From a posting to the Math Fun mailing list by R. W. Gosper, Dec 03 2009:

(Begin)
Gosper: I just reordered checks.
Bank Lady: Where would you like the numbering to start?
Gosper: What was my last one?
Bank Lady: 1093
Gosper: Well then obviously 3511 .
Bank Lady: [Opens mouth. Decides not to ask. Resumes typing.]
(End)


My question : How many checks can you order for Mr. Gosper's checkbook, using his numbering system?

The numbers 1093 and 3511 are known as Wieferich Primes, such that p^2 divides 2^(p-1)-1 and p is a prime.  They are indeed the only two such numbers that are known.  Therefore these are the only 2 checks that can be ordered using this numbering system.

Posted by Chris, PhD on 2012-09-20 11:56:14