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.
If the first number is more than two digits long, independently of the number base, you could write
111111....11 = 11 x 10101...1
so the number would be composite.
If the first number is two digits long, the number base could be either 1 (not possible, because then a "2" wouldn't be used in the second number) or a prime, greater than 2, minus 1, which works out to an even number.
In the latter case, call the base B; the second number would equal a multiple of B, + 2, which would be even, so it couldn't be a prime.
Solution
