Show that the product of two of the numbers 651000 82001 31

Show that the product of two of the numbers 65^1000 8^2001 + 3^177 , 79^1212 9^2399 + 2^2001 , and 24^4493

5^8192 + 7^1777 is nonnegative. Is your proof constructive or nonconstructive? [Hint: Do not try to evaluate these numbers!]

Solution

Let the numbers be p, q, r. Now either p is negative , zero or positive. So are q and r.

Take any two

if they are both positive, then product is positive.
if they are both negative, then product is positive
if one is zero the product is zero.

In the event that one is negative and the other positive then the third number is either negative, positive or zero and so by a judicious choice of the two numbers you can always get a non-negative result.

This was a nonconstructive proof, since we have not identified which product is non-negative.