Twin Peaks

For an integer n>=2, let lpf(n) denote the least prime factor of n. A pair of integers (x,y) is called a twin peak if

1. x<y,

2. lpf(x)=lpf(y),

3. For all z, x<z<y implies lpf(z)<lpf(x).

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 (x,y) by p=lpf(x)=lpf(y). By definition of the least prime factor function, p must be prime.

Call the distance between two twin peaks (x,y)


Then s must be an even multiple of p; that is, s=kp where k is even. A twin peak with s=kp is called a kp-twin peak. Thus we can speak of 2p-twin peaks, 4p-twin peaks, etc. A kp-twin peak is fully specified by k, p, and x, from which we can easily compute y=x+kp.

The set of kp-twin peaks is periodic with period q=p#, where p# is the primorial of p. That is, if (x,y) is a kp-twin peak, then so is (x+q,y+q). A fundamental kp-twin peak is a twin peak having x in the fundamental period [0,q). The set of fundamental kp-twin peaks is symmetric with respect to the fundamental period; that is, if (x,y) is a twin peak on [0,q), then so is (q-y,q-x).

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 p<=13 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 p=113 admits the 2p-twin peak


which settled the existence question. Immediately thereafter, Fred Helenius found the smaller 2p-twin peak with p=89 and


The effort now shifted to finding the least prime p admitting a 2p-twin peak. On Feb. 12, 1997, Fred Helenius found p=71, which admits 240 fundamental 2p-twin peaks, the least being


Helenius's results were confirmed by Dan Hoey, who also computed the least 2p-twin peak L(2p) and number of fundamental 2p-twin peaks N(2p) for p=73, 79, and 83. His results are summarized in the following table (OEIS A009190).


The 2p-twin peak of height p=73 is the smallest known twin peak. Wilson found the smallest known 4p-twin peak with p=1327, as well as another very large 4p-twin peak with p=3203. 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 [0,p#) is smaller.

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

1. What is the smallest twin peak (smallest n)?

2. What is the least prime p admitting a 4p-twin peak?

3. Do 6p-twin peaks exist?

4. Is there, as Conway has argued, an upper bound on the span of twin peaks?

5. Let p<q<r be prime. If p and r each admit kp-twin peaks, does q then necessarily admit a kp-twin peak?

See also

Andrica's Conjecture, Divisor Function, Least Common Multiple, Least Prime Factor

Explore with Wolfram|Alpha


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

Referenced on Wolfram|Alpha

Twin Peaks

Cite this as:

Weisstein, Eric W. "Twin Peaks." From MathWorld--A Wolfram Web Resource.

Subject classifications