Let pn be the nth prime. Let 0 < a < 1 be a real. Define the sequence xn by x0 = a, xn = the fractional part of pn/xn-1 if xn ≠ 0, or 0 if xn-1 = 0. Find all a for which the sequence is eventually zero.
Solution
Let {x} denote the fractional part of x. {x} = x minus some integer, so {x} is rational iff x is rational. Hence if xn is irrational, then pn+1/xn is irrational and hence xn+1 is irrational. So if a is irrational, then the sequence is never zero.
Suppose xn = r/s with 0 < r < s relatively prime integers. Then xn+1 = u/r, where u is the remainder on dividing spn+1 by r. So, when expressed as a fraction in lowest terms, the denominator of xn+1 is less than that for xn. So if a is rational and has denominator b, then after at most b iterations we get zero.
Comment: the fact that the pn are primes is a red herring.
© John Scholes
jscholes@kalva.demon.co.uk
11 May 2002