On a Problem of Ruderman

نویسندگان

  • M. Ram Murty
  • V. Kumar Murty
چکیده

In 1974, Harry Ruderman proposed the following problem in the problem section of this MONTHLY: if m > n ≥ 0 are integers such that 2m − 2n divides 3m − 3n , then show that 2m − 2n divides xm − xn for all natural numbers x . This problem is still open. We prove that there are only finitely many pairs of natural numbers m, n such that 2m − 2n divides 3m − 3n . Since the proof involves the Schmidt subspace theorem, our bounds on m and n are ineffective. We discuss how an effective version of the abc conjecture can be used to derive effective bounds on m and n.

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عنوان ژورنال:
  • The American Mathematical Monthly

دوره 118  شماره 

صفحات  -

تاریخ انتشار 2011