Some years ago, I read that Kolmogorov was so enraged that Karatsuba disproved one of his conjectures that he terminated his seminar shortly thereafter.

This Wikipedia page claims that Kolmogorov was just "very excited".

Do we have any testimony of what really happened?


1 Answer 1


Karatsuba's own report can be found in his 1995 paper Сложность вычислений (Complexity of computations). The phrasing he uses is "сильно взволновалo", which Google does translate as "very excited" or "greatly agitated". But we can see the usual dynamic of an anecdote in the making on the Reddit thread, where Kolmogorov is "very upset", and writing on the image linked by one of the users already has him "angry". In any case, the difference between "excited", "upset" and "enraged" is a matter of opinion even for those observing first hand.

Kolmogorov did do something unusual afterwards: abruptly ended the seminar and two years later published a paper with the result under Karatsuba's and Ofman's names (Kolmogorov's student) without letting Karatsuba know, see Why did Kolmogorov publish Karatsuba's algorithm? on CS SE. The joint authorship led to a confusion about the credit for Karatsuba's algorithm, although it was clearly credited to him in the paper. As many commenters noted, Kolmogorov's was the opposite of the usual misbehavior where people take somebody else's result and publish it under their own name.

Why Kolmogorov did what he did one can only speculate. Perhaps, he was embarassed that his conjecture of four years was disproved so easily, and not by him. Perhaps he wanted to bundle Karatsuba's result with Ofman's with which he was more involved. Or the discovery made his plans for the seminar obsolete. Or perhaps he felt he misjudged the nature of the problems to be discussed: they were not safely publicizable long shots but doable problems that "like parables, are only explained to the own disciples in private" (as Arnold put it, alluding to the gospel of Mark).

Here is Karatsuba's own telling in edited Google translation:

"In the fall of 1960 the Department of Mechanics and Mathematics of Moscow University opened a seminar on mathematical issues of cybernetics began under the leadership of A.N. Kolmogorov, where A.N. Kolmogorov formulated the $n^2$ conjecture and a number of problems were posed on the estimation of the complexity of solutions to linear systems of equations and other similar calculations. I actively began to reflect on the $n^2$ conjecture and exactly one week later I discovered that the algorithm with which I was hoping to get a lower bound for $M(n)$ gives an estimate of the form...

After a regular session of the seminar I told A.N. Kolmogorov about the new multiplication algorithm and refutation of the $n^2$ conjecture. This greatly agitated A.N. Kolmogorov, since it contradicted his rather plausible conjecture. On the next session of the seminar my method of multiplication was reported by A.N. Kolmogorov, and at this the seminar stopped its work. Later, in 1962, A.N. Kolmogorov wrote (perhaps with the participation of Yu.P. Ofman) a short article and published it in the Reports of the Academy of Sciences of the USSR. The article was titled as follows: A. Karatsuba, Yu. Ofman, Multiplication of Multidigit Numbers by Automata (Dokl. AN SSSR. 1962. V. 145, No. 2. P. 293-294). I learned about this article only when I was given its reprints. The unusual way of publication is also emphasized by the fact that both articles [5] and [8] were presented by A.N. Kolmogorov for publication simultaneously on November 13, 1962." [emphasis added]

[5] is the Karatsuba-Ofman article, and [8] is an article Об алгоритмической сложности дискретных функций (On algorithmic complexity of discrete functions) by Ofman alone. Here is the Russian original:

"Осенью 1960 г. в Московском университете на механико-математическом факультете начал работать семинар по математическим вопросам кибернетики под руководством А.Н. Колмогорова, где А.Н. Колмогоровым была сформулирована гипотеза $n^2$ и поставлен ряд задач об оценке сложности решений линейных систем уравнений и других сходных вычислений. Я активно стал размышлять над гипотезой $n^2$ и ровно через неделю обнаружил, что алгоритм, которым я надеялся получить нижнюю оценку величины $M(n)$, дает оценку вида..."

После очередного заседания семинара я сообщил А.Н. Колмогорову о новом алгоритме умножения и об опровержении гипотезы $n^2$. Это сильно взволновало А.Н. Колмогорова, так как противоречило его довольно правдоподобной гипотезе. На следующем заседании семинара мой метод умножения был рассказан самим А.Н. Колмогоровым, и на этом семинар прекратил свою работу. Позднее, в 1962 г., А.Н. Колмогоров написал (может быть, при участии Ю.П. Офмана) небольшую статью и опубликовал ее в Докладах АН СССР. Статья называлась так: А. Карацуба, Ю. Офман, Умножение многозначных чисел на автоматах (Докл. АН СССР. 1962. Т. 145, № 2. С. 293-294). Об этой статье я узнал только тогда, когда мне были даны ее оттиски. Необычность способа публикации подчеркивается и тем, что обе статьи [5] и [8] представлены А.Н. Колмогоровым к опубликованию одновременно 13.11 1962 г."

  • $\begingroup$ I kind of wonder how the fast fourier method of fast multiplication escaped both Kolmogorov and his students working on his multiplication conjecture. Wasn't the fft known in the soviet mathematical community at that time(as it was discovered by some major mathematicians whose work was possibly accessible to them)? $\endgroup$
    – GEP
    Commented Aug 14, 2020 at 6:59
  • 8
    $\begingroup$ @GEP I don't think Fast Fourier Transforms were well known in 1960. Gauss' first discovery of it was unpublished and in spite of several partial rediscoveries it only became popular after Cooley and Tukey published their own method in 1965 en.wikipedia.org/wiki/Cooley%E2%80%93Tukey_FFT_algorithm $\endgroup$
    – Evargalo
    Commented Aug 14, 2020 at 9:42
  • 13
    $\begingroup$ IMO "сильно взволновало" is closer to "made him very worried". It absolutely does not mean "agitated", and "excited" is also a stretch. $\endgroup$ Commented Aug 14, 2020 at 10:16

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