Take care: the contrary of "for any integer isn't divisible by " is "There exists an integer such that "
What can we say about modulo ? (Check the two cases: is odd / is even)
For any integer n, (n^2 - 2) is not divisible by 4 ... Prove by contradiction
Proof: Suppose the contrary, that is for any integer m, (m^2 - 2) is divisible by 4. By definition of divide, n = dk. Thus (m^2 - 2) = 4k.
i don't know how to finish this. i know that if n squared is even then n is even (long way is n=2k for even, n=2k+1 for odd) and n has to be even and greater than 2 for my thus statement to be true. just can't quite piece it together.
I'm studying this topic as well and I need some help with the proofs:
So, first we assume that is false
...and this is where you need the modulo thing, I guess, because you need the term to have a remainder of two for i to be solvable. For example but I don't know how to prove this. It would be good to see a finished proof! Can someone write the proof in the correct way as well, because I need to learn how to write a proof correctly. Actually, that sounds a little bit rude- but I'd like to be able to use the symbols correctly etc. and any help would be much appreciated!
(These math symbols take some getting used to.)
The modulo thing allows you to write very correctly what you want to say, i.e. for any integer, the remainder of the division of by 4 is never 2.
So you've assumed the contrary, which is, under a "modular form":
Since for any integer a way to prove the statement is to compute and see that it only can be or
Therefore no square can be congruent to 2 modulo 4. ( because )
does that mean you need a Lemma? This is a formal question about how you would present or show the results of the "modular form." "Modulo" is similar to the 'mod' [and 'div'] from programming? How would you prove the mod 4 pattern for all n? (it seems clear that it is 1, 0, 1 ...), ...?
is not divisible by 4
Assume the contrary, namely,
Then for this value of
is even is even
This contradicts the assumption that .