top ten coolest numbers

This is an attempt to give a count-down of the top ten coolestnumbers. Let’s first admit that this is a highly subjectiveordering–one person’s 14.38 is another’s $\\frac{\\pi^{2}}{6}$ . Theastute (or probably simply “awake”) reader will notice, for example,a definite bias toward numbers interesting to a number theorist in thebelow list. (On the other hand, who better to gauge the coolness ofnumbers than a number-theorist…) But who knows? Maybe I can beconvinced that I’ve left something out, or that my ordering should beswitched in some cases. But let’s first set down some ground rules.

What’s in the list? What makes a number cool? I think aword that sums up the key characteristic of cool numbers is“canonicity”. Numbers that appear in this list should be somehowfundamental to the nature of mathematics. They could represent afundamental fact or theorem of mathematics, be the first instance ofan amazing class of numbers, be omnipresent in modern mathematics, orsimply have an eerily long list of interesting properties. Perhaps amore appropriate question to ask is the following:

What’s not in the list? There are some really awesomenumbers that I didn’t include in the list. I’ll go through severalexamples to get a feel for what sorts of numbers don’t fit thecharacteristics mentioned above.

Shocking as it may seem, I first disqualify the constants appearing inEuler’s formula $e^{i\\pi}+1=0$ . This was a tough decision. Perhapsthese five ( $e$ , $i$ , $\\pi$ , 1, and 0) belong at the top of the list, orperhaps they’re just too fundamentally important to be consideredexceptionally cool. Or maybe they’re just so cliché’d thatwe’ll get a significantly more interesting list by excluding them.

Also disqualified are numbers whose primary significance is cultural,rather than mathematical: despite being the answer to life, theuniverse, and everything, 42 is a comparatively uninteresting number. Similarly not included in the list were876-5309, 666, and the first illegal prime number. Similarly disqualified were constants of nature like Newton’s $g$ and $G$ , the fine structure constant, Avogadro’s number, etc.

Finally, I disqualified number that were highly non-canonical inconstruction. For example, the prime constant and Champoleon’sconstant are both mathematically interesting, but only because theywere, at least in an admittedly vague sense, constructed to be assuch. Also along these lines are numbers like G63 and Skewe’sconstant, which while mathematically interesting because of rolesthey’ve played in proofs, are not inherently interesting in and ofthemselves.

That said, I felt free to ignore any of these disqualifications when Ifelt like it. I hope you enjoy the following list, and I welcomefeedback.

Honorable Mentions

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65,537 - This number is arguably the number with the mostpotential. It’s currently the largest Fermat prime known. If itturns out to be the largest Fermat prime, it might earn itselfa place on the list, by virtue of thus also being the largest oddvalue of $n$ for which an $n$ -gon is constructible using only a ruleand compass.
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Conway’s constant - The construction of the number can be foundhere http://mathworld.wolfram.com/ConwaysConstant.html. Thoughthis number has some remarkable properties (not the least of which isbeing unexpectedly algebraic), it’s completely non-canonicalconstruction kept it from overtaking any of our list’s currentmembers.
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1728 and 1729 - This pair just didn’t have quite enough goingfor them to make it. 1728 is an important $j$ -invariant of ellipticcurves and modular forms, and is a perfect cube. 1729 happens to bethe third Carmichael number, but the primary motivation forincluding 1729 is because of the mathematical folklore associated itto being the first taxicab number, making it more interesting(math-)historically than mathematically.
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28 - Aside from being a perfect number, a fairly interestingfact in and of itself, the number 28 has some extra interesting“aliquot” properties that propels it beyond other perfect numbers.Specifically, the largest known collection of sociable numbers hascardinality 28, and though this might seem a silly feat in and ofitself, the fact that sociable numbers and perfect numbers areso closely related may reveal something slightly more profound about28 than it just being perfect.
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26 - being the only number between a square and a cube is pretty cool; as well as that to Actuaries, this number has relavance to life expectancy - in than it is a turning point. (this will change over time as is just a tenuous arguement to support giving 26 a mention!).

And now, on to the top 10:

#10) The Golden Ratio, $\\phi$
This was a toughone. Yes, it’s cool that it satisfies the property that itsreciprocal is one less than it, but this merely reflects that it’s aroot of the wholly generic polynomial $x^{2}-x-1=0$ . Yes, it’s coolthat it may have an aesthetic quality revered by the Greeks, but thisis void from consideration for being non-mathematical. Only slightlyless canonical is that it gives the limiting ratio of subsequentFibonacci numbers. Redeeming it, however, is that this generalizes toall “Fibonacci-like” sequences, and is the solution to twosort of canonical operations:

\\displaystyle\\frac{1}{1+\\frac{1}{1+\\frac{1}{1+\\frac{1}{\\ddots}}}}

and

\\displaystyle\\sqrt{1+\\sqrt{1+\\sqrt{1+\\cdots}}}

Also, this number plays an important role in the hstory of algebraicnumber theory. The field it generates is the first known example of afield in which unique factorization fails. Trying to come to grips withthis fact led to the invention of ideal theory, class nubers, etc.

