# Examples of mathematical discoveries which were kept as a secret

There could be several personal, social, philosophical and even political reasons to keep a mathematical discovery as a secret.

For example it is completely expected that if some mathematician find a proof of $P=NP$, he is not allowed by the government to publish it as same as a usual theorem in a well-known public journal because of high importance and possible uses of this special proof in breaking security codes which gives an undeniable upper hand to the state intelligence services with respect to other countries. Also by some social reasons publishing such a proof publicly is not suitable because many hackers and companies may use it to access confidential information which could make a total chaos in the community and economy.

The example shows that in principle it is possible to have some very significant brilliant mathematical proofs by some genius mathematicians which we are not even aware of. But in some cases these "secrets" unfold by an accident or just because they lost their importance when some time passed and the situation changed.

Question: What are examples of mathematical discoveries which were kept as a secret when they discovered and then became unfolded after a while by any reasons?

• A nonmathematical example is Robert Hooke's discovery of Hooke's law; he published a statement of the law in 1660 with the letters sorted into alphabetical order so that the statement was unintelligible, but so that he could plausibly claim later to have known it in 1660. This is itself a very early example of cryptographic hashing.
– MJD
Nov 3, 2014 at 14:36
• I can't believe I'm the first to comment with "I could tell you but then I'd have to kill you" :-) (despite popular belief, that far predates Top Gun) Nov 4, 2014 at 20:31
• @Taladris In an interview, Don Knuth explains why he believes that $P = NP$ and continues to remake that My main point, however, is that I don't believe that the equality $P = NP$ will turn out to be helpful even if it is proved, because such a proof will almost surely be nonconstructive. If so, nobody would be afraid of the result. Nov 5, 2014 at 1:51
• I have a proof of the Riemann Hypothesis; I'm just sitting on it though ;) Nov 5, 2014 at 3:56

Niccolò “Tartaglia” Fontana invented the first general method to find the roots of an arbitrary cubic equation (based on earlier work by himself and others on how to solve cubics of particular forms), but kept his method secret so as to preserve his advantage in problem-solving competitions with other mathematicians.

He divulged the secret to his student Gerolamo Cardano on condition that Cardano keep the secret, and there was a bitter dispute between the two when Cardano broke his promise.

• Excellent example, because Tartaglia was apparently second to keep it a secret. I was apparently kept secret by Scipione del Ferro who was first to discover it, at least according to Gerolamo Cardano, after visiting his inheritors to see his notebooks. Incidentally, I do not believe Cardano was a student of Tartaglia, but a competitor. Cardan got the secret in exchange for a recommandation letter (wikipedia). Nov 6, 2014 at 17:28

William Sealy Gosset, while working at Guinness, developed a way of gauging the quality of raw materials with very few samples ($\implies$ less lab work $\implies$ cheaper!). As the story goes, company policy at Guinness forbade its chemists from publishing their findings. Thus Gosset had to publish under the pseudonym "Student".

The results of his work are known as Student's t-test and Student's distribution.

An example is Pythagorians discovery of irrationality of $\sqrt{2}$. They kept it as a secret for a while because of their special philosophical point of view about the rationality of all numbers in the world. In fact their cosmology were based on a presumption that everything in the nature is made of numbers and their ratios. Some stories say that finally a student of Pythagoras' academy left the society and revealed this secret to public.

Read more on their philosophical point of view here, and the controversy over irrational numbers here. Also you can find additional information on the history of unfolding the secret of irrationality of $\sqrt{2}$ in many texts in history of philosophy and mathematics including Russell's "A History of Western Philosophy".

