Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: A Fibonacci-like sequence of composite numbers. | A Fibonacci-like sequence of composite numbers John W. Nicol University of Illinois Urbana IL nicol@ Submitted June 17 1998 Submitted in revised form October 12 1999 Accepted November 6 1999. Abstract In 1964 Ronald Graham proved that there exist relatively prime natural numbers a and b such that the sequence An defined by An An-1 An-2 n 2 A0 a A1 b contains no prime numbers and constructed a 34-digit pair satisfying this condition. In 1990 Donald Knuth found a 17-digit pair satisfying the same conditions. That same year noting an improvement to Knuth s computation Herbert Wilf found a yet smaller 17-digit pair. Here we improve Graham s construction and generalize Wilf s note and show that the 12-digit pair a b 407389224418 76343678551 also defines such a sequence. Mathematical Reviews Subject Numbers 11B39 11N99. 1 Introduction Ronald Graham 2 proved that there exist relatively prime natural numbers a and b such that the sequence An defined by An An-1 An-2 n 2 A0 a A1 b 1 THE ELECTRONIC .JOURNAL OF COmBINATORICS 6 1999 R44 2 contains no prime numbers. Graham s pair a b was 331635635998274737472200656430763 1510028911088401971189590305498785 . Donald Knuth 4 found the smaller pair a b 62638280004239857 49463435743205655 . Herbert Wilf 8 noted that Knuth s computation could be improved and so found the pair a b 20615674205555510 3794765361567513 . Here we show that Wilf actually generalized the system of congruences given by Graham. By further generalizing the system improving Graham s construction and applying Knuth s method to the cases that are generated we show that the pair a b 407389224418 76343678551 also defines such a sequence. Graham constructed the following 18 triples of numbers pk mk rk such that 1. pk is prime 2. pk I Fn o n I mk Fn is the Fibonacci number. 3. The congruences x rk mod mk cover the integers . for every number n there exists a k 1 k 18 such that n rk mod mk 3 4 1 2 3 2 5 5 1 7 8 3 17 9 4 11 10 2 47 16 7 19 18 10 61 15
