**Pafnuty Chebyshev** is largely remembered for his investigations in number theory. Chebyshev was also interested in mechanics and is famous for the orthogonal polynomials he invented.

- Pafnuty was born in Okatovo, a small town in western Russia, south-west of Moscow.
- Let us say a little about life in Russia at the time Pafnuty Lvovich was growing up.
- Later in life Pafnuty Lvovich would greatly benefit from his fluency in French, for it would make France a natural place to visit, French a natural language in which to communicate mathematics on an international stage, and provide a link with the leading European mathematicians.
- Chebyshev was, therefore, well prepared for his study of the mathematical sciences when he entered Moscow University in 1837.
- The Russian university system that Chebyshev entered had undergone considerable change.
- At Moscow University the person who was to influence Chebyshev most was Nikolai Dmetrievich Brashman who had been professor of applied mathematics at the university since 1834.
- Chebyshev always acknowledged the great influence Brashman had been on him while studying at university, and credited him as the main influence in directing his research interests, referring to their "precious personal talks".
- The department of physics and mathematics in which Chebyshev studied announced a prize competition for the year 1840-41.
- Chebyshev graduated with his first degree in 1841 and continued to study for his Master's degree under Brashman's supervision.
- Once, much later in his career, Chebyshev objected to being described as a "splendid Russian mathematician" and said that surely he was a "world-wide mathematician" rather than a Russian mathematician.
- It is very clear that right from the time he began his studies for his Master's degree that Chebyshev aimed at international recognition.
- There is no conclusive evidence, but it must be highly likely that if Chebyshev did not personally visit Paris in 1842 then he sent his paper to Liouville via Chikhachev.
- Chebyshev continued to aim at international recognition with his second paper, written again in French, appearing in 1844 published by Crelle in his journal.
- In the summer of 1846 Chebyshev was examined on his Master's thesis and in the same year published a paper based on that thesis, again in Crelle's journal.
- During 1843 Chebyshev produced a first draft of a thesis which he intended to submit to obtain his right to lecture once he found a suitable position.
- Times were hard and Moscow had no suitable positions available for Chebyshev but, in 1847, he was appointed to the University of St Petersburg submitting his thesis On integration by means of logarithms.
- Although Chebyshev's thesis was not published until after his death, he published a paper containing some of its results in 1853.
- Between arriving in St Petersburg and this 1853 publication Chebyshev published some of his most famous results on number theory.
- Chebyshev's work on prime numbers included the determination of the number of primes not exceeding a given number, published in 1848, and a proof of Bertrand's conjecture.
- Chebyshev proved Bertrand's conjecture in 1850.
- The proof of this result was only completed two years after Chebyshev's death by Hadamard and (independently) de la Vallée Poussin.
- Chebyshev was promoted to extraordinary professor at St Petersburg in 1850.
- Chebyshev's interest both in the theory of mechanisms and in the theory of approximation stem from his 1852 trip.
- set the foundations of the Russian school of approximation theory: we show the relation of Chebyshev's ideas in approximation theory to applied problems (theory of mechanisms and computational mathematics).
- It was in this work that his famous Chebyshev polynomials appeared for the first time but he later went on to develop a general theory of orthogonal polynomials.
- It was Chebyshev who saw the possibility of a general theory and its applications.
- Geronimus has pointed out that in his first paper on orthogonal polynomials, Chebyshev already had the Christoffel-Darboux formula.
- The trip Chebyshev undertook in 1852 was one of many.
- Almost every summer Chebyshev travelled in Western Europe, but when he did not, he spent the summer in Catherinenthal near Reval (now known as Tallinn in Estonia).
- We have mentioned some contributions that Chebyshev made to the theory of probability.
- As a result of his work on this topic the inequality today is often known as the Bienaymé-Chebyshev inequality.
- Twenty years later Chebyshev published On two theorems concerning probability which gives the basis for applying the theory of probability to statistical data, generalising the central limit theorem of de Moivre and Laplace.
- Further, Chebyshev was the first to estimate clearly and make use of such notions as "random quantity" and its "expectation (mean) value".
- Let us mention a few further aspects of Chebyshev's work.
- The Chebyshev parallel motion is three linked bars approximating rectilinear motion.
- A number of famous mathematicians were taught by Chebyshev and gave a descriptions of him as a lecturer.
- The first quote we give is by Lyapunov who attended lectures by Chebyshev in the 1870s.
- Let us quote from a lecture given by Chebyshev in 1856 where he explained how he saw the interaction of the pure and applied sides of mathematics.
- Chebyshev retired from his professorship at St Petersburg University in 1882; he had been appointed to this particular post 22 years earlier.

Born 16 May 1821, Okatovo, Kaluga Region, Russia. Died 8 December 1894, St Petersburg, Russia.

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Origin Russia, Number Theory, Special Numbers And Numerals

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive