Joseph Louis François Bertrand (1822-1900), for many years a member and secretary of the prestigious Paris Academy of Sciences, is not one of those mathematicians who is immortalized in large format to adorn the walls of faculties and departments and it is unusual that his colleagues current mathematicians let us know snippets of his Biography. However, his name is linked to some statements well known by professionals, one of them even by a wide audience. Is he Bertrand’s postulate which states that between a number and its double there is always at least one prime. For example, between 2023 and 4046 there is 2027 and many others.
Bertrand stated his conjecture in 1845 in a paper on symmetries of functions, within what we would call group theory today. Pafnuty Chebyshev He proved it seven years later and, furthermore, he proved that the number of primes between an arbitrarily large number and its double increases without limit. For example, between 100 and 200 there are only 21 primes and between 1000 and 2000 there are already 135. Chebyshev’s proof, a bit complicated, was masterfully simplified in 1932 by the mathematical singular Paul Erdos. A fundamental point in his proof is the fact that the product of the first n numbers always divides the next n. For example, 1·2·3·4=24 divides 5·6·7·8=1680. Therefore, the primes between n and 2n will always appear as factors in the quotient. Also Shrinivasa Ramanujan gave a proof of Bertrand’s postulate, although it is less elementary and elegant than Erdős’s.
Our current understanding of the distribution of prime numbers allows us to go much further. If we consider a real number α greater than 1 and less than two, from a certain number n we know that between n and αn there will always be a prime. For example, for α = 1.00025 there are no primes between 80000 and 80020 = 80000α, but one can ensure that there will be between n and αn for n greater than 400000. On the other hand, the recent advances due to James Maynard, which have earned him the Fields medal of 2022, imply that it is possible to sporadically find many cumulative primes in really small intervals.
Another curious contribution by Bertrand is known as Bertrand’s paradox, a simple and beautiful probabilistic paradox that seems to challenge the power of mathematics to provide unambiguous solutions. We consider an equilateral triangle, pointing upwards, and its circumscribed circle (the one that passes through its three vertices). The problem is to find the probability that a chord (a segment that joins two points on the circle) chosen at random is longer than the side of the triangle.
One solution is to reason by saying that the rope could always be rotated so that it started from the upper vertex. The triangle divides the circumference into three similar arcs of 120° and that the chord is greater than the side is equivalent to ending in the arc below the base, therefore, the probability sought is ⅓. Another solution is obtained if we rotate the string until it is below the center and parallel to the base. It will be longer than the base, just as you cut the top half of the radius perpendicular to it, leading to the ½ result.
Bertrand obtained a third solution in his work, and there are others. Escaping from this paradox involves understanding that it is necessary to specify mathematically what “random” means when formulating the problem, because there are an infinite number of ways to do it. In 1933, Andrei Kolmogorov laid the foundations of modern probability theory by establishing some axioms, minimum requirements, which must be satisfied by all probability models regardless of the situation in which they are applied. The different solutions in the paradox correspond to each model.
Finally, Bertrand also has a theorem with his name. We all know that the force of gravity is attractive and inversely proportional to the square of the distance. Through complex mathematical magic, this implies that a planet always revolves around a massive sun in an elliptical orbit (it’s Kepler’s first law). The ellipse is traversed in a non-uniform way, the planet advances faster the closer it is to its sun located in one of the focuses; however, the movement is periodic: in a constant time, one year in the case of the Earth, it returns to the starting point and repeats the movement, which favors a watchmaker’s precision in the study of the Solar System, especially by incorporating corrections due to mutual disturbances.
Suppose that in a hypothetical universe the suns attract the planets with a different force of gravity than usual, is it still possible that all planetary orbits are periodic? Bertrand’s theorem gives a complete answer: they are ellipses when they are ellipses and this occurs only when the gravitational force is normal or proportional to distance, the type of force that appears when stretching a spring (Hooke’s law). Curiously, Isaac Newton already considered in his principle this second possibility. He showed which corresponds to the sun being at the center of the ellipse, rather than at a focus.
If we dream of another type of attractive gravitational force that only depends on the distance, for example, inversely proportional to it, Bertrand’s theorem ensures that the orbits are not generally going to be periodic. In reality, general relativity provides an effective modification of the original gravitation that contributes to the precession of Mercury’s perihelion.
Serve these three beautiful statements associated with Bertrand to remember, a year late, the bicentenary of the birth, on March 11, of a not-so-known mathematician, but who has the singular honor of linking his name to a postulate, a paradox and a theorem.
Fernando Chamizo He is a professor at the Autonomous University of Madrid and a member of the Institute of Mathematical Sciences
Coffee and Theorems is a section dedicated to mathematics and the environment in which they are created, coordinated by the Institute of Mathematical Sciences (ICMAT), in which researchers and members of the center describe the latest advances in this discipline, share meeting points between the mathematics and other social and cultural expressions and remember those who marked their development and knew how to transform coffee into theorems. The name evokes the definition of the Hungarian mathematician Alfred Rényi: “A mathematician is a machine that transforms coffee into theorems.”
You can follow SUBJECT in Facebook, Twitter and instagramor sign up here to receive our weekly newsletter.
#mathematical #gems #Joseph #Bertrand