Little Goldbach
How many of the numbers 1 to 20 are not the sum of two primes?
Age
11 to 14
Challenge level
Problem
Both 4 and 8 can be written as the sum of two prime numbers ($4=2+2$, $8=5+3$). How many numbers less than 20 cannot be written as the sum of two prime numbers?
If you liked this problem, here is an NRICH task which challenges you to use similar mathematical ideas.
Student Solutions
Answer: 5
The numbers are 1, 2, 3, 11 and 17. Note that, because 2 is the only even prime, the only way we can sum two primes to give an odd number is to sum the odd number before it (if it is a prime) and 2.
One of the oldest unsolved problems in number theory is Goldbach's conjecture, which says that every even number (except 2) can be written as the sum of two primes. Perhaps you could be the first to prove it...