Prove that when all expessions of the form:
+/- √1 +/- √2 +/- ....... +/- √100 are multiplied together, the result is an integer.
As an example, multiplying all expressions of the form: +/- √1 +/- √2 is equivalent to finding the result of this product:
(√1 +√2)( √1 - √2)( - √1 + √2)( - √1 - √2)

**** Extra Challenge: Solve this puzzle without the aid of a computer program.

(In reply to computer exploration -- far from a proof by Charlie)

Actually, since the second half of the computation just duplicates the first half and as I have shown the result is a square, it suffices to compute half of the terms.

Each of these amounts in the case of n=6 to around 7981444995488.94, near enough to 7981444995489. We can do a check by comparing nearby squares (though there are a lot of them, they are quite far apart by this stage):

63703464216000440340358144    7981444995488^2
63703464216016403230349121    7981444995489^2
63703464216032366120340100    7981444995490^2

7981444995489 is closest by far. So for n=6, the result is 63703464216016403230349121. Unfortunately, there is not enough information to calculate the result for n=7 with any exactitude.

 

Edited on September 19, 2015, 3:21 am

Posted by broll on 2015-09-19 03:03:04