An earthling with a superpower telescope observed a chalkboard on a distant planet. On it were some mathematical statements. After months of translating, he successfully translated all the words and digits. Unfortunately, due to the complexity of the language, he couldn’t figure out the exact number of ones in each number. All he knows is that they each have at least 2 ones and the first number (but not necessarily the second) has an even number of ones. Other than the ones, the only other digit is a single two. The following is the furthest he could translate it:

1…1 [with an even number of ones] is a prime number
1…12 is a prime number

Assuming both numbers use the same base number, prove that someone or something made a mistake.

It looks to me like the number base must be odd. Also, it looks to me like the binomial theorem will be a useful tool to show that at least one of the two numbers must be even, especially in the case where the base is of the form 4k+3.

Posted by Richard on 2004-01-19 13:42:34