Original Author:Anna Kramer is a contributing writer for Quantum magazine.
Translated by:math001, a group member of the Doda Math Network.
Follow Dida Math Network to get more math fun articles every day
Many people complete their first math proof while in secondary school. This is the proposition proved by the ancient Greek mathematician Euclid: there are infinitely many prime numbers. Just a few lines of text, using only the simple concepts of integers and multiplication.
The proof is so. Assuming that there are finite numbers of primes, then multiplying them all by adding 1 will give you a new integer. This number will cause a contradiction. This contradiction illustrates that prime numbers can only be infinitely many.
Later mathematicians had a fascination with giving different proofs of the proposition in different ways.
Why?One of the reasons for this is fun. And for a more important reason, "I think the line between recreational math and serious math is very small," said William Gasarch, a professor of computer science at the University of Maryland. " Earlier this year, he posted a new testimonial online.
Gasaqi's proof is just the latest in a series of new proofs. In 2018, Romeo Mestrovic of the University of Montenegro compiled a review of the history of mathematics with nearly 200 "infinitely many prime numbers." In fact, in the field of analytic number theory, mathematicians study integers with continuous variables. Such a history began around 1737, when the mathematician Leonhard Euler once again proved the existence of an infinite number of prime numbers by using the infinite series 1 + 1 2 + 1 3 + 1 4 + 1 5 + divergence.
Christian Elsholtz, a mathematician at the Graz University of Technology in Austria, also recently published a new proof. He said that this is the opposite of the process by which mathematicians combine simple lemmas into complex theorems and finally arrive at a conclusion. He reversed the process. "I use Fermat's theorem, which is actually a very difficult theorem to prove. Then I use it to draw a very simple inference. Such reverse work may reveal hidden connections between different areas of mathematics, he said.
People seem to be on each other's feet, and they all want to come up with some "absurd high-level" proof. Andrew Granville, a mathematician at the University of Montreal, said he also provided two new proofs. "It has to be fun. Doing something technically showy isn't the point. You can only want to "show off" because it's interesting. ”
Granville said the friendly contest actually had a serious purpose. The job of a mathematician is not just to solve a specific mathematical problem thrown at him. "The creative process of mathematics is not just a mechanical process of taking a problem and then solving it mechanically. Mathematics is the way in which people create techniques and develop ideas based on known outcomes. ”
As Gasage puts it, "All of them start with the idea of proving that prime numbers are infinite, and then transition to serious mathematics." You're only looking at prime numbers today, and you're studying square density tomorrow. ”
Gasatchi's proof is based on the fact that if you dye a natural number with a finite variety of colors, there will always be a pair of numbers of the same color, and the sum of them will also be dyed the same color. This is a theorem proved by Issai Schur in 1916. Using Schul's theorem, Gasage proves that if prime numbers are finite, then there will be a perfect cubic number (i.e., a natural number shaped like the cubic of an integer) equal to the sum of two other perfect cubic numbers. However, as early as 1770, Euler had already demonstrated that there was no such thing as a three-cube number. This is the case of n = 3 of Fermat's theorem, which assumes that there is no positive integer solution for the equation x n + y n = z n when n>2. Based on this contradiction, Gasaki deduced that prime numbers must be infinitely many.
In a 2017 proof, Granville used another of Fermat's theorems. Granville first cites a theorem proposed by Bartel Leendert van der Waerden in 1927 that states that if you dye integers with a finite number of colors, then there is a sequence of homochromatic difference of any length. Similar to Gasage, Granville starts with the assumption that prime numbers are finite. He then used van der Walden's theorem to find a column of perfectly squared numbers with a tolerance of 4 and the same color. But Fermat has shown that such a sequence does not exist. Contradiction!Since such a sequence exists if the prime number is finite, but it cannot exist, there must be an infinite number of prime numbers. Granville's proof is the second in recent times to make use of Van der Walden's theorem, which was also used by Levent Alpge in a 2015 article that he published as an undergraduate and now a postdoc at Harvard.
Granville was particularly fond of Elscholz's article, which also used Fermat's theorem and the counterproof, which assumes that there are only a finite number of prime numbers. Like Gassage, El Scholz incorporates Shure's theorem, albeit in a slightly different way. Erscholz also provided an additional proof, using a theorem proposed by Klaus Roth in 1953, which stated that a subset of integers with a sufficiently large density must contain a set of equal difference sequences of length 3.
Through these aspects of work, some of the deeper and even practical mathematical problems are expected to be answered. For example, in an environment with only a finite number of prime numbers, would a public-key cryptography system based on the difficulty of factorization of large numbers become very easy to crack? Elscholz wondered if there was some connection between proving that prime numbers are infinity and proving how difficult it would be to crack such a cryptographic system. "It seems like there's a little bit of a weak correlation right now, but it's interesting to see a deeper connection and win the award," El Scholz said. ”
Granville says that the most brilliant mathematics can often be developed from a singular combination of different fields and disciplines, and is often born after mathematicians have spent years thinking about lower-level but interesting problems. He was obsessed with applying seemingly distant disciplines to the discipline of number theory. In a recent one, Granville praised Hillel Furstenberg's 1955 proof method that uses point-set topology. Like Albog, Furstenberg was still an undergraduate at the time of the publication of his proof. He has since achieved remarkable results, achieving excellent results in all branches of mathematics.
Granville asked himself if his obsession with new proofs of an infinite number of prime numbers was "just out of curiosity, or was he playing a big game of long-term chess?"He asked himself, "I can't tell you!"”
Follow Dida Math Network to get more math fun articles every day