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Problem #524
Solution
Goldbach’s Conjecture states that every even N ≥ 4 is a sum of two primes.
Every even number N ≥ 6 is a sum of two even composite numbers because
N = 4 + N − 4. What is the largest even number that is not a sum of two
odd composite numbers?
Answer: The largest even number that is not the sum of two odd composite
numbers is 38.
Proof. We first observe that 38 − x is not a composite number for
x = 9, 15, 21, 24, 27, 33, or 35.
Therefore 38 is not a sum of two composite numbers.
Now suppose that n is an even integer with n > 38. We will show that n is
sum of two odd composite numbers.
Case 1. n ≡ 4 (mod 6). Then n = 40 + 6k for some k ≥ 0, and we may
write n = (6k + 15) + 25.
Case 2. n ≡ 0 (mod 6). Then n = 42 + 6k for some k ≥ 0, and we may
write n = (6k + 21) + 21.
Case 3. n ≡ 2 (mod 6). Then n = 44 + 6k for some k ≥ 0, and we may
write n = (6k + 9) + 35.
Source: Which Way Did the Bicycle Go, by J.D.E. Kornhauser, D. Velleman,
and S. Wagon.
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