Albert said:
(1)$a,b,k\in\mathbb{N}$
(2)$a>b$
(3)$k=\dfrac {a^2+ab+b^2}{ab-1}$
please find :
(i) $\max(k)$
(ii) all possible values of $k$
$k>\dfrac{\sqrt[3]{3a^3b^3}}{ab}=3---(1)$(AP >GP)
if a,b both are odd numbers then k does not exist
if a,b both are even numbers then k is also a even number (ex:a=4,b=2 then k=4)
so min(k)=4
let a=2m, b=2n,here $m,n \in N$
then :
$k=\dfrac {4(m^2+mn+n^2)}{(4mn-1)}=4x (x\in N)$
for all a,b being even numbers ,we want to prove k=4 then we must prove x=1,that is to prove :
$x=\dfrac{(m^2+mn+n^2)}{(4mn-1)}=1$------(2)(if $x\in N)$
if a odd and b even then k must be odd (ex:a=11 ,b=2 then k=7)
also if a even and b odd then k must be odd
let a=b+d (here b:even and d odd ,so a must be odd)
$k=\dfrac {3b^2+3bd-3+d^2+3}{b^2+bd-1}=3+\dfrac{d^2+3}{b^2+bd-1}$
here $\dfrac{d^2+3}{b^2+bd-1}$ must be even
now we must prove :
$y=\dfrac{d^2+3}{b^2+bd-1}=4$-----(3)(if $y\in N)$
if (2) and (3) can be proved then all is done ,that is min(k)=4 ,and max(k)=7
all the possible valus of k=4 and 7
the proof (2) and (3):I am still thinking ----
By using a program (10001>a>b>0) the possible valus of k=4 and 7
I know (2) and (3) must be true , but how to prove it ?
