# Delicate numbers. Mathematicians announced a new class of prime numbers

Scientists have proven that there are special primes that are so sensitive that a change in any of them digits converts such numbers into composite numbers. However, researchers have not yet found specific examples.

Take the numbers 294 001, 505 447 and 584 141. Did you notice anything special about them? You can guess that they are all simple (they can be divided without remainder only by themselves and by one). But the primes mentioned above are even more unusual!

If you select any digit in each of these numbers and change it, the new number will be composite and therefore no longer prime. Let’s change, for example, the number 1 in the number 294 001 by 7, and the resulting number will be divisible by 7; change 1 to 9, and the resulting number is divisible by 3.

Such primes are called digitally delicate (“Sensitive to digit substitution”) is a relatively recent mathematical discovery. In 1978 Murray Klamkin, a mathematician, as well as an active author and editor of complex mathematical problems, was carried away by the question of whether such numbers exist. His question was quickly answered by the “wandering mathematician” Pal Erdos, one of the strongest problem solvers. Erdos proved that “sensitive” primes do exist, and he also established that there are many such numbers. The result obtained by Pal will be true not only for decimal, but also for any other number system.

Since then, mathematicians have gone further, developing Erdёs’ ideas. Among them is the Fields Prize laureate Terence Tao: in article 2011 the scientist proved that the “positive proportion” of prime numbers is digitally delicate, that is, it is sensitive to the replacement of digits (for all number systems). This means that the spacing between consecutive sensitive primes is virtually unchanged – in other words, digit-sensitive primes will not appear less frequently as they increase.

In two recent articles Michael Filaseta from the University of South Carolina took the idea further, proposing an even more sparse class of prime numbers that are sensitive to digit substitution.

“This is a wonderful result”, – noted Paul Pollack from the University of Georgia.

Inspired by the work of Erdёs and Tao, Filaseta wondered what would happen if we added an infinite sequence of zeros as the first part of a prime number. The values ​​of numbers 53 and… 0000000053 are the same. Could replacing any of those infinite zeros added to a prime number automatically make it composite?

Filaceta decided to call such numbers, assuming that they exist, “highly sensitive to digit substitution” (widely digitally delicate), and investigated their properties in articlereleased in November 2020 with former graduate student Jeremiah Southwick.

Unsurprisingly, the new condition makes it difficult to find similar numbers. “294 001 is digitally sensitive, but not very sensitive,” said Pollack, “because if we replace … 000 294 001 with … 010 294 001, we get 10 294 001 – another prime number.”

In fact, Philaseta and Southwick were unable to find a single example of a prime number highly sensitive to digit substitution in the decimal system, despite checking all integers up to 1,000,000,000. But that did not stop them from making some compelling claims about the likes. even hypothetical numbers.

First, they showed that such numbers in the decimal system are indeed possible, and, moreover, there are an infinite number of them. Scientists have gone further and proved that the positive proportion of prime numbers is highly sensitive to digit substitution, just as Tao had previously done with prime numbers sensitive to digit substitution. In his doctoral dissertation, Southwick achieved the same results for the number systems ranging from binary to nine, as well as eleven and thirty-one.

Pollack was impressed with the results. “We can do infinitely many different operations with these numbers, but no matter what we do, we will get a composite number anyway,” he said.

The proof is based on two tools. The first, using the concept of “covering system”, was invented by Erdрдs in 1950 to solve another problem in number theory. “The covering system,” says Southwick, “gives us a large number of segments and also ensures that every positive integer is in at least one of those segments.” If, for example, you divide all positive integers by 2, you get two segments: one containing even numbers where the remainder is 0, and the other containing odd numbers where the remainder is 1. Thus, all positive integers were Are “covered”, and numbers in the same segment are considered “congruent” to each other.

The situation with prime numbers that are highly sensitive to digit substitution is, of course, more confusing. You will need a lot more segments, about 1025,000, and in one of these segments, each prime is guaranteed to become composite if any of its digits, including its leading zeros, are incremented.

A prime that is highly sensitive to digit substitution should also become composite if any of its digits decrease. Here the second tool comes to the rescue – the “sieve”. The sieve theory (by the way, appeared in ancient Greece) offers a way of counting and evaluating for integers satisfying certain properties. Philaseta and Southwick took an approach similar to that used by Tao in 2011 to show that if you take the primes in the aforementioned segment and decrease one of the digits, the positive proportion of those primes becomes composite. In other words, the positive proportion of these primes is highly sensitive to digit substitution.

“The Philaseta-Southwick theorem,” notes Pollack, “is a beautiful and unexpected illustration of the strength of a covering system.”

IN January article Filaceta and his current graduate student Jacob Jouyer made an even more startling claim: there are long, sequential series of primes, each highly sensitive to digit substitution. For example, you can find 10 consecutive primes that are highly sensitive to digit substitution. But to do this, you will have to explore as many primes as there are not even atoms in the universe, says Filaseta. He compared this process to winning the lottery 10 times in a row: the chances of winning are extremely small, but still not zero.

Filaceta and Jouyer proved the theorem in two stages. First, they used covering systems to prove that there is a segment containing infinitely many primes, all of which are highly sensitive to digit substitution.

In the second step, they applied the theorem, proven in 2000 Daniel Shiu to show that somewhere in the list of all primes, there is an arbitrary number of consecutive primes contained in that segment. These consecutive primes, because they are in this segment, are necessarily highly sensitive to digit substitution.

Karl Pomerans from Dartmouth College liked these articles very much, he called Philaset a master in the application of covering systems to intriguing problems in number theory. Math can be more than just an exercise with sophisticated tools, it can also be real fun. “

At the same time, as noted by Pomerans, representing a number in the form of its digits in decimal notation can be convenient, but it has nothing to do with what the number actually is. ” He concludes that there are more fundamental ways of representing numbers, for example, Mersenne primes – primes in the form 2p – 1 for a prime number p.

Filaceta agreed. However, recent articles raise questions that we still have to explore. Filaseta wonders if there are prime numbers in every number system that are highly sensitive to digit substitution. Jouiera, for her part, wonders if there are “infinitely many primes that become composite by inserting a digit between two digits, rather than simply replacing a digit.”

Another nagging question from Pomeranz: Do all primes end up becoming digitally sensitive or highly sensitive as they approach infinity? Is there a limited number of primes that are sensitive to digit substitution (or highly sensitive to digit substitution)? Pomerans believes that the answer to this question, however formulated, should be negative. But both Pomerance and Philaceta see these claims as an intriguing hypothesis that neither of them knows how to prove without relying on another unproven hypothesis.

“The history of mathematical research is such that you cannot know in advance whether you can solve a difficult problem and whether it will lead to something important,” said Pomerans. – You cannot determine in advance: today I am going to do something meaningful. Although, of course, it’s great when everything goes like this. “