-------------------------------------------------------------- --------------------------------------------------------------------------

Ad

Ad

the ten coolest numbers

by ater




The Ten Coolest Numbers

Cam McLeman

This is an attempt to give a count-down of the top ten coolest numbers. Let's first concede that this is a highly subjective ordering -- one person's 14.38 is another's $ \frac{\pi^2}{6}$ . The astute (or probably simply ``awake'') reader will notice, for example, a definite bias toward numbers interesting to a number theorist in the below list. (On the other hand, who better to gauge the coolness of numbers than a number-theorist...) But who knows? Maybe I can be convinced that I've left something out, or that my ordering should be switched in some cases. But let's first set down some ground rules.

What's in the list?

What makes a number cool? I think a word that sums up the key characteristic of cool numbers is ``canonicality.'' Numbers that appear in this list should be somehow fundamental to the nature of mathematics. They could represent a fundamental fact or theorem of mathematics, be the first instance of an amazing class of numbers, be omnipresent in modern mathematics, or simply have an eerily long list of interesting properties. Perhaps a more appropriate question to ask is the following:

What's not in the list?

There are some really awesome numbers that I didn't include in the list. I'll go through several examples to get a feel for what sorts of numbers don't fit the characteristics mentioned above.

Shocking as it may seem, I first disqualify the constants appearing in Euler's formula $ e^{i\pi}+1=0$ . This was a tough decision. Perhaps these five ($ e$ , $ i$ , $ \pi$ , 1, and 0) belong at the top of the list, or perhaps they're just too fundamentally important to be considered exceptionally cool. Or maybe they're just so cliché'd that we'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, the universe, and everything, 42 is (comparatively) a mathematically uninteresting number. Similarly not included in the list were 867-5309, 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 in construction. For example, the prime constant and Champernowne's constant are both mathematically interesting, but only because they were, at least in an admittedly vague sense, constructed to be as such. Also along these lines are numbers like G63 and Skewe's constant, which while mathematically interesting because of roles they've played in proofs, are not inherently interesting in and of themselves.

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

Honorable Mentions

  • 65,537 - This number is arguably the number with the most potential. It's currently the largest Fermat prime known. If it turns out to be the largest Fermat prime, it might earn itself a place on the list, by virtue of thus also being the largest prime value of $ n$ for which an $ n$ -gon is constructible using only a rule and compass.
  • Conway's constant - The construction of the number can be found here http://mathworld.wolfram.com/ConwaysConstant.html. Though this number has some remarkable properties (not the least of which is being unexpectedly algebraic), it's completely non-canonical construction kept it from overtaking any of our list's current members.
  • 1728 and 1729 - This pair just didn't have quite enough going for them to make it. 1728 is an important $ j$ -invariant of elliptic curves and modular forms, and is a perfect cube. 1729 happens to be the third Carmichael number, but the primary motivation for including 1729 is because of the mathematical folklore associated it to being the first taxicab number, making it more interesting (math-)historically than mathematically.
  • 28 - Aside from being a perfect number, a fairly interesting fact 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 has cardinality 28, and though this might seem a silly feat in and of itself, the fact that sociable numbers and perfect numbers are so closely related may reveal something slightly more profound about 28 than it just being perfect.
  • 4 - The problem with 4 is the difficulty in distinguishing between cool properties of 2 and cool properties of 4. Where, for example, should we include the (trivial, but not uninteresting) relations $ 2^2=2*2=2+2=4$ ? If this were all, there would be no question that 4 doesn't even belong on the Honorable Mentions list,but 4 does have a particularly poignant claim to fame: It is the unique $ n$ such that $ \mathbb{R}^n$ admits more than one differential structure, and indeed admits uncountably many so. The space $ \mathbb{R}^4$ (and 4-dimensional geometry) seems to persistently crop up as a pathology in differential geometry.
  • Chaitin's Constant $ \Omega\approx $ ???. The question marks themselves form part of the reason this constant could be included, $ \Omega$ being a nice example of a number which is definable but not computable. Chaitin's constant can loosely be described as the probability that a Turing machine will halt on a randomly-provided string. There is no doubt that such a constant would represent something fundamental, but there are some unfortunate ambiguities in the definition, largely stemming from the ambiguity in ordering/encoding the set of all Turing machines. Alternate encodings define different constants, and it's difficult to say that any particular encoding is more canonical than any other.

