how come it only works with n=3? there must be some sort of reasoning behind it? because say if i had not known that n=3 is a solution (by inputting values into excel) then how would i have come about the answer?
also, in the proof for n>3 this has been used:
CASE I: n is even,
(n^{2}+\frac{n}{2})^{2}<\frac{n^{5}-1}{n-1}< (n^{2}+\frac{n}{2}+1)^{2}
CASE II: n is odd,
(n^{2}+\frac{n-1}{2})^{2}<\frac{n^{5}-1}{n-1}< (n^{2}+\frac{n+1}{2})^{2}
Where have the expressions for the boundaries come from? Are they related to n=3 in some way? Because if not, one could also say that this holds for n > 2 or n > 5 etc.
Basically, whats the relationship between the boundaries used and n > 3?
ah that's really good! (yes 9 was a typo sorry)
how come only the upper boundary of n being odd is related to n > 3? i.e. with n^2 - 2n - 3 > 0
Also, how would you come up with these equations from scratch!? I get how I worked back from fardeen's work to get the case...