#9) 691
The prime number 691 made it on here fora couple of reasons: First, it’s prime, but more importantly, it’s thefirst example of an irregular prime, a class of primes ofimmense importance in algebraic number theory. (A word of caution:it’s not the smallest irregular prime, but it’s the one thatcorresponds to the earliest Bernoulli number, $B_{12}$ , so 691 is only“first” in that sense). It also shows up as a coefficient of everynon-constant term in the $q$ -expansion of the modular form $E_{12}(z)$ . Further testimony to the arithmetic significance is itsseemingly magical appearance in the algebraic $K$ -theory: It’s knownthat $K_{22}(\\mathbb{Z})$ surjects onto 691.

#8) 78,557
The number 78,557 is here to representan amazing class of numbers called Sierpinski numbers, definedto be numbers $k$ such that $k2^{n}+1$ is composite for every $n\\geq 1$ . That such numbers exist is flabbergasting…we know fromDirichlet’s theorem that primes occur infinitely often in non-trivialarithmetic sequences. Though the sequence formed by $78557\\cdot 2^{n}+1$ isn’t arithmetic, it certainly doesn’t behave multiplicativelyeither, and there’s no apparent reason why there shouldn’t be a large(or infinite) number of primes in every such sequence. Thisnotwithstanding, Sierpinski’s composite number theorem proves thereare in fact infinitely many odd such numbers $k$ . As a smalldisclaimer, though it’s proven that 78,557 is indeed a Sierpinskinumber, it is not quite yet known that it is the smallest. There are17 positive integers smaller than 78,557 not yet known to benon-Sierpinski.

#7) $\\frac{\\pi^{2}}{6}$
Perhaps the first strikingthis about this number is that it is the sum of the reciprocals of thepositive integer squares:

\\displaystyle 1+\\frac{1}{4}+\\frac{1}{9}+\\cdots+\\frac{1}{n^{2}}+\\cdots=\\frac{%\\pi^{2}}{6}.

Though the choice of $2$ here for the exponent is somewhatnon-canonical (i.e. we’ve just noted that $\\zeta(2)=\\frac{\\pi^{2}}{6}$ ,where $\\zeta$ stands for the Riemann zeta function), and that this islargely interesting for math-historical reasons (it was the first sumof this type that Euler computed), we can at least include it here torepresent the amazing array of numbers of the form $\\zeta(n)$ for $n$ a positive integer. This class of numbers incorporates two amazingand seemingly disparate collections, depending on whether $n$ is even(in which case $\\zeta(n)$ is known to be a rational multiple of $\\pi^{n}$ ) or odd (in which case extremely little is known, even for $\\zeta(3)$ .

Further, there’s something slightly more canonical about the fact thatits reciprocal, $\\frac{6}{\\pi^{2}}$ , gives the “probability” (in asuitably-defined sense) that two randomly chosen positive integers arerelatively prime.

#6) Feigenbaum’s constant
- This is the entry onthis the list with which I have the least familiarity. The one thinggoing for it is that it seems to be highly canonical, representingthe limiting ratio of distance between bifurcation intervals for afairly large class of one-dimensional maps. In other words, all mapsthat fall in to this category will bifurcate at the same rate, givingus a glimpse of order in the realm of chaotic systems.

#5) 2
This number caused quite a bit ofcontroversy in discussions leading up to the construction of this list.The question here is canonicality. The first argument of “It’s theonly even prime” is merely a re-wording of “It’s the only primedivisible by 2,” which could uniquely characterizes any prime(e.g. 5 is the only prime divisible by 5, etc.). Of debatablecanonicality is the immensely prevalent notion of “working inbinary.” To a computer scientist, this may seem extremely canonical,but to a mathematician, it may simply be an (not quite) arbitrarychoice of a finite field over which to work.

Yet 2 has some remarkable features even ignoring aspects relating toits primality. For instance, the somewhat canonicalfield of real numbers $\\mathbb{R}$ has index 2 in its algebraicclosure $\\mathbb{C}$ . The factor $2\\pi i$ is prevalent enough incomplex and Fourier analysis that I’ve heard people lament that $\\pi$ should have been defined to be twice its current value. It’s also the only prime number $p$ such that $x^{p}+y^{p}=z^{p}$ has any rational solutions.

Finally, if nothing else, it is certainly the first prime, and couldat least be included for being the first representative of such anamazing class of numbers.

#4) 808017424794512875886459904961710757005754368 $\\times 10^{9}$

The above integer is the size of the monster group, the largestof the sporadic groups. This gives it a relatively high degree ofcanonicality. It’s unclear (at least to me) why there should beany sporadic groups, or why, given that they exist, thereshould only be finitely many. Since there is, however, theremust be something fairly special about the largest possible one.

