What is the largest even positive integer that cannot be expressed as the sum of two composite odd numbers?
Hey Charlie, get rid of the stupid computer program! All you need are three arithmetic sequences.
18, 24, 30, 36, .... These are all numbers of the form 6k. Each can be written as 3*3+3*k for k>=3. So they are all the sum of two composite odd numbers.
34, 40, 46, 52, .... These are all numbers of the form 6k+4. Each can be written as 5*5+3*k for k>=3. So they are all the sum of two composite odd numbers.
44, 50, 56, 62, .... These are all numbers of the form 6k+2. Each can be written as 5*7+3*k for k>=3. So they are all the sum of two composite odd numbers.
The largest even number not in any of these sequences is 38. All the possible decompositions of 38 into a sum of odd positive integers are 38 = 1+37 = 3+35 = 5+33 = 7+31 = 9+29 = 11+27 = 13+25 = 15+23 = 17+21 = 19+19. All these sums have a prime number so the answer is 38.
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