea seems to be a kind of Heegner point construction, combined with an really ingenious descent for passing from the modular curves to the Fermat curve. The really difficult part of the argument seems to be to show that the field of definition of the solution (which, a priori, is some ring class field of an imgainary quadratic field) actually descends to Q. I wasn't able to get all the details, which were quite intricate...
So it seems that the Shimura Taniyama conjecture is not true after all. The experts think that it can still be salvaged, by extending the concept of automorphic representation, and introducing a notion of ``anomalous curves" that would still give rise to a ``quasi-automorphic representation".

The email reached Gian-Carlo Rota at MIT, who took it at face value and circulated it more widely. Eventually David Feldman posted it to the Usenet group sci.math. The thread is here.

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    $\begingroup$ While this was already mentioned by KConrad in a comment, it is good to have it as answer, too. $\endgroup$ – user9072 Apr 1 '16 at 13:04
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    $\begingroup$ @quid (and KConrad): My apologies. I'd quite overlooked the comment. $\endgroup$ – Steven Landsburg Apr 1 '16 at 13:26
  • $\begingroup$ I like this one because, in my opinion, there is nothing in it which hints at this being a joke. If I saw this by the time FLT wasn't proven yet then I would probably believe it (provided I forgot what day it is :P). $\endgroup$ – Wojowu Apr 1 '16 at 20:28
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    $\begingroup$ Worth noting that this was (IIRC) after Andrew Wiles had announced his proof, but before he fixed the final gap that somebody found while proof-reading it. Bit of a cruel joke really, from Wiles's point of view at least. $\endgroup$ – Rand al'Thor Apr 3 '16 at 13:45
  • $\begingroup$ I confess to having been briefly convinced by this one, which a colleague emailed to me. $\endgroup$ – Steve Kass Apr 3 '16 at 22:51
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This one is my favorite (especially a mixture of anyons and morons with opposite spins):

> From: Enrico Bombieri <eb@IAS.EDU> Tue, 1 Apr 1997 12:35:12 -0500
> Date: Tue, 1 Apr 1997 12:35:12 -0500 To: eb@IAS.EDU,
> zeilberg@euclid.math.temple.edu
> 
> Dear Doron,
> 
> There are fantastic developments to Alain Connes's lecture at IAS last
> Wednesday. Connes gave an account of how to obtain a trace formula
> involving zeroes of L-functions only on the critical line, and the
> hope was that one could obtain also Weil's explicit formula in the
> same context; this would solve the Riemann hypothesis for all
> L-functions at one stroke. Thus there cannot be even a single zeroe(1)
> off the critical line!
> 
> Well, a young physicist at the lecture saw in a flash that one could
> set the whole thing in a combinatorial setting using supersymmetric
> fermionic-bosonic systems (the physics corresponds to a near absolute
> zero ensemble of a mixture of anyons and morons with opposite spins)
> and, using the C-based meta-language MISPAR, after six days of
> uninterrupted work, computed the logdet of the resolvent Laplacian,
> removed the infinities using renormalization, and, lo and behold, he
> got the required positivity of Weil's explicit formula! Wow!
> 
> Regards also from Paula Cohen. Please give this the highest diffusion.
> Best,
> 
> Enrico
> 
> 
> (1) This is the correct spelling, according to vicepresident Dan
> Quayle.
> --------------------------------------------------------------
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    $\begingroup$ "...mixture of anyons, morons, and dorons..." :-) $\endgroup$ – M.G. Apr 1 '16 at 8:31
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[only borderline mathematical]

Today Ali Frolop and Douglas Scott published a paper (http://arxiv.org/abs/1603.09703) in which they found:

... there is a remarkable correspondence between each type of peculiarity in the digits of π and the anomalies in the CMB.

enter image description here

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  • $\begingroup$ This is not submitted on April first, probably due to timezone issues. Is the paper provably a hoax? $\endgroup$ – joro Apr 1 '16 at 12:58
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    $\begingroup$ @joro: here are two proofs. 1. Permute the letters of "Ali Frolop". 2. Look here: arxiv.org/abs/1504.00108 $\endgroup$ – potentially dense Apr 1 '16 at 13:05
  • $\begingroup$ @joro 3. proof: April fool is actually mentioned in the text $\endgroup$ – user89746 Apr 1 '16 at 13:27
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    $\begingroup$ arxiv.org/find/astro-ph/1/au:+Frolop_A/0/1/0/all/0/1 $\endgroup$ – Tobias Kienzler Apr 1 '16 at 13:57
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    $\begingroup$ @joro And for "not submitted on April first": well, is has to be submitted a bit early to make it into the daily summary e-mails from arxiv $\endgroup$ – user89746 Apr 4 '16 at 9:27
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https://en.wikipedia.org/wiki/Heegner_number#Almost_integers_and_Ramanujan.27s_constant

This concerns the number $e^{\pi\sqrt{163}}$, and says "In a 1975 April Fool article in Scientific American magazine,[7] "Mathematical Games" columnist Martin Gardner made the (hoax) claim that the number was in fact an integer, and that the Indian mathematical genius Srinivasa Ramanujan had predicted it—hence its name."

The punchline is tied into complex multiplication, though I don't know the details.

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    $\begingroup$ This is the second hoax of Gardner here as of now. $\endgroup$ – joro Apr 1 '16 at 16:11
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Well, there's the April 1, 1997 paper by Doron Zeilberger, The Transcendence of E plus Pi and E times Pi (the following quote is snipped a bit; full text available at the link).

The purpose of this note is to announce that Hermite's[H] celebrated result that $e$ is transcendental, combined with an amazing (but apparently overlooked) statement of Goodwin[G], imply the transcendence of both $e + \pi$ and $e \pi$.

But even more interesting than the above implication is the way by which it was arrived, via computer-generated deduction.

We first developed a C-based meta-language, MISPAR, that has built-in number-theoretical deduction capabilities, that inputs suitably formatted statements about numbers (especially targeted to handle transcendence theory), and outputs new statements. Then, using ten diligent graduate students, many results that appeared in papers on the subject were entered in the appropriate format. Then we used a genetic algorithm to deduce million of new results, most of them either trivial or uninteresting (or both!).

Then we made a long list of open problems. Whenever the computer made a new deduction, it was compared against the statements in the list, looking for possible matches.

While we sure hoped to obtain new interesting results, even in our wildest dreams we did not anticipate such a spectacular deduction.

We are sure that MISPAR would make many more interesting deductions in the future. The package itself, and implementation details, will be eventually published at the author's website (http://www.math.temple.edu/~zeilberg).

References

[G] E. J. Goodwin, Amer. Math. Monthly, 1 (1894), 246-247.

[H] C. Hermite, Comptes Rend. Acad. Sci. Paris, 77 (1873), 18-24, 74-79, 285-293.

The explanation is here; apparently, due to some careful phrasing, the statement of the paper is actually technically correct (or at least so says Zeilberger).

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    $\begingroup$ The "careful phrasing" is that Zeilberger claims the result follows from a "statement of Goodwin", but does not claim that said statement of Goodwin is true! E. J. Goodwin is better known as the author of the Indiana Pi Bill, and the title of the unreviewed note in the Monthly cited here is "Quadrature of the Circle". If it was anything like the Pi Bill, I don't think it would take more than one graduate student to derive whatever you'd like from it. (Hooray for explosion). $\endgroup$ – Henning Makholm Apr 2 '16 at 8:46
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One year maybe about twenty years ago an April 1 story circulated on email, giving the news that TeX had been sold to Microsoft (and would therefore no longer be free). It included a pretty convincing firsthand account, complete with embarrassing technical glitches when Bill Gates took the stage at the grand public announcement.

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Paul Taylor, who has often contributed to MO, once posted this to the categories mailing list; the original thread can be found here:

From cat-dist Thu Apr 1 10:24:48 1999
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by mailserv.mta.ca (8.9.3/8.9.3) id IAA04316
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Date: Thu, 1 Apr 1999 13:10:47 +0100 (BST)
From: Paul Taylor
Message-Id: <199904011210.NAA21705@wax.dcs.qmw.ac.uk>
To: categories@mta.ca
Subject: categories: Is Zermelo-Fraenkel set theory inconsistent?
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  IS ZERMELO-FRAENKEL SET THEORY INCONSISTENT?

