# Prove $18080108080 \sum_{k=0}^{1560-1} 10^{10k}+1$ is prime

I saw this fact on twitter:

I would like to know how one would show this number is prime. Is there an elementary way to show that this number is prime? Is there a simplified primality testing algorithm in this case that possibly I could code and run on my computer? Is there a reference to a paper that can be given?

• It could have passed a probable prime test. Commented Jul 18, 2014 at 1:12
• author en.wikipedia.org/wiki/Clifford_A._Pickover probably originally from one of his books. The pattern is sufficiently special that a special-purpose test may prove it prime. I'm guessing you can ask questions on Twitter, ask the original source. Commented Jul 18, 2014 at 1:39
• It's not clear to me that it's a "special-purpose" prime at all. If we choose $a,b,c,d,e\in\{0,1,8\}$ and consider the numbers written $(1abcdedcba)^n1$, then from the prime number theorem we should expect about a 3% chance that one of them with $1000<n<2000$ is prime. It doesn't seem to be remarkable that there is some of the $3^5=243$ combinations of $a,b,c,d,e$ where one of them is prime, for no particular reason (and it would probably be easy to find another one by a brute-force search on computer). Commented Jul 18, 2014 at 2:07
• Consider the fact that something like this could be posted on twitter without anyone actually verifying whether or not it is true. This is the code I ran, feel free to see if there are any mistakes.$\quad$ s = ""; PrimeQ[ 10*ToExpression[Do[s = StringJoin["1808010808", s], {1560}]] + 1]