What is the biggest even number N that can't be written as a sum of odd composite numbers?
Bonus: Find another feature unique to N.
I decided to look at the last digit of even numbers.
Every odd ending in 5 (except 5 itself) is composite.
The smallest odd composite ending in 5 is 15. So ending in zero, 15+15=30 and above is the sum of two odd composites.
The smallest odd composite ending in 7 is 27. So ending in two, 15+27=42 and above is the sum of two odd composites.
The smallest odd composite ending in 9 is 9. So ending in four, 15+9=24 and above is the sum of two odd composites.
The smallest odd composite ending in 1 is 21. So ending in six, 15+21=36 and above is the sum of two odd composites.
The smallest odd composite ending in 3 is 33. So ending in eight, 15+33=48 and above is the sum of two odd composites.
Now I can just work backward through the few remaining evens. The first candidate is 38. This cannot be written as the sum of two odd composites, so it is the solution.
What else is special about 38? In Roman numerals, it is XXXVIII which would make it last if you put them all in alphabetical order.
So 38 is the last number of two sequences. (Does it make sense to be the last in an infinite sequence?)

Posted by Jer
on 20180729 11:13:33 