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a a t(a admissible value argument assumed AVERAGE B,b)-1 We begin b2 To prove becomes begin with 11 bl b2 California Cartesian product completes the proof consecutive fractions Consequently consider constant contains covers the unit Cycle denominator denote the number dimensional equation 9 equation Xb evaluate exactly one solution fewer Fn_i follows fractions x/B gases help of Lemmas Hence i)da implies B>n-b>b+l induction inequalities initial inner sum interval I(a/b interval l(0/b length minimum molecules neighborhood neighbors number of integers number of prime pairs paradox PENNSYLVANIA STATE UNIVERSITY positive integer prime factors prime to n problem r+l b rational number a/b readily verified real number recurrence region defined Replace result Returning reversible satisfied smaller smallest positive value sum ranges ta-m theory unit interval UNIVERSITY LIBRARY write