Cunningham numbers

The Cunningham project is described in the following excerpt from a sci.math posting by Bob Silverman, who has contributed many of the factorizations.

In 1925 Lt.-Col. Alan J.C. Cunningham and H.J. Woodall gathered together all that was known about the primality and factorization of such numbers and published a small book of tables. "These tables collected from scattered sources the known prime factors for the bases 2 and 10 and also presented the authors' results of thirty years' work with these and the other bases" (see Ref 1)

Since 1925 many people have worked on filling in these tables. It is likely that this project is the longest, ongoing computational project in history. D.H. Lehmer, a well known mathematician who passed away in 1991 was for many years a leader of these efforts. Professor Lehmer was a mathematician who was at the forefront of computing as modern electronic computers became a reality. He was also known as the inventor of some ingenious pre-electronic computing devices specifically designed for factoring numbers. These devices are currently in storage at the Computer Museum in Boston.

For a history of this project I suggest you obtain a copy of:

Ref 1: J. Brillhart, D.H. Lehmer, J. Selfridge, S.S. Wagstaff Jr., & B. Tuckerman
Contemporary Mathematics vol 22,
"Factorizations of b n 1, b = 2,3,5,6,7,10,11,12 up to high powers",
published by the American Math. Society 1983, 2nd Edition 1988.

The factorizations of bn + 1, for b = 2, 3, 5, 6, 7, 10, 11, 12 are held in the files 2+, 3+, ... linked to in the table below. Likewise, the factorizations of bn - 1 are held in 2-, 3- etc. These files contain only primitive factors. Some Aurifeuillean factorizations have been listed separately; others are amalgamated. By and large, the smaller ones are amalgamated. For example, in the 10+ file, 1050 + 1 is given as

50      101.7019801.14103673319201.1680588011350901

but 10150 + 1 as

150L    261301.38654658795718156456729958859629701
150M    601.3903901.168290119201.25074091038628125301

All files abide by my standard format, designed to be both human-readable and easily parsed by programs.

I believe these tables are accurate and complete up to the end of February 2011 but would appreciate being told of any corrections and updates. With over 8000 entries in total, there is a fair chance that I have made transcription and other errors.

bn + 1 2+ 3+ 5+ 6+ 7+ 10+ 11+ 12+
bn - 1 2- 3- 5- 6- 7- 10- 11- 12-

Acknowledgements: As well as all the factorers, far too many to mention individually, I'd particularly like to thank Sam Wagstaff for making his extensive tables available to me. Indeed, my versions are essentially his tables, reformatted to make them easier to parse by factoring programs.

There is one small addition to the regular Cunningham tables: the 3LM extension table.

To make monitoring changes to these tables easier, the file UPDATE contains a list of changes to the files made since the date specified in UPDATE. Every now and again, that date will be reset and the update started afresh. I hope that makes sense.


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