The late Professor Paul Erdös, about whom the book The Man Who Loved Only Numbers by Paul Hoffman was written, penned the following closed couplet:
Chebyshev said it, and I say it again,
There is always a prime between n and 2n.The point of this little bit of poesy was that Erdös had just devised a new proof of what is called Bertrand's Postulate, which is simply the second line of the couplet. It was a conjecture of Bertrand that had first been proved by the Russian mathematician Pafnuty Chebyshev (1821-1894). The conjecture (now a theorem) says, for example, that there is a prime between your age and double your age.
The story in all this is that in 1845, the mathematician Joseph Bertrand had conjectured that if n > 3, then there is at least one prime number between n and 2n - 2, and the Russian mathematician Pafnuty Chebyshev (1821-1894) had succeeded in proving the conjecture in 1850 as usually stated in the equivalent form
If n > 1, then there is at least one prime number p for which n < p < 2n
or as in the more folksy second line of Erdös's couplet.
Lest the above story give the impression of trivializing Bertrand, be assured that he was a formidable mathematician. And though he didn't prove his conjecture, it is a very good one, for many people have tried to improve on it, with some little, but complicated, success. Nothing as easy as, say, that there is always a prime between n and 1.9n, for example. It needs to be pointed out that Bertrand did not make his conjecture without good reason, and his reason was that he had verified it for all numbers n from 4 to three million!
Joseph Bertrand was born in 1822 in Paris to well-educated parents. His father, a science writer, died when Joseph was but nine, and he was reared by a sister of his father and her husband, who happened to be the mathematician Jean-Marie Duhamel. Young Joseph was a prodigy who at the time of his father's death had a grasp of algebra and geometry. At age 11, he unofficially attended lectures at the illustrious École Polytechnique, obtaining a degree when he was 16, and he received his doctorate in mathematics at age 17 (his thesis was on thermodynamics). He was then able to attend classes officially! He spent the rest of his life in various academic posts (including 50 years at the École Polytechnique) except for a period when he was in the National Guard during the French Revolution. He suffered many hardships in those times.
One of the many books that Bertrand wrote was a two-volume work entitled (in English), Treatise on Differential and Integral Calculus, for which he is famous (work on a third volume was lost in a wartime fire that burned down his house.) He wrote other books and research papers on topics such as thermodynamics, probability theory, and mechanics. He was reportedly a beloved teacher.
Bertrand was a member of the Paris Academy of Sciences, serving as one of its officers from 1874 to the time of his death in 1900 in Paris.