At the end of this message is a sketch of an argument that leads to the conclusion that Zermelo-Fraenkel set theory is inconsistent.

The impact on mathematics is not as devastating as the incautious observer might suppose. Recall that ZERMELO set theory (1908), which is essentially equivalent to the categorists' notion of ELEMENTARY TOPOS with natural numbers and the axiom of choice, is adequate for most of the purposes of mathematics, though not, as I shall try to explain, logic (and theoretical computer science).

ZERMELO-FRAENKEL set theory is the extension of this system by the axiom-scheme of REPLACEMENT, which was first formulated by Adolf (later Abraham) Fraenkel, Nels Lennes and Thoralf Skolem in 1922, although Dimitry Mirimanoff already had something of the idea in 1917. Notice that this is some two decades after the appearance of the famous "antinomies" of set theory, so presumably the set theorists' guard had dropped by that time, and they had begun again to assert megalomaniac axioms. On the other hand, it is a decade before the second generation of paradoxical results, Godel's incompleteness theorem and Turing's unsolvability of the Halting Problem.

Whenever I see set theory books in a library or bookshop, I turn to the index to find out what they have to say about Replacement. Usually there is some trivial result, such as the existence of what categorists call image factorisation, that could have been proved from Zermelo's axioms with a little more facility in set-theoretic constructions.

The basic use of Replacement, that you will find in the better set theory books, is the recursive construction of sets (in substance -- types or objects to type-theorists and categorists -- rather than their names).

For example, Mostowski's theorem states that every well founded extensional binary relation < is isomorphic to the membership relation for a unique transitive set. This is found, recursively, by means of the formula f(x) = { f(y) | y < x }, which also provides the extensional reflection (quotient) of any well founded relation. In fact the latter result (where the quotient relation is merely another relation, rather than a membership relation) can be proved using the topos or Zermelo axioms alone, and not Replacement [1], although there are categorical generalisations of this that certainly do need Replacement.

Richard Montague [2] proved a result that should have been taken as a warning of the perilous nature of Replacement, though I suspect that Montague's personal eccentricities may have been the reason why he was ignored. ZF can prove the consistency, not only of Zermelo set theory (Z) itself, but also of Z extended by any single theorem of ZF.

Adrian Mathias has claimed [3] that Bourbaki was "ignorant" of Replacement, ie that it did not occur in "Theorie des Ensembles" [4]. Although Bourbaki is hardly very clear on this matter, it does include a version of Replacement in its axioms, indeed one that is in widespread use in category theory and other parts of mathematics, namely that one can form the UNION of any SET-INDEXED FAMILY of SETS.

One application of N-indexed unions in theoretical computer science is Scott's "D-infinity" construction of models of the untyped lambda calculus. Starting from any domain D0=D, one may form its function-space D1=(D0->D0), and similarly D2=(D1->D1), etc., linking these together with embedding- projection pairs. If D was one of the examples of L-domains, having a pair of elements with infinitely many minimal upper bounds, then one can show (classically) that D-infinity has the cardinality of a model of Zermelo set theory, so need not exist within such a model unless it also satisfies Replacement.

These two ways of seeing Replacement have a common theme: we use N-indexed or transfinite unions to unfold a free(ish) model of one logic within a model of another.

Having seen this in the context of a messy domain-theoretic construction, we may think in a more disciplined way about free models of the lambda calculus, the topos axioms, etc. In fact, there is no difficulty in constructing these models, as they are merely TERM ALGEBRAS. The problem lies in proving that the term algebra has the universal (initiality) property that qualifies it as "free":

Let S be the universe (the category of ZF-sets, for example) and F the term algebra (internal to S) for the logic L. Suppose that S itself is a model of L.

Then there is a unique interpretation functor []:F->S that takes each syntactic operation of F (eg prod(a,b)) to the semantics ([a] x [b]) in S.

It is merely unique up to unique isomorphism if the L-structure in S is defined by universal properties rather than being chosen.

This initiality property may also be expressed type-theoretically. Per Martin-Lof [5] introduced objects with such a property, called UNIVERSES, observing the analogy with Replacement. This point of view stresses that the above property is a RECURSION SCHEME.

Let me explain how I came to realise that the existence of []:F->S depends, in general, on Replacement.

