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Alladi approximation to 9 Arithmetic progression arranged in ascending ascending order belong best approximation best possible constant closed interval conjugate fractions conjugate stages consecutive points consecutive rationals Consider continued fraction contradiction according convergent to 9 corresponding fractions corresponding intervals corresponding numbers deduce Define recursively defined an interval Definition denominator denoted difference in suffixes DIOPHANTINE APPROXIMATIONS eoual established exist infinitely f.f. series F.Fn f.fR Farey series Fibonacci numbers Fibonacci sequence fraction p/q Further let generalisation Hence the theorem implies increasing values INSTITUTE OF MATHEMATICAL interval arranged Krishnaswami Lemma mediants of consecutive middle point nearest integer function noint number of rationals number of terms Number Theory one-one points of symmetry positive integers Proof properties of Farey properties of mediants quadratic irrational r=l r sequence of distance series of order series with Fibonacci Topics in Number