A nice property of primes
A while ago, a friend of mine told me that he once knew a number with the following property: “You can prepend any number before that number and will always obtain a prime number.”
Of course, he was just kidding: Such a number can’t exist, because a number is divisible by 3 if and only if its digit sum is divisible by 3. So, it is no problem to find a number to prepend such the digit sum of the resulting number is divisible by 3. That implies that the resulting number is not a prime.
However, it’s possible to prove a similar theorem…
We prove the following: For any number , there is a number with digits, such there are infinitely many primes ending with this -digit number.
To prove this, let us first consider the case . This means, that we want to show that there is a number such there are infinitely many primes ending in this number .
It’s a well-known fact that there are infinitely many primes. Thus, due to the pigeonhole principle, there must be a number in to occur infinitely many often. Thus, such a number exists.
Now, it’s no big step to generalize this to more digits: Assume we found our number with the desired property. I.e. there are infinitely many primes ending with . Now, we “cross out” all primes ending not in . So, we are now only considering the infinitely many primes ending in
But now, according to the pigeonhole principle, there must be a number such that there are infinitely many primes whose next-to-last digit is . That means, there are infinitely many primes ending in the digits .
Now, we iterate the procedure: We cross out all primes not ending in . This yields – again – infinitely many primes ending in . So we can argument the same way as before: There must be – due to pigeonhole principle – a number $n”$ such that infinitely many primes end in .
We can continue this procedure as long as we want – since there are infinitely many primes.
So, this means, there are arbitrarily large numbers (concatenated digits ) such that there are infinitely many primes ending with exactly the digits .
I like this proof because it is – once again – a proof of existence that does not explicitly construct a solution to the problem, but just shows that there is one.