Prove that the numbers that begin with 1 and end with 1, with any number of 2s in the middle, and all 1s and 2s separated by 00 are composite.

For example:
1002001
1002002001,
1002002002001,
1002002002002001

(In reply to Simple solution, with no words! by Federico Kereki)

Unfortunately, your simple solution without words does not seem to me to be a solution at all, at least not without some words to explain it. Lets call a number a 1-2 number if it begins with 1, ends with 1, has any number of 2's in the middle, and 'adjacent' 1's and 2's are always separated by two 0's. Let us call a number a 1-1 number if it is the result of replacing every 2 in a 1-2 number with a 1. Then your computation indicates in a schematic way that, given any 1-1 number, if we multiply it by 1001, we get a 1-2 number. What is required by the problem, however, is (something like) a demonstration that, given any 1-2 number, it is the multiple of a 1-1 number and 1001.

To see the difference, compare the idea of proving that the sum of any two primes greater than 2 is an even number greater than 4 (easy), versus the idea of proving that every even number greater than 4 is the sum of two primes greater than 2 (still not proven, by anyone!).

Of course, one can easily adapt your computation to a correct solution, but, unfortunately for your subject heading, such an adaptation requires some words.

Posted by RoyCook on 2003-10-13 13:12:52