Two primes that add up to 101

Question

Can 101 be written as the sum of two prime numbers? Either find the two primes or prove there are none, without listing all the primes below 101. Then: for which odd numbers is such a representation possible at all, and why is the same question for even numbers so much harder?

Accepted Answer

No. Parity. Two numbers adding to an odd total must be one even and one odd. The only even prime is 2. So if 101 is a sum of two primes, one of them is 2 and the other is 101 minus 2 = 99. But 99 = 9 × 11 is not prime. So no such pair exists. Odd numbers in general. An odd number n is a sum of two primes exactly when n minus 2 is prime. So 5 = 2 + 3, 7 = 2 + 5, 9 = 2 + 7, 13 = 2 + 11, 15 = 2 + 13, 19 = 2 + 17 all work, while 11, 17, 23, 27 and 101 do not. The contrast with even numbers. For even numbers the question is famously open: Goldbach's conjecture says every even number greater than 2…

Two primes that add up to 101

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