Consider six consecutive positive integers. Show that there is a prime number that divides exactly one of them.

Steve conjectured that the puzzle-statement holds not only for six numbers, but for every sequence of n numbers where n>2. The way to solve the original puzzle, was to observe that of six consecutive numbers, there is always one that is not divisible by 2, 3 or 5. This process may or may not work in the general case.

This motivates the following problems that I challenge you to prove or disprove for general (or some) numbers n:

1. Given a sequence of n consecutive numbers where n>1. Let p_1=2, p_2=3,...,p_k be the prime numbers smaller or equal to n. Then one of the consecutive numbers cannot be divided by one of the primes p_1,..,p_k.

2. Given a sequence of n consecutive numbers. Show that there is a prime number that divides exactly one of them.

Obviously 2. would follow from 1. But maybe there are counter examples. I don't know. Anyone cares to tackle this?

Posted by JLo on 2006-09-01 14:10:47