## On a Special Kind of Prime Numbers

March 31, 2021

### Prime Numbers

Today I would like to write a brief blog post about the “atoms” of the natural numbers: prime numbers. Prime numbers are those numbers which are evenly divisible by exactly two numbers, namely by $1$ and by the number itself. For example, $17$ is a prime number because it is only divisible by $1$ and $17$, whereas $42$ is not a prime because it is, for instance, also divisible by $2$, $3$ and $7$. The first few prime numbers are given by $$2,3,5,7,11,13,17,19,23,29,\ldots$$ As already mentioned, prime numbers are sometimes called the “atoms” of the natural numbers, because every natural number can be written as the product of primes: $42 = 2 \cdot 3 \cdot 7$. So far, so easy.

In the course of time, mathematicians have asked many questions concerning different properties of prime numbers. Some of these questions have beautiful answers (e.g., the infinitude of primes or the prime number theorem on the frequency of primes). Over the past few decades, prime numbers have also become increasingly important in our everyday life. In particular, they play a substantial role in the encryption and secure transmission of messages and data of different types. Although the concept of prime numbers seems to be quite simple and elementary, some of hardest and most famous maths problems concern them (e.g., the famous twin prime conjecture or the Riemann hypothesis, which is also connected to primes). So primes are still a bit of a mystery for mathematicians. The famous mathematician Paul Erdős once appropriately said that “God may not play dice with the universe, but something strange is going on with the prime numbers”, with reference to the well-known quote of Albert Einstein.

### Digitally Delicate Primes

However, I actually wanted to write about some recent progress on a particular type of prime numbers: so-called digitally delicate primes. These numbers are prime numbers that cannot have any digit changed and remain prime. The number $294001$ is the smallest example of such a number: It is prime itself, but if we change precisely one of its digits, say we change the $9$ to a $7$, then it is not prime anymore. (On the contrary, the prime number $191$ is not a digitally delicate prime because if we change the digit $9$ to $8$, then we obtain the number $181$, which is again a prime number.) Even though the smallest of these numbers (i.e., $294001$) is quite large, Paul Erdős showed that there exist infinitely many of them. The first few digitally delicate primes are listed as A050249 in the Online Encyclopedia of Integer Sequences.

The mathematicians Michael Filaseta and Jeremiah Southwick have recently proven the infinitude of an even more rarefied class of digitally delicate primes [1]. They call a number widely digitally delicate if after adding infinitely many leading zeros to the number, you can still change any of its (now infinitely many) digits without obtaining a prime number again. Imagine that: There are infinitely many possible digits you’re allowed to change, and no matter which one you change, the obtained number is always guaranteed to be a non-prime number. And Filaseta and Southwick showed that there are infinitely many of such numbers—even though nobody has ever found such a number!

Impressed? More details and examples can be found in a very exciting article of the Quanta Magazine!

### References

1. [1] M. Filaseta, J. Southwick, Primes that become composite after changing an arbitrary digit, Math. Comp. 90 (2021) 979–993. https://doi.org/10.1090/mcom/3593.

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