How Many Of This Weird Prime Are There?

A friend made me aware of a neat little unsolved problem in number theory. I know it seems like number theory is nothing but unsolved problems, but this is an unfair reputation. There are as many as four solved problems in number theory. It’s a tough field.

The question started with the observation that 11 is a prime number. And so is 101. But 1,001 is not; nor is 10,001. How many prime numbers are there that have the form 10^n + 1 , for whole-number values of n? Are there infinitely many? Finitely many? If there’s finitely many, how many are there?

It turns out this is an open question. We know of three prime numbers that you can write as 10^n + 1 . I’ll leave the third for you to find.

One neat bit is that if there are more 10^n + 1 prime numbers, they have to be ones where n is itself a whole power of 2. That is, where the number is 10^{2^k} + 1 for some whole number k. They’ve been tested up to 10^{16,777,216} + 1 at least, so this subset of the Generalized Fermat Numbers seems to be rare. But wouldn’t it be just our luck if from 10^{16,777,217} + 1 onward they were nothing but primes?

Author: Joseph Nebus

I was born 198 years to the day after Johnny Appleseed. The differences between us do not end there. He/him.

3 thoughts on “How Many Of This Weird Prime Are There?”

  1. A similar-looking problem which is much easier (kind of fun, actually) is to determine how many primes occur in the sequence 101, 10101, 1010101, 101010101, …


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