“Twin Peaks”的概念、定义、习题解题方法及翻译-数学辞典

单词

Twin Peaks

释义

Twin Peaks

For an Integer , let denote the Least Prime Factor of . A Pair ofIntegers 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 peakconsists of two mountains of equal height with no mountain of equal or greater height between them. Denote the height oftwin peak

. By definition of the Least Prime Factor function,

must be Prime.

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 specifiedby

, 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 thefundamental 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 onFeb. 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 ManjulBhargava 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 Heleniusfound , 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 notedthat the latter twin peak is at the high end of its fundamental period and that its reflection within the fundamentalperiod 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?

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

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