Why do mathematicians like to prove the same result in different ways?
The concentration of primes, indicated by yellow dots on this hexagonal spiral of positive integers, decreases with distance from the beginning of the number line. This many times proven regularity is described by the theorem on the distribution of prime numbers.
"You can not believe in God, but you need to believe in the Book," - once said the Hungarian mathematician Pal Erdos... The Book, which exists only in theory, contains the most elegant proofs of the most important theorems. Erds' assertion hints at the motivation of mathematicians to continue looking for new proofs of already proven theorems. One of their favorites is the theorem on the distribution of primes, such that they are divisible only by themselves and by 1. And although mathematicians do not know whether the proof will get into the Book, two rivals compete for the first place, proofs that simultaneously and independently found in 1896 by Jacques Hadamard and Charles Jean de La Vallée-Poussin .
So what exactly does this theorem state?
The prime number theorem makes it possible to approximate the number of prime numbers not exceeding a given number n. This value is called π (n), where π is the distribution function of prime numbers [ not related to the number π / approx. transl.]. For example, π (10) = 4 because there are 4 primes up to 10 (2, 3, 5, and 7). Similarly, π (100) = 25, since there are 25 primes among the first 100 numbers. Among the first 1000 numbers, there are 168 primes, so π (1000) = 168, and so on. Note that when looking at the first 10, 100, and 1000 integers, the percentage of prime numbers in them dropped from 40% to 25% and 16.8%, respectively. These examples hint, and the prime number theorem confirms that the density of primes not exceeding a given number decreases as that number increases.
But even if you had an ordered list of integers up to, say, a trillion, who would want to manually calculate π (1,000,000,000,000)? The prime number theorem is a way to save energy.
It says that π (n) is "asymptotically equal" to n / ln (n), where ln is the natural logarithm. Asymptotic equality can be thought of as rough equality, although this is not entirely true. For example, let's estimate the number of primes not exceeding a trillion. Instead of counting individual primes to calculate π (1,000,000,000,000), you can use this theorem and find out that there are approximately 1,000,000,000,000 / ln (1,000,000,000,000), which equals 36,191,206 825 when rounded to the nearest integer. And this estimate differs from their real number, 37 607 912 018, by only 4%.
With asymptotic equality, the accuracy improves with increasing numbers substituted in the formula. Basically, the more we move closer to infinity - which is not a number in itself, but simply something more than any number - asymptotic equality approaches real equality. And although the real number of prime numbers will always be expressed as an integer, the value on the other side of asymptotic equality, that is, the fraction in which the natural logarithm appears, can take any value on the real line. This connection between real and integer numbers is counterintuitive to say the least.
All this blows the mind a little, even for mathematicians. And what is most unpleasant, the statement of the theorem on the distribution of prime numbers says nothing about why such a relation holds.
“The theorem has never been valuable on its own. It's all about proof, ”said Michael Bode , professor of mathematics at Queensland University of Technology in Australia.
While the original proofs of Hadamard and La Vallée-Poussin were elegant, they were based on complex analysis - the study of functions of complex numbers - which some people dislike, since the assertion of the theorem itself has nothing to do with complex numbers. However, Godfrey Harold Hardy in 1921 heralded the emergence of non-analytical evidence - the so-called. elementary proof - the theorem on the distribution of prime numbers " extremely unlikely ", and stated that if someone finds it, "will have to rewrite the theory."
Atle Selbergand Erdös himself took up the challenge, and in 1948 each published a new, independent, elementary proof of the prime number theorem using the properties of logarithms. This evidence prompted other mathematicians to consider similar approaches to number theory hypotheses that were previously considered too simple for such complex statements. As a result, many interesting results were obtained, including the elementary proof of Helmut Meier in 1985 about unexpected inhomogeneities in the distribution of primes.
"The prime number theorem has a lot of unsolved questions," said Florian Richter , a mathematician at Northwestern University who recently published a new rudimentary proofthis famous statement. Richter found it while trying to prove far-reaching consequences of the prime number theorem.
Over time, number theorists have helped establish a culture in which mathematicians prove and re-prove theorems not only to test claims, but also to improve their theorem-proving skills and understanding of the mathematics used.
This is outside the scope of the prime number theorem. Paulo Ribenboim collected at least 7 proofs of the infinity of prime numbers. Stephen Kifovit and Terra Stamps identified 20 pieces of evidence showing that the harmonic series 1+ 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + ... does not converge to a finite number, while Kifovit added 28 more to them... Bruce Ratner lists over 371 proofs of the Pythagorean Theorem , including great examples by Euclid, Leonardo da Vinci, and 20th US President James Abram Garfield, who was then an Ohio Congressman.
The habit of seeking duplicate evidence is so ingrained in the community that mathematicians can practically count on it. Tom Edgar and Yajun Anh noted that the quadratic law of reciprocity , in addition to the original proof of Gauss from 1796, has 246 more proofs . They plotted the amount of evidence versus time, and extrapolated that by 2050, the 300th proof of this law could be expected.
“I like new proofs of old theorems for the same reason I like new roads and detours that lead to places I know,” said Sofia Restad , a graduate student at the University of Kansas. These new roads give mathematicians a spatial sense of the place in which their intellectual pursuits take place.
Mathematicians may never stop looking for new, clearer ways to prove both the prime number theorem and their other favorite theorems. If you are lucky, some of them will even be honored to be included in the "Book".