Smooth Neighbors

Abstract

We give a new algorithm that quickly finds z-smooth neighbors, where a number is z-smooth if none of its prime factors exceeds z, and if b is a solution of p∣b(b+1)⇒p⩽z, then the pair (b, b+1) are called z-smooth neighbors.

Keywords:

• smooth neighbors
• ABC-conjecture
• Stormer’s problem

2000 AMS Subject Classification:

• Primary 11Y50
• Secondary 11N25

Acknowledgments

The authors would like to thank the referee for many helpful comments.

Notes

¹These numbers may be downloaded from http://www.aimath.org/conrey/SmoothNeighbors/ .

²Available at abcathome.com .