10th Brazil 1988 Problem 1 Find all primes which can be written both as a sum of two primes and as a difference of two primes.
5 = 3+2 = 7-2 Solution 2 is the smallest prime, so it cannot be written as the sum of two primes. All other primes p are odd, so if p can be written as the sum of two primes and the difference of two primes, then in each case one of the primes must be 2. Thus we have p = q - 2 = r + 2 for some primes q, r. If r = 1 mod 3, then p is a multiple of 3, so p must be 3. But then r = 1, which is not a prime. So there are no solutions with r = 1 mod 3. If r = 2 mod 3, then q is a multiple of 3, so q must be 3, but then p = 1, which is not a prime. So there are no solutions with r = 2 mod 3. Hence r must be a multiple of 3. But r is prime, so r must be 3. Hence q = 7 and p = 5. Thanks to Suat Namli . 10th Brazil 1988 © John Scholes jscholes@kalva.demon.co.uk 5 February 2004 Last corrected/updated 5 Feb 04