There is an amazingly simple but incredibly powerful argument, due to Peter Freyd and known variously as (Artin-Wraith) glu(e)ing, sconing, the Freyd cover, logical relations and other names. It is based on some very elementary categorical investigations of a certain comma category involving F and S. This argument has been developed rather a long way (the most recent paper that I know of is [6]), and we are pretty close to having a purely categorical proof of the strong normalisation theorem for lambda calculi that, unlike the syntactic proofs, is completely generic with regard to the calculus in question.

Freyd originally showed that the terminal object (1) of the free topos (F) is projective, and more generally the "global sections functor" F(1,-) : F -> S preserves colimits. In particular, it preserves the initial object (0), which is categorical jargon for saying that S proves the consistency of F, because the S-set of F-morphisms 1->0 is the initial (empty) S-set.

I found this suspicious, because the punch-line of Andre Joyal's 1973 (but as yet unpublished and unavailable) categorical proof of Godel's incompleteness theorem is that such a functor F(1,-) : F -> S does not preserve the initial object.

The more careful amongst categorists ought also to be suspicious when I speak of "a functor F(1,-) or [] : F -> S" where F is an INTERNAL category in S. The meaning that we must give to this phrase is that it is "syntactic sugar" for a certain FIBRATION p: V -> F, where V is also an internal category and p and internal functor in S.

This brings us back to the relationship between Replacement as a recursive construction of objects and Replacement as infinitary colimits: "p: V -> F" is the colimit (in a 2-category whose objects are fibrations) of a recursively defined diagram vaguely similar to that which gives Scott's D-infinity.

I have come to the conclusion that attempts to define "colimits" such as this are inherently circular: what, after all, does it mean to have a "cocone" to test such an alleged colimit?

My categorical formulation of Replacement speaks about fibrations and smaller colimits defined internally in the style of Benabou. This is to be found in the final section (9.5) of my book [7], Section 7.7 of which also gives an account of Freyd's gluing construction.

This book is officially due to be published in mid-May, but it is already in stock (and I have my own copy in front of me), and is available direct from the publishers at 50 pounds (inclusive of overland postage and packing). Please contact Richard Knott, email: rknott@cup.cam.ac.uk fax: +44 1223 315 052 tel: +44 1223 325 916 (but other methods are preferable) snail: Cambridge University Press, The Edinburgh Building, Shaftesbury Road, Cambridge, CB2 2RU, UK with your address and credit card number. (2.50 pounds extra for airmail.)

Having seen that Replacement provides a UNIFORM way of proving consistency of any fragment of logic, we come at last to the inconsistency argument:

Let L(0) be Zermelo set theory (or the axioms for an elementary topos).

For each n, let L(n+1) be L(n) plus as much of the axiom-scheme of replacement as is needed to justify the gluing construction that shows that

  L(n+1) |-  ``L(n) is consistent.''

Now let L(infinity) be the union of L(n) over n:N.

If L(infinity) |- false then L(n) |- false for some n.

But L(infinity) |- ``L(n) is consistent,''

so L(infinity) proves its OWN consistency, contradicting Godel's theorem.

However, L(infinity) has a standard non-trivial interpretation in Zermelo--Fraenkel set theory, which is therefore inconsistent.

[1] Paul Taylor, Intuitionistic Sets and Ordinals, JSL 61 (1996) 705-44

[2] Richard Montague, Fraenkel's Addition to the Axioms of Zermelo, pp 91--114 of Bar-Hillel, et al., eds., Essays on the Foundations of Mathematics, Magnes Press, Hebrew University, 1966 (distributed by Oxford University Press).

[3] Adrian Mathias, The Ignorance of Bourbaki, Mathematical Intelligencer, 14 (1992) 4-13.

[4] Nicolas Bourbaki, Elements de Mathematique XXII: Theories des Ensembles, Livre I, Structures, Hermann, 1957 (English translation 1968).

[5] Per Martin-Lof, An Intuitionistic Theory of Types: Predicative part, pp 73--118 in Rose and Sheperdson, eds., Logic Colloquium '73, North-Holland, Studies in Logic and the Foundations of Mathematics #80, 1975

[6] Djordje Cubric, Peter Dybjer and Philip Scott, Normalisation and the Yoneda Embedding, MSCS 8 (1998) 153--192.

