2002 Indonesia MO Problems/Problem 6
Problem
Find all prime number such that and are also prime.
Solution
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 .
See Also
2002 Indonesia MO (Problems) | ||
Preceded by Problem 5 |
1 • 2 • 3 • 4 • 5 • 6 • 7 | Followed by Problem 7 |
All Indonesia MO Problems and Solutions |