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 answer: THERE IS A MISTAKE SOMEWHERE...
The second number may be written as 11111...2=
q*(1111...0)+2= which implies:
q (the base of the counting system)>2
q is odd - OTHERWISE the second number is even THEREFORE NOT A PRIME
the first number 1+q+q^2+q^3+...q^(2k-1)=
is even for an odd q
if the first number is odd than the second is even
conclusion: the two number differ in parity therefore cannot be both primes.