[7] Paul Taylor, Practical Foundations of Mathematics, Cambridge University Press, Cambridge Studies in Advanced Mathematics #59, xii+572pp, 1999.

http://www.dcs.qmw.ac.uk/~pt/Practical_Foundations/

Paul Taylor 19990401

This message may be copied elsewhere, ON CONDITION that it is quoted in its ENTIRETY.

(I did quote it in its ENTIRETY, including all the headers at the top!)

(Quite a few got the joke, but I think also quite a few missed it and took Paul at his word.)

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  • $\begingroup$ I don't get it; what was the joke? $\endgroup$ – goblin Apr 4 '16 at 10:21
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    $\begingroup$ He purports to prove that ZFC is inconsistent? And succeeded in getting some people to take him at his word? $\endgroup$ – Todd Trimble Apr 4 '16 at 10:45
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    $\begingroup$ oh. This just seems like a "mere" hoax, without very much in the way of humour (to me at least). It would be better if it were more explicitly tongue-in-cheek, I think. $\endgroup$ – goblin Apr 4 '16 at 11:00
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    $\begingroup$ Oh no, of course it was a hoax. When I said 'joke', I'm referring to his joking around, i.e., not being serious. $\endgroup$ – Todd Trimble Apr 4 '16 at 11:07
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The most recent additions to our Seeley G. Mudd Manuscript Library feature contributions from the estate of Oliver Wendell Holmes, Jr., which include letters and legal manuscripts of Pierre de Fermat (a lawyer by vocation). It is in the density of Fermat's litigation records during the period 1660-1662 that his lost mathematical proof is finally to be found.

It turns out that Fermat's proof employs what is now known as the Mason-Stothers theorem (proved independently by Stothers [2] and Mason [3] in the late 20th century). In the discovered manuscript, Fermat himself gave an elementary proof of the Mason-Stothers theorem, but his approach resembles that presented in An alternate proof of Mason's theorem by Snyder [4]. For this reason we here omit Fermat's proof of the Mason-Stothers theorem, and only reproduce the subsequent part of his proof of his last theorem, paraphrased in modern terminology.

http://www.princeton.edu/~aloo/fermat

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More Physicsy than Maths, but there's Don Schneider's "discovery" of a quasar with redshift 4.1 (NB - the largest quasar red-shift known at that time was 3.7), announced at Institute of Advanced Study, Princeton on 1st April.

The number 4.1 was chosen to be a subtle hint, that this whole presentation was a prank. Few people got it right away, most others didn't and were particularly curious regarding the finer nuances of the discovery, which Schneider did happen to address convincingly on the course of his "report". It was a well-cooked up prank!

This is chronicled in Ed Regis - ``Who Got Einstein's Office'', Addition-Wesley (1987):

... But Schneider hands out his charts to the audience, and there's no disbelieving the data. Wavelength plotted against energy flux, the graph looks like a distorted view of lower Manhattan, with sharp peaks and valleys, and one very sharp spike, looking like the World Trade Center. That's the quasar, with its record-breaking redshift. The whole room is abuzz. People are talking to each other a mile a minute, and John Bahcall has the devil of a time moderating the question period. They want to know everything: Where's the object located? What's its coordinates? What's the exact time the observations were taken? But Don answers them all, every last one. . . until it's clear that the thing has gone far enough, and he brings it all to a close. There's another speaker to be heard from, poor fellow. Schneider is going to be one tough act to follow.

Indeed. Some people are even now getting the drift, an inkling of what's actually been going on here. A redshift of four point one, and today is April first. Can this be? . . . Oh, Jesus! It must be. And in fact, yes, it is! It's all . . . an April Fool's joke! Don Schneider has just pulled off the coup of the decade, getting the combined astrophysical brains of Princeton University, Bell Labs, and the Institute for Advanced Study to believe that in the space of a few hours in the morning, at an Institute with absolutely no observing facilities whatsoever, not even so much as a pair of binoculars, he's discovered the world's farthest object smack in the middle of a gravitational lens.

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This list is missing one of the greatest mathematics April Fools, the Mandelbrot Monk article by Girvan from 1999. It is beautifully done and famous enough to have its own Wikipedia article.

