Researcher interview questions shared by candidates
For any prime number larger than 3, prove that p^2 - 1 is always divisible by 24.
The trick is to realize P^2-1 = (P+1)(P-1). P is prime so it must be odd so P+1 and P-1 are both even and therefore one of them is divisible by 4 and the other by 2. Of any 3 consecutive numbers, one must be divisible by 3 so if we take P-1, P, and P+1; P cannot be divisible by 3 since it's a prime greater than 3 so either P+1 or P-1 is divisible by 3. Therefore we have the factors: 3*2*4 = 24.
Another method is as follows. All primes must be equivalent to +/- 1 or +/- 5 mod 12. All other mod will have some factor in common with 12. In the case of +1: (12k+1)^2 - 1 = 144k^2 + 24k = 24(6k^2+1) has a factor of 24. The same is true for -1. In the case of +5: (12k+5)^2-1 = 144k^2+120k+24 = 24(6k^2+5k+1). The same is true for -5.