This question appeared in a national mathematics olympiad a few years ago.
It was the easiest question on the test.
I remember that when I participated, I couldn't solve this one. But now I was looking through old papers, I found this question again, thought about it for 3 minutes - and solved it. :)
Here is the riddle in its original phrasing (translated to English).
"For which natural numbers n will the numbers n+1, n+11, n+111 all be prime?"
(Hint: The solution is quick and elegant.)
For anyone who's interested but can't solve it right away, please go ahead and post your comments/thoughts anyway. :)
n=2
2+1=3, Prime.
2+11=13, Prime.
2+111=113, Prime.
That's the only one that I could find, perhaps I am missing a few.
could N= any even number?
n=0
Quote from: Meh on October 19, 2004, 01:14 PM
could N= any even number?
No, take 6 for an example. 6+1 is 7, which is a prime number. 11+6 is 17, which is also a prime number, but 111+6=117 which is not a prime number.
Quick answer, check the first line for big clue...
Quote
n + 1, n + 2 + 3*3, n + 3*37
One of those will be divisible by three, so the only possibility is when that one is equal to three (since three itself is prime).
Incidentally, Natural numbers are 1+, Whole numbers are 0+, and Integers are positive or negative with no decimal (I forget how it's defined).
n=infinity
Quote from: TheMinistered on October 25, 2004, 09:42 PM
n=infinity
Hmmmmmmmmmmmmm, no.
(Adron's post is the answer, read it for a spoiler)
Yoni, I recently returned to the forums, sorry about this belated post - I understand the answer, but not how you came to the conclusion.