I have been working through D. L. Johnson's "Elements of Logic via Numbers and Sets" and have come across this problem, amongst others, which has me stumped.

"Give a proof by contradiction of the statement:

: the sum of the squares of three consecutive integers cannot leave remainder

on division by

."

Therefore, I have suggested that we assume (for contradiction) ¬P ('not' P), so

,

where

is the lowest integer in the consecutive series and

is an integer.

Then,

.

I think for the purposes of our proof, I wish to show that

cannot be an integer (so providing the necessary contradiction). I'm not sure, however, how to do this, and so whether this former assertion is correct. Any suggestions much appreciated.