a. Unfortunately I had to google what a Keith number is, and the search result produced a list of Keith numbers which immediately revealed that 197 is NOT the only 3digit Keith number. 742 is also a Keith number, being found in the sequence 7 4 2 13 19 34 66 119 219 404 742...
b. Since 6^3 > 197, if 197 is the sum of three cubes they must all be <= 5. A little inspection and it's easy to see that 2^3 + 4^3 + 5^3 = 8 + 64 + 125 = 197.
c. Every odd number can be expressed as the difference of two consecutive squares, since n^2  (n1)^2 = n^2  (n^2  2n + 1) = 2n  1 for all n. Specifically, 197 = 99^2  98^2.

Posted by tomarken
on 20140717 10:24:09 