Find the total number of positive integer solutions of:
Have you tried graphing the function?
You would probably have to enter it as
.
I can see straight away that so .
Since needs to be a positive integer, we know therefore that .
Now you just need to be a perfect square. works, since , so .
See if you can come up with any other solutions...
Hello, dhiab!
Here is a primitive solution . . .
Consecutive squares differ by an odd number: .Find the number of positive integer solutions of: .
We have: .
. . Then: .
. . Hence: .
. . Hence: .
Therefore, there are exactly two solutions.
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
I came up with this procedure while in college.
Example: .
is the difference between and
. . Therefore: .
We have: .
. .
. . Therefore: .
We have: .
. .
. . Therefore: .
But this cannot be expressed as a sum of conscutive positive odd numbers.
Therefore, there are three solutions: .