For an Integer , let denote the Least Prime Factor of . A Pair of Integers is called a twin peak if

- 1. ,
- 2. ,
- 3. For all , Implies .

Call the distance between two twin peaks

Then must be an Even multiple of ; that is, where is Even. A twin peak with is called a -twin peak. Thus we can speak of -twin peaks, -twin peaks, etc. A -twin peak is fully specified by , , and , from which we can easily compute .

The set of -twin peaks is periodic with period , where is the Primorial of . That is, if is a -twin peak, then so is . A fundamental -twin peak is a twin peak having in the fundamental period . The set of fundamental -twin peaks is symmetric with respect to the fundamental period; that is, if is a twin peak on , then so is .

The question of the Existence of twin peaks was first raised by David Wilson in the math-fun mailing list on Feb. 10, 1997. Wilson already had privately showed the Existence of twin peaks of height to be unlikely, but was unable to rule them out altogether. Later that same day, John H. Conway, Johan de Jong, Derek Smith, and Manjul Bhargava collaborated to discover the first twin peak. Two hours at the blackboard revealed that admits the -twin peak

which settled the Existence question. Immediately thereafter, Fred Helenius found the smaller -twin peak with
and

The effort now shifted to finding the least Prime admitting a -twin peak. On Feb. 12, 1997, Fred Helenius
found , which admits 240 fundamental -twin peaks, the least being

Helenius's results were confirmed by Dan Hoey, who also computed the least -twin peak and number of fundamental -twin peaks for , 79, and 83. His results are summarized in the following table.

71 | 7310131732015251470110369 | 240 |

73 | 2061519317176132799110061 | 40296 |

79 | 3756800873017263196139951 | 164440 |

83 | 6316254452384500173544921 | 6625240 |

The -twin peak of height is the smallest known twin peak. Wilson found the smallest known -twin peak with , as well as another very large -twin peak with . Richard Schroeppel noted that the latter twin peak is at the high end of its fundamental period and that its reflection within the fundamental period is smaller.

Many open questions remain concerning twin peaks, e.g.,

- 1. What is the smallest twin peak (smallest )?
- 2. What is the least Prime admitting a -twin peak?
- 3. Do -twin peaks exist?
- 4. Is there, as Conway has argued, an upper bound on the span of twin peaks?
- 5. Let be Prime. If and each admit -twin peaks, does then necessarily admit a -twin peak?

© 1996-9

1999-05-26