I've found (on the web) a short list of "SPECIAL FEATURES " of the number 197:

a. It is the only 3-digit Keith* number.
b. It is a sum of three cubes.
c. It is a difference of two squares.

One (and only one) of the above is not true.

Prove it.

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 3-digit 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 - (n-1)^2 = n^2 - (n^2 - 2n + 1) = 2n - 1 for all n.  Specifically, 197 = 99^2 - 98^2.

 

 

Posted by tomarken on 2014-07-17 10:24:09