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I saw Doug Ravenel give a talk that began with him announcing a proof of the Riemann Hypothesis. It was beautifully done, I thought. I didn't even realize what date it was.

http://www.math.wayne.edu/~rrb/MTS/archive/1104.html

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    $\begingroup$ Friedlander once started a talk with "Theorem: $\zeta(s)\neq 0$ for $\Re s>1/2$". When the audience started breathing again, he turned around and said: "This is a theorem. We just do not know yet who will prove it." $\endgroup$ – Jan-Christoph Schlage-Puchta Jan 27 '17 at 10:04
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This is perhaps only borderline mathematical, but the ChessBase website has historically made a fine art of the April Fool's joke, and arguably its most successful prank was also the most mathematical: A claim that the King's Gambit had been solved. They were careful not to claim that the opening had been mathematically solved, but that a generous cutoff for the score function had been set (i.e., if the computer thought that one side was "far enough ahead" then the assumption was made that that side really did have a win) and that the game tree was exhausted under this assumption. It was clever enough that even after knowing that it might have been an April Fool's joke, I was still uncertain for a while.

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  • $\begingroup$ That's actually not far (even in many subsidiary details) from yet another item in Martin Gardner's famous April column that already accounts for two answers here. $\endgroup$ – Noam D. Elkies Apr 1 '16 at 18:01
  • $\begingroup$ Is it certain that the hoax is really false? Any chance it is open now? $\endgroup$ – joro Apr 2 '16 at 5:53
  • $\begingroup$ @joro : You mean the Chessbase hoax? Yes, of course it is false. Did you follow the link? What do you mean "Any chance it is open now?"? Does "it" refer to the hoax? If so, what does it mean for a hoax to be "open"? $\endgroup$ – Timothy Chow Apr 7 '16 at 15:30
  • $\begingroup$ I read most of the link and realize it claims unproved result. The question is "Is the result false?" as in the hoax for counterexample of FLT. $\endgroup$ – joro Apr 7 '16 at 16:12
  • $\begingroup$ @joro : I still don't fully understand, but perhaps the following remarks may help. Given any computer technology in the foreseeable future, only a tiny fraction of chess positions have the property that their value (i.e., win/loss/draw) can be established rigorously, and only a slightly less tiny fraction of chess positions can be be analyzed "semi-rigorously" in the sense of the hoax article. No "interesting" opening position such as the King's Gambit falls into either category. Note that many positions whose outcome is "morally certain" still cannot be analyzed even semi-rigorously. $\endgroup$ – Timothy Chow Apr 7 '16 at 17:53
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Here is a counterexample to Fermat's Last Theorem, which is correct according to double precision calculations:

Counterexample

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    $\begingroup$ This is funny, but what is the relation with April first? $\endgroup$ – joro Apr 2 '16 at 6:15
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    $\begingroup$ @joro: Perhaps the corresponding episode was aired on an April 1. BTW, the number in the LHS of the purported equality is (obviously) divisible by 9, whereas the number in the RHS of it is congruent to 1 modulo 9, Q.E.A. $\endgroup$ – José Hdz. Stgo. Apr 2 '16 at 19:08
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    $\begingroup$ Double precision (the usual 64-bit binary floating point of IEEE 754) has enough precision to show immediately that this is wrong. With single precision, each of the twelfth powers are rounded to positive infinity (which would not convince anyone the identity was correct). Some calculators, however, might claim this relation is correct, $3987^{12}+4365^{12}=4472^{12}$. $\endgroup$ – Jeppe Stig Nielsen Apr 3 '16 at 0:17
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    $\begingroup$ @J.H.S. - Unfortunately, the episode, The Wizard of Evergreen Terrace was aired on September 20, 1998. Damn Shame. Also, from Fermat's Last Theorem in fiction: These agree to 10 of 44 decimal digits, but notice that simple divisibility rules show 3987 and 4365 are multiples of 3 so that a sum of their powers is also. The same rule reveals that 4472 is not divisible by 3, so that this "equation" cannot hold either. $\endgroup$ – Greenonline Apr 7 '16 at 16:37
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    $\begingroup$ Observe that the doughnut is undergoing a "Homertopy". $\endgroup$ – silvascientist Apr 8 '16 at 6:46
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A breathless announcement of a weakness in ubiquitous cryptography based on elliptic curves:

this result will require a major increase in parameter for elliptic curve cryptosystems ... we recommend increasing elliptic curve key sizes from 256 bits to 3072 bits

This 2016 claim seems almost plausible until one considers how sphere packing really relates to faster discrete logarithm computation. It deserves bonus points for using recent actual papers by Viazovska et al. to construct its argument.