• There is a short story,in The Tree of possible, by French novelist Bernard Werber about a society organized about the religion of numbers, social classes depending on how far you can count (the great leader being able to count until... 20). Nov 5, 2014 at 1:47
• Similar point to be made about "Infinitesimals." nytimes.com/2014/04/08/science/… Nov 6, 2014 at 19:53
• Was there a practical reason why the Pythagoreans went to such lengths, even murder? It's easy to say due to "religious extremism" but it's interesting to consider whether there were rational motivations. Nov 7, 2014 at 1:57
• @yters and others, apparently, according to Martínez's The cult of Pythagorus (books.google.com/books?id=yCd6nAEACAAJ)—to which I don't have access, but see paragraph 2 on p. 747 of jstor.org/stable/10.4169/amermathmont.121.08.746—this story is not true. Nov 8, 2014 at 4:03
• @yters: religious extremism can be quite rational. If you are a religious leader and everyone knows your faith is based on demonstrably false claims, your authority takes a huge blow. Maybe not fatal as we can imagine by looking at today's religions (some of which insist on literal interpretation of some rather dubious statements), but still. Nov 26, 2014 at 8:39

Actually, I think the best example is of a mathematician who is alive nowadays and proved many results which were unpublished for many years just because he wanted to present them all together in order to solve a famous problem: Fermat's last theorem.
His name is Andrew Wiles.

Cryptography is often a good source of such instances:

At GCHQ, Cocks was told about James H. Ellis' "non-secret encryption" and further that since it had been suggested in the late 1960s, no one had been able to find a way to actually implement the concept. Cocks was intrigued, and developed, in 1973, what has become known as the RSA encryption algorithm, realising Ellis' idea. GCHQ appears not to have been able to find a way to use the idea, and in any case, treated it as classified information, so that when it was rediscovered and published by Rivest, Shamir, and Adleman in 1977, Cocks' prior achievement remained unknown until 1997. Source: Wikipedia

• Differential cryptanalysis is another good example, which I think should be added. (My edit adding this was rejected by people other than the original author.) Nov 3, 2014 at 18:11
• @DavidRicherby I got unreasonably excited by the term "Differential cryptanalysis", thinking it was going to be some form of cipher breaking based somehow on real analysis... Nov 3, 2014 at 18:49
• @DavidRicherby I think you could add it as a separate answer. While belonging to the same area (security/cryptography), it is a different discovery. Nov 3, 2014 at 19:01
• It's also worth noting that the Diffie--Hellman key exchange was discovered first at GCHQ by Malcolm Williamson and later publicly by Diffie and Hellman. In the public arena, Diffie--Hellman (1976) was found before RSA (1978), but in the classified world it was RSA by Cocks (1973) that came before Diffie--Hellman by Williamson (1976). See cryptocellar.web.cern.ch/cryptocellar/cesg/ellis.pdf.
– KCd
Nov 3, 2014 at 20:56
• Related, interesting link: golem.ph.utexas.edu/category/2014/10/… If you view algorithms to break encryptions as a mathematical discovery (and why shouldn't you?), this is certainly a good (and very scary) example. Nov 5, 2014 at 8:28

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Added later: I posted the "aenrsw" above in cryptic form as a bit of secretive levity, but in case the link gets broken, the character string is an anagram used by Newton in a 1677 letter to Leibniz to lay claim to the fundamental theorem of calculus without revealing it.

A second, famous example, from an 1829 letter from Gauss to Bessel alluding to the discovery of non-euclidean geometry:

"[I]t will likely be quite a while before I get around to preparing my very extensive investigations on this for publication; perhaps this will never happen in my lifetime since I fear the cry of the Boetians if I were to voice my views."

• Here Boetians is a euphemism for "stupid people". Jan 2, 2015 at 11:26

The writing of Stephen Wolfram's book A New Kind of Science involved Wolfram Research employees. Wolfram considers proofs done by these employees to be covered by their NDAs, which is exemplified by Matthew Cook's proof that the cellular automata Rule 110 is Turing complete. That proof was first written in 1994, but Matthew Cook was dissuaded from trying to publish it until 1998. When he did Wolfram successfully moved to block the proof from being presented at the conference Cook had intended to use for this purpose, and the proof was not published until 2004. Reference.

