If

æ ç è 5 4 ö ÷ ø m   » æ ç è 2 1 ö ÷ ø n

then

mlog 5 4 » n log2

and we shall find

m n » log2 log5/4 .

Using Euclid's algorithm the result can be represented as a continued fraction.

log2 log5/4 = 3 + Q log5/4

where

0 <

Q log5/4

< 1

. So

log2 log5/4 = 3 + 1 log 5/4 Q = 3 + 1 9.4087788 = 3 + 1 9 + 1 2.4463109

giving the first approximation

28 9

. Repeating the algorithm:

log2 log5/4 = 3 + 1 9 + 1 2 + 1 2 + 1 4 +...

Truncating this continued fraction gives the successive approximations

59 19

,

146 47

and

643 207

.