# What exclusive club is this?

You stumbled across a letter accidentally sent to you. It bears an official looking (though archaic) seal and the contents are as follows:

TO whom it may concern:

Thank you for your interest in joining our club. Our roots date back to the 17th century, though not many people know about us. Fewer still know who our members are. In fact, we have only 47 members to date, although we have 4 more applicants who wish to be verified. If you think you are worthy of joining our club, you have to wait in queue! However, do stay in touch as we may need to verify your credentials soon.

Regards, N

Below the fanciful signature was the following postscript:

A few fun facts about our club:

• We were founded by a friar
• We call those who hunt us by a nickname, "those who are feeble and contemptible"
• The rest of the club calls me "a lesser one" -- but I don't think so do you?
• The founding members are all very close to each other (though if I may say so the newer members aren't as close
• Our club has no member number limits for now (but these might be decided on later)

Well, for a secretive and exclusive club, they don't seem to guard or encrypt their facts very well do they? (Maybe their club address directory needs an update). With this information, even I could figure out what club this is!

Can you figure out what club N is from, and, as a bonus, find the clues in the above text?

• @JMP Although the riddle tag is often wrongly used, this one actually is a riddle: it's describing an unknown thing using cryptic clues and we should identify the solution. – Rand al'Thor Apr 26 at 12:55

You are

perfect numbers, or equivalently Mersenne primes.

We were founded by a friar

We call those who hunt us by a nickname, "those who are feeble and contemptible"

The rest of the club calls me "a lesser one" -- but I don't think so do you?

Maybe the unknown signatory $$N$$ is a potential odd perfect number (of which none are yet known to exist)?

The founding members are all very close to each other (though if I may say so the newer members aren't as close

See the list of perfect numbers: 6, 28, 496, 8128 are all relatively close compared with the others.

Our club has no member number limits for now (but these might be decided on later)

It is not known if there are infinitely many perfect numbers or not.

Thank you for your interest in joining our club. Our roots date back to the 17th century, though not many people know about us. Fewer still know who our members are. In fact, we have only 47 members to date, although we have 4 more applicants who wish to be verified.

Mersenne lived in the 17th century. The list of perfect numbers includes 51, but apparently the last 4 are not fully verified yet, leaving 47.

• About the "lesser one" line: is it because rot13(Zrefraar cevzrf ner whfg bar yrff guna cbjref bs gjb)? – oAlt Apr 26 at 11:23
• @oAlt Nah, I don't think so. It says "the rest of the club calls me", so I think the signatory $N$ is somehow a "lesser one" than other members. – Rand al'Thor Apr 26 at 11:24
• Perhaps ‘lesser one’ literally just means the smallest one, so N is the smallest one in this set? – Beastly Gerbil Apr 26 at 11:44
• Nice one on GIMPS, I didn't think that would be that obvious. About "lesser one" -- rot13(unir lbh pbafvqrerq n Zrefraar cevzr juvpu vf... na "bqq bar bhg"?) Also: one last clue in the text pointing to the answer you haven't caught. Good luck! – titaniummorro Apr 26 at 11:52
• "they don't seem to guard or encrypt their facts very well do they?" seems to be a clue related to this – scubbo Apr 26 at 22:50