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  • $\begingroup$ :-) For 256 bit prime, the complexity is about O(2^11). $\endgroup$ – joro Apr 2 '16 at 16:15
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The following is output from Maple:

A := 34816783:  
B := 29698715047:  
C := 120979604904878607889:  
D := 103195600023374741883001:

isprime (A);   true    
isprime (B);   true    
isprime (C);   true  
isprime (D);   true 

AxD;  3592938812568633315821457205783  
BxC;  3592938812568633315821457205783  
AxD-BxC; 0 

Thus AxD and BxC are two prime factorizations of 3592938812568633315821457205783.

This was widely distributed decades ago by a German mathematician. In the current version of Maple versions this no longer works,but a similar hoax with different numbers can probably be found.

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    $\begingroup$ Yet another reason not to assume unique factorization. Gerhard "Let This Be A Warning" Paseman, 2016.04.05. $\endgroup$ – Gerhard Paseman Apr 5 '16 at 22:44
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    $\begingroup$ This is interesting, but is the relation with April 1? Probably it will be on topic in this question: mathoverflow.net/questions/11517/computer-algebra-errors $\endgroup$ – joro Apr 6 '16 at 5:26
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There is also Daniel Schoch's article in which he give way how to obtain number of Gods in our universe using Euler characteristic.

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The mathematician philosopher Hilary Putnam otherwise not known for lightheadedness revealed on april 1, 1980 that subtle logical phenomena in the context of the Lowenheim-Skolem theorem imply the impossibility of fixing an intended model of anything, including the natural numbers (!) and the reals. This actually got published the same year:

Putnam, Hilary. Models and reality. J. Symbolic Logic 45 (1980), no. 3, 464-482.

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  • $\begingroup$ Is there online reference? $\endgroup$ – joro Apr 11 '16 at 16:02
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    $\begingroup$ @joro, the paper is available at princeton.edu/~hhalvors/teaching/phi520_f2012/putnam1980.pdf But, katz, are you saying that something intended as an April Fools joke got published as a serious paper, with no disclaimer, in the Journal of Symbolic Logic? $\endgroup$ – Gerry Myerson Apr 12 '16 at 0:18
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    $\begingroup$ @GerryMyerson, this is for april fools' jokes, no? $\endgroup$ – Mikhail Katz Apr 12 '16 at 6:48
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    $\begingroup$ It's for reporting on them, not for making them (and, anyway, the canonical day for making them has come and gone). $\endgroup$ – Gerry Myerson Apr 12 '16 at 7:09
  • $\begingroup$ @Gerry, sorry. In my defense I could add that I am less than my usual two weeks late with this assignment (also the question was closed for a while). $\endgroup$ – Mikhail Katz Apr 12 '16 at 7:17
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There is an "April Fools" issue of the Math Horizons, published in 2007:

http://www.maa.org/sites/default/files/pdf/horizonsarchive/Horizons-Apr07lores.pdf

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https://arxiv.org/abs/1504.00108

A Farewell to Falsifiability Douglas Scott, Ali Frolop, Ali Narimani, Andrei Frolov (Submitted on 1 Apr 2015)

Some of the most obviously correct physical theories - namely string theory and the multiverse - make no testable predictions, leading many to question whether we should accept something as scientific even if it makes no testable predictions and hence is not refutable. However, some far-thinking physicists have proposed instead that we should give up on the notion of Falsifiability itself. We endorse this suggestion but think it does not go nearly far enough. We believe that we should also dispense with other outdated ideas, such as Fidelity, Frugality, Factuality and other "F" words. And we quote a lot of famous people to support this view.

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I think this short note by D. Zeilberger counts, I am not sure though if it was intended for an April 1st joke.

Note: Zeilberger's conjecture is equivalent with the Collatz conjecture.

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