Twin Peaks

For an Integer $n\geq 2$, let $\mathop{\rm lpf}\nolimits (x)$ denote the Least Prime Factor of $n$. A Pair of Integers $(x,y)$ is called a twin peak if

1. $x < y$,

2. $\mathop{\rm lpf}\nolimits (x) = \mathop{\rm lpf}\nolimits (y)$,

3. For all $z$, $x < z < y$ Implies $\mathop{\rm lpf}\nolimits (z) < \mathop{\rm lpf}\nolimits (x)$.

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 $p = \mathop{\rm lpf}\nolimits (x) = \mathop{\rm lpf}\nolimits (y)$. By definition of the Least Prime Factor function, $p$ must be Prime.

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

$$s \equiv y-x.$$

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 $y \equiv x+kp$.

The set of $kp$-twin peaks is periodic with period $q = p\char93$, where $p\char93$ 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 in the math-fun mailing list on Feb. 10, 1997. Wilson already had privately showed the Existence of twin peaks of height $p\leq 13$ 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

$$x = 126972592296404970720882679404584182254788131,$$

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

$$x = 9503844926749390990454854843625839.$$

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

$$x = 7310131732015251470110369.$$

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.

$p$	$L(2p)$	$N(2p)$
71	7310131732015251470110369	240
73	2061519317176132799110061	40296
79	3756800873017263196139951	164440
83	6316254452384500173544921	6625240

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 $[0, p\char93 )$ 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