I am indebted to Simon Singh's excellent new book The Simpsons and their Mathematical Secrets, which I recently reviewed, for providing me with what must be the longest trivially memorable prime number at a whopping 31 digits.
Prime numbers are much loved by mathematicians. Of itself this is no great achievement - mathematicians are routinely besotted with numbers that only their parents could love - but primes are genuinely interesting. (For a start, the RSA mechanism that keeps your banking details safe when you buy online depends on them.) You will probably remember from school that primes are the whole positive numbers that are only divisible by themselves and 1 - so they begin 2, 3, 5, 7, 11, 13, 17...
You may wonder why 1 is not a prime and you would not be alone. In fact it was until a couple of hundred years ago, when mathematicians decided it was too unique (they probably missed the pun) and excluded it. Mathematicians can do this (unlike physicists), as they make up their own rules.
The number revealed in the book is Belphegor's prime, named after one of the princes of Hell, though I think it would be much better called the devilish prime. One thing that helps make it memorable is that it is a palindrome - it reads the same forwards as backwards - but the main (dare I say, the prime) reason it springs to mind so easily is its devilish construction.
Start with the number of the beast, 666 and stick a horribly unlucky 13 zeroes either side. Finally cap it off with bookends of 1 and you get
... a 31 digit prime number that is entrancingly memorable. Thanks, Simon!