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.

The first of the two numbers, in order to be prime, must actually be written in an even base. In an odd base, the value would be the sum of an even number of odd numbers, resulting in an even number, which can't be prime if it's larger than 2, which this would be. So by the first number, the base must be even. (for example 11 is prime in base 10, 12, 16 or 18).

But if the base is even, then the second number is even, which again, can't be prime, as it is also larger than 2, the only even prime.

Posted by Charlie on 2004-01-19 14:21:12