A351243 Counterexamples to a conjecture of Selfridge and Lacampagne.

247, 277, 967, 977, 1211, 1219, 1895, 1937, 1951, 1961, 2183, 2191, 2911, 2921, 3029, 3641, 3649

Offset: 1

Author

Jeffrey Shallit, Feb 05 2022

Keywords

• nonn
• base
• more

Comments

The conjecture was that every natural number k not divisible by 3 can be written as the quotient of two terms chosen from A147991.

For every specific k, the problem of representing k as the quotient of two terms of A147991 can be decided by using a queue-based breadth-first search algorithm on the transition diagram of a finite automaton that on input j in base 3 computes j*k and checks to see if both j and j*k are in A147991.

It is not known if there are infinitely many counterexamples to the conjecture, but perhaps 3^m+4, for m >= 5 and odd, are.

References

• R. K. Guy, Unsolved Problems in Number Theory, Springer, 2004. In Section F31, the conjecture of Selfridge and Lacampagne is mentioned, and it is stated that Don Coppersmith found the counterexample 247.

Links

• James Haoyu Bai, Joseph Meleshko, Samin Riasat, and Jeffrey Shallit, Quotients of Palindromic and Antipalindromic Numbers, INTEGERS 22 (2022), #A96.
• J. H. Loxton and A. J. van der Poorten, An Awful Problem About Integers in Base Four, Acta Arithmetica, volume 49, 1987, pages 193-203. In section 7, Selfridge and Lacampagne ask whether every k != 0 (mod 3) is the quotient of two terms of this sequence.

Crossrefs

Cf. A147991.