Prove or disprove (discrete math)Prove or disprove the following statement. The difference of the square...
Prove or disprove (discrete math)
Prove or disprove the following statement. The difference of the square of any two consecutive integers is odd
This is working step: let be 2 consective integers:
If m is odd then , if m is even then , then adding 1 will make it odd.
Can you please advise me if my working is the right step and could I answer like this?
Answer & Explanation
Your proof looks correct. You might want to make it more clear that you are saying when you do your arithmetic.
You also don't need to consider the cases where m is even and odd separately: since 2m is a multiple of 2, it must be even, and so you can conclude that is not evenly divisible by 2, so it is odd.
What does it mean for an integer ℓ to be odd exactly? It means that we may express ℓ in the form , where (note that n can be any integer).
Now consider your question:
The difference of the square of any two consecutive integers is odd.
Thus, assume (as you did in your first attempt) that m is an arbitrary integer in Z; then, we are dealing with the difference
Now expand (1) as Strants did in his answer (except with a slight modification):
where . We know because adding two integers, namely 2m and 1, yields an integer, namely ℓ. Here's the important part: notice what form ℓ takes. We have that where m is an arbitrary integer in Z. Thus, by definition, we can see that ℓ is an odd integer.
Maybe this will clear anything up you did not quite get before.