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 (1111...11) is in base P, it is (1+)(1+P^2+P^4...); if it has more than two digits, it will be a composite number. Therefore, the first number is just 11, and P must be one less than a prime. Also, P cannot be 2, since that digit is used in the second number, so P is even.
The second number is (P+P^2+P^3+...) + 2, obviously even, so it cannot be a prime number.
Posted by e.g.
on 2004-01-19 14:08:53