P/q =? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ------?1
q/p=1/ ?=?/-2
2q/p=?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?------- 2
adding 1 and 2?
\sqrt{n+1} [/tex] = p/q + 2q/p = p +? /2pq --3
subtracting 1 from 2
?
=? ?-------?4
from 3 and?4
and? are rational numbers as? and ?
are rational for integer p and q .
here n+1 and n-1 are perfect square of positive integer.
now,(n+1)-(n-1)=2 which is not possible since any perfect square differ by
?
at?least 3.
thus there is no positive integer n which??and ?is rational.
