{P-Q}{P^(n-1) + K1[P^(n-2)]Q + K2[P^(n-3)]Q^2 + ....+ Q^(n-1)} cannot equal C^n - B^n
where all the variables are positive integers, n>2, P>Q, C>B
unless all the Ks are equal to 1 or in some instances the Ks have the same value.
{P-Q}{P^(n-1) + K1[P^(n-2)]Q + K2[P^(n-3)]Q^2 + ....+ Q^(n-1)} cannot equal C^n - B^n
where all the variables are positive integers, n>2, P>Q, C>B
unless all the Ks are equal to 1 or in some instances the Ks have the same value.
Can i ask why my post was deleted? It was a completely legitimate post providing a proper counterexample.
PS: I have another counterexample with the new constraint given, but if someone is deleting links for no reason, there is no reason to post it.
EDIT:
Forget my above comments. It is just the TS created 2 exact topics and i answered to the other:
Proof of a Conjecture or Counter Examples
First, take a look at the example in the other thread and see if you can come up with another one based on ChessTal's method.
This is why we don't double post....
-Dan
For:
P=3
Q=2
n=3
K1=83
C=8
B=1 we have:
$\displaystyle (P - Q) \cdot \left( {{P^{n - 1}} + {K_1} \cdot {P^{n - 2}} \cdot {Q^1} + {K_2} \cdot {P^{n - 3}} \cdot {Q^2} + ... + {K_{n - 2}} \cdot {P^1} \cdot {Q^{n - 2}} + {K_{n - 1}} \cdot {Q^{n - 1}}} \right) \ne {C^n} - {B^n} \Leftrightarrow $
$\displaystyle (3 - 2) \cdot ({3^{3 - 1}} + 83 \cdot {3^{3 - 2}} \cdot {2^1} + {2^{3 - 1}}) \ne {8^3} - {1^3} \Leftrightarrow $
$\displaystyle 9 + 498 + 4 \ne 511 \Leftrightarrow $
$\displaystyle 511 \ne 511$