• Wow, that seems low. Nov 6, 2014 at 9:41
• There's been quite a bit of comment that Wolfram's work was not entirely original, and that this was little more than a "land grab" for already established proofs. vserver1.cscs.lsa.umich.edu/~crshalizi/reviews/wolfram Nov 6, 2014 at 19:55

Onsager announced in 1948 that he and Kaufman had found a proof for the fact that the spontaneous magnetization of the Ising model on the square lattice with couplings $J_1$ and $J_2$ is given by

$M = \left(1 - \left[\sinh 2\beta J_1 \sinh 2\beta J_2\right]^{-2}\right)^{\frac{1}{8}}$

But he kept the proof a secret as a challenge to the physics community. The proof was obtained by Yang in 1951

• Is there anything like an article or something, where it'd be apparent that Onsager indeed made such an announcement? Nov 4, 2014 at 16:25
• I've updated the answer. Nov 5, 2014 at 20:13

I do not know whether this is appropriate for an answer: the work being secret, I cannot tell what it is (this is not a joke). And I would not be enough of a mathematician to describe it anyway.

It concerns Alexander Grothendieck, 86 today, winner of the Fields Medal in 1966. My original information is from the French magazine La Recherche, n°486, April 2014 (Unfortunately I do not have my copy with me to check details). Another account is available on the web, also in French. Apparently this is in relation with a documentary film made recently about him.

He is now retired in a small village in the Pyrenean Mountains since 1991, at the age of 63. At that date he left the University of Montpellier where "*he left with Jean Malgoire 20,000 pages of notes and letters written in the course of 15 years". Malgoire and two colleagues, Matthias Künzer and Georges Maltsiniotis, analyzed some of it and extracted Grothendieck's concept of derivator.

But this is only a small part of the 20,000 pages of mathematical writing, most of which has not been examined, and which he does not want to be published, and would rather have destroyed, since he started opposing scientific research for ideological reasons.

According to Michel Demazure, it may take fifty years or more to figure out all these unpublished documents.

I think there are supposed to be more documents in his current home, but I do not have my source handy to check.

Descriptive Geometry was invented by Gaspard Monge in 1765 (he was eighteen) for military applications. For 15 years (possibly more) it was classified and kept as a military secret.

It is a geometric technique to represent 3-dimensional objects through projections on planes. It allows 3D geometric constructions similar to what is usually done in 2D, through correspondence of points between the projections. It was used at the time for fortification design.

• For a superb statue of Monge, go to Beaune, France - also famous as one of the birthplaces of cinematography, but, tremendously more importantly for your visit, the world's fine wine production centre :) Nov 6, 2014 at 17:10

As a probabilist, I cannot not mention the following, very fascinating to me, story of the life of Döblin and his relationship with the formula (almost) universally known as Itō formula http://en.wikipedia.org/wiki/Wolfgang_D%C3%B6blin

In 1994, a member of the original IBM DES team, Don Coppersmith, published a paper stating that differential cryptanalysis was known to IBM as early as 1974, and that defending against differential cryptanalysis had been a design goal. According to author Steven Levy, IBM had discovered differential cryptanalysis on its own, and the NSA was apparently well aware of the technique. IBM kept some secrets, as Coppersmith explains: "After discussions with NSA, it was decided that disclosure of the design considerations would reveal the technique of differential cryptanalysis, a powerful technique that could be used against many ciphers. This in turn would weaken the competitive advantage the United States enjoyed over other countries in the field of cryptography." Within IBM, differential cryptanalysis was known as the "T-attack" or "Tickle attack".

The scheme was first published by Whitfield Diffie and Martin Hellman in 1976, although it had been separately invented a few years earlier within GCHQ, the British signals intelligence agency, by James H. Ellis, Clifford Cocks and Malcolm J. Williamson but was kept classified.

Two cases of mathematical secret from the Second World War: Abraham Wald and Alan Turing.

Though apparently technically related (according to Wikipedia), these two cases were actually independent of each other, and the mathematics used for very different purposes.

