Question

Prove that any odd number can be written as the difference of two squares

Answer 1

One way is by intuition

lets list all the perfect squares
1
4
9
16
25
36
49
etc


if we list their differences, we notice a pattern
4-1=3
9-4=5
16-9=7
25-16=9
36-25=11
49-36=13
etc
we notice that the difference between 2 consecutive squares is always an odd number, also the odd numbers go up


before, I said that it was false because we can't get 1, but we can do 1²-0²=1

to get negatives, reverse it
-1=0²-1²

so we can see that we can get all the odd numbers





another way I saw in another answer by Syed514 in this question: brainly.com/question/2284978

anyway
so what syed did was

any odd number can be written as 2n-1 where n is an integer


odd=2n-1
odd=0+2n1
odd=n²-n²+2n-1
odd=n²-(n²-2n+1)
odd=n²-(n-1)²
we can now see that any odd number can be written as a difference of 2 squares
we also have shown that any odd number can be written as the difference of 2 consecutive square numbers