## A Puzzle

What relationship do 5 and 6 have that make them unlike any other pair of distinct positive integers under 1000? (1000 was as far as I tested.)

WARNING: Don't read the rest of this thread if you want to solve the puzzle for yourself. It's not that hard--my dad got something rather close to the answer pretty quickly, and he's a computer programmer, not a mathematician. You can PM me if you want a hint.

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 is the only semi-prime number between twin prime numbers under 1000?
 Please don't put any more answers in this thread. Since you answer was wrong, I may as well respond to it. Both 4 and 6 are semiprime numbers that have prime numbers on both sides. Besides, you were supposed to find a relationship between the two numbers.

## A Puzzle

you're right, there are one more case... but seems to be the only 2 cases... (edited because I put some wrong statements)

if you show that thoses cases are not the only cases, then you'll be famous, because this could be a proof of the twin primes conjecture

ps: why we cannot post here???

 I don't want anyone to post the answer here so that people can work out the answer for themselves. But on second thought, maybe I will just insert a disclaimer in the first post.
 well... ok... I think your puzzle is a little bit "generic", I mean, at least for me, as a first look, the puzzle can have multiple answers... is some information missing?
 No. I'm sorry it looks generic. Why don't you tell me all the answers you can think of, and I'll tell you whether you got the one I thought of. Also, keep in mind that your answer has to express a unique relationship *between* the numbers. So just saying "6 is unique because x, and 5 is unique because y" is not going to cut it. The relationship I am talking about does not exist between any other pair of distinct numbers under 1000.
 What about this: The two positive integers x, y we are looking for satisfy x^2-y^2 = (x-y)(x+y) = 11. x=6, y=5 is clearly the unique solution. There is no need for the restriction that the integers be less than 1000.

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 Quote by Surreal Ike What relationship do 5 and 6 have that make them unlike any other pair of distinct positive integers under 1000? (1000 was as far as I tested.)
That is the only (unordered) pair of distinct integers lying between 4 and 7.

 Umm ... they're the only two sequential integers with the ratio 5/6?
 The only pair of integers that equals (5,6) ....
 Recognitions: Homework Help Science Advisor Let W be the set of Wilson primes, and let $S=W\cup\{n:2n+1\in W\}$. Then (5, 6) is the only pair of consecutive elements in S up to 500 million.

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