Some fifty years ago, I read that Bellman optimization principle had been kept a secret. It took me some time to sort this out, with the help of the internet, as the information was not fully accurate. But it was apparently almost true.

Actually, the mathematical technique kept secret was the related work of Abraham Wald on sequential analysis.

Here is a (hopefully) more accurate version of the story, translated from the French, from the page 145, in chapter 4 on Dynamic Programming of a Swiss engineering book: Recherche opérationnelle pour ingénieurs, Volume 2 By Jean-François Hêche, Thomas M. Liebling, Dominique de Werra, PPUR (Presses Polytechniques et Universitaires Romandes), 2003,ISBN: 2 88074 459 8.

Like most models and methods of Operations Resarch, Dynamic Programming is, too, a by-product od the Second World War. It is indeed during that conflict, with the concern of improving the effectiveness of quality control in the production of ammunition, that the American mathematician Abraham Wald proposes a new method that he name sequential analysis. The main idea of his approach resides in the replacement of traditionnal sampling with fixed size by a process in which it is decided dynamycally whether the accumulated evidence is sufficient, or whether, to the contrary, further test must be performed. This method being particularly efficient, it remains classified as a military secret even after the war, and it is only in 1947 that it becomes awailable to the public. Very soon afterwards, at the Rand Corporation of California, independently from Wald, but also in a military context, Richard Bellman becomes interested in optimal control, and, more generally, in sequential optimization problems. He originated the name dynamic programming and made the methodology known in his reference work of 1957. It is in this book that Bellman proposes his optimality principle which states that any shortest path is itself composed of shortest paths.

This is confirmed by looking in Wikipedia at the page on sequential analysis, which lists other contributors, but has no reference to Bellman principle. I do not know sequential analysis, and cannot assess at this time whether it is appropriate to disconnect sequential analysis from the later work of Bellman, as is done in wikipedia.

But this same wikipedia page raises another case of secrecy in mathematics, concerning a similar approach used by Alan Turing in his cryptographic work, which remained secret until the 1980s :

A similar approach was independently developed at the same time by Alan Turing, as part of the Banburismus technique used at Bletchley Park, to test hypotheses about whether different messages coded by German Enigma machines should be connected and analysed together. This work remained secret until the early 1980s.

Boldface used in citations is mine, not from the original text.

The Decimal System of representation of Numbers and Rational and Irrational Fractions were well guarded secret. Indian Astronomers used it for Astronomical and Astrological calculations. The formulas are written in symbolic form in the Vedas, believed to have been composed around 1500 BC. From this Wikipedia page,

... Evidence of early use of a zero glyph present in Bakhshali manuscript, a text of uncertain date, possibly a copy of a text composed as early as the 2nd century BC.

Only around 9th Century AD, Khwārizmī translated this Hindu decimal positional number system in Persian which was further introduced to Western world in twelfth century through Latin translation.

In the article "Overview of Indian mathematics", Laplace wrote:-

The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. the importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.

• added citations.. Nov 6, 2014 at 18:23
• I would be interested to see a reference relating to the claim of secrecy. The article about the Bakhshali manuscript does not mention a hypothesis of secrecy in its source materials. Nov 7, 2014 at 10:52

Ed Thorp is thought to have discovered the Black-Scholes formula a few months/years prior to Black and Scholes.

Instead of sharing it, he created one of the world's first quantitative hedge fund and used the formula to (quickly said) trade options better than the rest of the market. Some references :

The same guy had also developed a theory of counting cards in Blackjack, using it in numerous casinos, before publishing it in his best-selling book, Beat the dealer.

A lot of interesting answers above! I would add that a lot of the work done at Betchley Park during W.W. II as well as the work done by Oppenheimer's group at Los Alamos was of course kept secret until the war was over -- and even long after that!

Johann Bernoulli challenged the other mathematicians of his time to solve the Brachistochrone problem, even though he knew the solution already.