Also contributing to this number’s rank on this list is the remarkableproperties of the monster group itself, which has been realized(actually, was constructed as) a group of rotations in196,883-dimensional space, representing in some sense a limitto the amount of symmetry such a space can possess.

#3) Euler-Mascheroni Constant, $\\gamma$
One ofthe most amazing facts from elementary calculus is that the harmonicseries diverges, but that if you put an exponent on the denominatorseven just a hair above 1, the result is a convergent sequence.A refined statement says that the partial sums of the harmonic seriesgrow like $\\ln(n)$ , and a further refinement says that the error ofthis approximation approaches our constant:

\\displaystyle\\lim\\limits_{n\\rightarrow\\infty}1+\\frac{1}{2}+\\frac{1}{3}+\\cdots+%\\frac{1}{n}-\\ln(n)=\\gamma.

This seems to represent something fundamental about the harmonic series, andthus of the integers themselves.

Finally, perhaps due to importance inherited from the cruciallyimportant harmonic series, the Euler-Mascheroni constant appearsmagically all over mathematics.

#2) Khinchin’s constant, $K\\approx 2.685452...$
For a real number $x$ , we define a geometric mean function

\\displaystyle f(x)=\\lim\\limits_{n\\rightarrow\\infty}(a_{1}\\cdots a_{n})^{1/n},

where the $a_{i}$ are the terms of the simple continued fractionexpansion of $x$ . By nothing short of a miracle of mathematics, thisfunction of $x$ is almost everywhere (i.e. everywhere except for a setof measure 0) independent of $x$ !!! In other words, except fora “small” number of exceptions, this function $f(x)$ always outputsthe same value, which is called Khinchin’s constant and is denoted by $K$ . It’s hard to impress upon a casual reader just how astoundingthis is, but consider the following: Any infinite collection ofnon-negative integers $a_{0},a_{1},\\ldots$ forms a continued fraction,and indeed each continued fraction gives an infinite collection ofthat form. That the partial geometric means of these sequences isalmost everywhere constant tells us a great deal about thedistribution of sequences showing up as continued fraction sequences,in turn revealing something very fundamental about the structure ofreal numbers.

#1) 163
Well, we’ve come down to it, thisauthor’s humble opinion of the coolest number in existence. Though anunlikely candidate, I hope to show you that 163 satisfies so manyeerily related properties as to earn this title.

I’ll begin with something that most number theorists already knowabout this number – it is the largest value of $d$ such that thenumber field $\\mathbb{Q}(\\sqrt{-d})$ has class number 1, meaning that its ringof integers is a unique factorization domain. The issue offactorization in quadratic fields, and of number fields in general, isone of the principal driving forces of algebraic number theory, and tobe able to pinpoint the end of perfect factorization in the quadraticcase like this seems at least arguably fundamental.

But even if you don’t care about factorization in number fields, theabove fact has some amazing repercussions to more basic number theory.The two following facts in particular jump out:

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$e^{\\pi\\sqrt{163}}$ is within $10^{-12}$ of an integer.
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The polynomial $f(x)=x^{2}+x+41$ has the property that for integers $1\\leq x\\leq 41$ , $f(x)$ is prime.

Both of these are tied intimately (the former using deep properties ofthe $j$ -function, the latter using relatively simple argumentsconcerning the splitting of primes in number fields) to the abovequadratic imaginary number field having class number 1. Further,since $\\mathbb{Q}(\\sqrt{-d})$ is the last such field, the two listedproperties are in some sense the best possible.

Most striking to me, however, is the amazing frequency with which 163shows up in a wide variety of class number problems. In addition tobeing the last value of $d$ such that $\\mathbb{Q}(\\sqrt{-d})$ has class number1, it is the first value of $p$ such that $\\mathbb{Q}(\\zeta_{p}+\\zeta_{p}^{-1})$ (the maximal real subfield of the $p$ -thcyclotomic field) has class number greater than 1. That 163appears as the last instance of a quadratic field having uniquefactorization, and the first instance of a real cyclotomic fieldnot having unique factorization, seems too remarkable to becoincidental. This is (maybe) further substantiated by a couple ofother factoids

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Hasse asked for an example of a prime and an extension such thatthe prime splits completely into divisors which do not lie in acyclic subgroup of the class group. The first such example is any primeless than 163 which splits completely in the cubic field generated bythe polynomial $x^{3}=11x^{2}+14x+1$ . This field has discriminant $163^{2}$ . (See Shanks’ The Simplest Cubic Fields).
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The maximal conductor if an imaginary abelian number field ofclass number 1 corresponds to the field $\\mathbb{Q}(\\sqrt{-67},\\sqrt{-163})$ ,which has conductor $10921=67*163$ .

It is unclear whether or not these additional arithmetical propertiesreflect deeper properties of the $j$ -function or other modular forms,and remains a wide open field of study.

Originally posted on http://math.arizona.edu/ mclemanCam’s homepage