A question from a famous set of interview questions asks:
Pairs of primes separated by a single number are called prime pairs. Examples are 17 and 19. Prove that the number between a prime pair is always divisible by 6 (assuming both numbers in the pair are greater than 6). Now prove that there are no 'prime triples.'
Now try really hard to solve this for yourself, then click below to see an example solution.
Let's call the numbers a, b and c, where a and c are prime.
First we notice that 6 = 2 * 3, so if a number is divisible by six, it must also be divisible by two and three.
Since a and c are prime, they are not divisible by two. But every second number is even, therefore b must be even and divisible by two.
Similarly, every third number is divisible by three. But a and c are prime, forcing us to conclude that the number b must also be divisible by three.
Ergo, the number b is divisible by six.
It is now easy to see why prime triplets cannot exist. Given five numbers in increasing order:
a b c d e
Here, a, c and e are prime numbers. We can also say we have two prime pairs a, c and c, e. This means that b and d should be divisible by six, from the argument above.
The numbers a, c and e cannot be divisible by three, since they are prime. As for b and d, the distance between them is two, meaning only one of them can be divisible by three.
This leads to a contradiction, and therefore prime triplets cannot exist.