This of course gave rise to the Calculus of Variations.

In the 1920s Alfred Tarski found proofs that if a system of logic had either {CpCqp, CpCqCCpCqrr} or {CpCqp, CpCqCCpCqrCsr} as theses of the system, then it has a basis which consists of a single thesis. In other words, if both of {CpCqp, CpCqCCpCqrr} or both of {CpCqp, CpCqCCpCqrCsr} belong to some system S, then S has a single formula which can serve as the sole axiom for proving all other theses of the system. The proof never got published, but did seem known to several authors for a while. Then it got lost. A few years later, proofs of Tarski's results did get found.

• I find this explanation of Tarski's results completely opaque: it doesn't conform to any logical notation I'm familiar with. What are these sequences of letters supposed to mean? Nov 10, 2014 at 1:38
• @RyanReich Hi Ryan. The symbols are in Polish notation as invented by Jan Lukasiewicz. Lukasiewicz and Tarski used that notation in a 1930 paper called "Investigations into the Sentential Calculus", where one (maybe both, my memory is not precise here) of those pairs gets mentioned. The "C" stands for a conditional, and thus CpCqp can get read "if p, then if q, then p" and "CpCqCCpCqrr" can get read "If p, then if q, then if p implies that q implies r, then r". Parenthesizing them they would be C(p,C(q,p)), and C(p,C(q,C(C(p,C(q,r)),r))). Thus in an infix notation we might write Nov 10, 2014 at 4:23
• (p->(q->p)) and (p->(q->((p->(q->r))->r))). Nov 10, 2014 at 4:26

Fermat announced many of his results, but released very few details of his proofs, so his methods could be considered for the most part to have been kept secret.

Often he challenged others in written correspondence to solve problems that he claimed to be able to do, such as finding nontrivial integral solutions to some Pell equations (he offered the example of $x^2 - 61y^2 = 1$, where he said he chose the coefficient 61 because it is small and shouldn't be too much work, whereas in fact the first solution in positive integers is unusually large, so it was not a random example) or finding all solutions in positive integers to $y^2 = x^3 - 2$ (the only one is $(x,y) = (3,5)$) or showing every prime that is 1 mod 4 is a sum of two squares.

He hoped these public challenges would inspire others to get interested in number theory, but this goal was largely unsuccessful in his lifetime. It took about 100 more years for anyone to really pick up where Fermat left off, and that person was Euler. All he had to start off with were the claims by Fermat of what he could do, and proofs of everything had to be reconstructed, assuming Fermat had proofs at all. To this day only one proof of Fermat in his own writing is known (that 1 and 2 are not "congruent numbers" based on his method of descent).

Of course the most famous challenge by Fermat was Fermat's last theorem, but that is not in the same category as his usual challenges because it was a result he wrote to himself in one of his books and was not intended for others. So in a sense the statement of FLT itself could be considered secret, but it doesn't really fit the parameters of the question as Fermat was almost certainly wrong about having a proof in this case.

• I'm not anything near an expert in math history but you are saying that we have no evidence of Fermat proving for example that $x^4+y^4=z^4$ has no solutions or any of the many theorems that carry his name? Nov 7, 2014 at 14:49
• @ypercube: There is no written record of Fermat's reasoning except for a proof that 1 is not a congruent number. See Weil's "Number Theory: An Approach Through History from Hammurapi to Legendre." In the chapter on Fermat, Weil writes often about Fermat being cryptic in his letters. As for the impossibility of solving $x^4 + y^4 = z^4$ in positive integers, that is a consequence of the impossibility of solving $x^4 + y^2 = z^4$ in positive integers, and that follows from 1 not being a congruent number, so in a sense Fermat did leave to posterity a proof of his last theorem for exponent 4.
– KCd
Nov 7, 2014 at 18:43

I think Gauss anticipated non-euclidian geometries, and kept it quite because he was afraid of how controversial it was. He actually had a dispute with Janos Bolyai, who thought he was stealing his idea.