Twin Peaks

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 .

A broken-line graph of the least prime factor function resembles a jagged terrain of mountains. In terms of this terrain, a twin peak consists of two mountains of equal height with no mountain of equal or greater height between them. Denote the height of twin peak  by . By definition of the least prime factor function,  must be prime.

Call the distance between two twin peaks 

(1)

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 (pers. comm., 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

(2)

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

(3)

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

(4)

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 (OEIS A009190).

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?

REFERENCES:

Sloane, N. J. A. Sequence A009190 in "The On-Line Encyclopedia of Integer Sequences."