2002 Indonesia MO Problems/Problem 6
Find all prime number such that and are also prime.
If , then and . Since is not prime, can not be . If , then and . Both of the numbers are prime, so can be .
The rest of the prime numbers are congruent to ,,, and modulo , so is congruent to or modulo . If , then . If , then . That means if is congruent to ,,, or modulo , then either or can be written in the form .
The only way for to equal is when or , which are not prime numbers. Thus, the rest of the primes can not result in and both prime, so the only solution is .
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