#10) The Golden Ratio, $ \phi$

Thiss was a tough one. Yes, it's cool that it satisfies the property that its reciprocal is one less than it, but this merely reflects that it's a root of the wholly generic polynomial $ x^2-x-1=0$ . Yes, it's cool that it may have an aesthetic quality revered by the Greeks, but this is void from consideration for being non-mathematical. Only slightly less canonical is that it gives the limiting ratio of subsequent Fibonacci numbers. Redeeming it, however, is that this generalizes to all ``Fibonacci-like'' sequences, and is the solution to two sort of canonical operations:

$\displaystyle \phi$ $\displaystyle =\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}$ and $\displaystyle \quad\quad\phi=\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{\ddots}}}}.$

#9) 691

The prime number 691 made it on here for a couple of reasons: First, it's prime, but more importantly, it's the first example of an irregular prime, a class of primes of immense importance in algebraic number theory. (A word of caution: it's not the smallest irregular prime, but it's the one that corresponds to the earliest Bernoulli number, $ B_{12}$ , so 691 is only ``first'' in that sense). It also shows up as a coefficient of every non-constant term in the $ q$ -expansion of the modular form $ E_{12}(z)$ . Further testimony to the arithmetic significance is its seemingly magical appearance in the algebraic $ K$ -theory: It's known that $ K_{22}(\mathbb{Z})$ surjects onto 691.

#8) 78,557

The number 78,557 is here to represent an amazing class of numbers called Sierpinski numbers, defined to be numbers $ k$ such that $ k2^n+1$ is composite for every $ n\geq 1$ . That such numbers exist is flabbergasting...we know from Dirichlet's theorem that primes occur infinitely often in non-trivial arithmetic sequences. Though the sequence formed by $ 78557\cdot 2^n+1$ isn't arithmetic, it certainly doesn't behave multiplicatively either, and there's no apparent reason why there shouldn't be a large (or infinite) number of primes in every such sequence. This notwithstanding, Sierpinski's composite number theorem proves there are in fact infinitely many odd such numbers $ k$ . As a small disclaimer, though it's proven that 78,557 is indeed a Sierpinski number, it is not quite yet known that it is the smallest. There are 17 positive integers smaller than 78,557 not yet known to be non-Sierpinski.

#7) $ \frac{\pi^2}{6}$

Perhaps the first striking this about this number is that it is the sum of the reciprocals of the positive 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 somewhat non-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 is largely interesting for math-historical reasons (it was the first sum of this type that Euler computed), we can at least include it here to represent the amazing array of numbers of the form $ \zeta(n)$ for $ n$ a positive integer. This class of numbers incorporates two amazing and 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 that its reciprocal, $ \frac{6}{\pi^2}$ , gives the ``probability'' (in a suitably-defined sense) that two randomly chosen positive integers are relatively prime.

#6) Feigenbaum's constant $ \delta\approx 4.669201...$

This is the entry on this the list with which I have the least familiarity. The one thing going for it is that it seems to be highly canonical, representing the limiting ratio of distance between bifurcation intervals for a fairly large class of one-dimensional maps. In other words, all maps that fall in to this category will bifurcate at the same rate, giving us a glimpse of order in the realm of chaotic systems.

#5) The Oddest Prime: $ 2$

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

Yet 2 has some remarkable features even ignoring aspects relating to its primality. For instance, the somewhat canonical field of real numbers $ \mathbb{R}$ has index 2 in its algebraic closure $ \mathbb{C}$ . The factor $ 2\pi i$ is prevalent enough in complex 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 could at least be included for being the first representative of such an amazing class of numbers.

#4) The Monster $ \vert M\vert= $ 808017424794512875886459904961710757005754368000000000

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

Also contributing to this number's rank on this list is the remarkable properties of the monster group itself, which has been realized (actually, was constructed as) a group of rotations in 196,883-dimensional space, representing in some sense a limit to the amount of symmetry such a space can possess.

#3) The Euler-Mascheroni Constant, $ \gamma\approx 0.577215\ldots$

One of the most amazing facts from elementary calculus is that the harmonic series diverges, but that if you put an exponent on the denominators even just a hair above 1, the result is a convergent sequence. A refined statement says that the partial sums of the harmonic series grow like $ \ln(n)$ , and a further refinement says that the error of this 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, and thus of integers themselves.

Finally, perhaps due to importance inherited from the crucially important harmonic series, the Euler-Mascheroni constant appears magically all over mathematics. For some idea of $ \gamma$ 's ability to pop up in unforeseen places, see the MathWorld entry on the Euler-Mascheroni constant.

#2) Khinchin's constant, $ K\approx 2.685252\ldots$

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 fraction expansion of $ x$ . By nothing short of a miracle of mathematics, this function of $ x$ is almost everywhere (i.e. everywhere except for a set of measure 0) independent of $ x$ !!! In other words, except for a ``small'' number of exceptions, this function $ f(x)$ always outputs the same value, dubbed Khinchin's constant and denoted by $ K$ . It's hard to impress upon a casual reader just how astounding this is, but consider the following: Any infinite collection of non-negative integers $ a_0, a_1, \ldots$ forms a continued fraction, and indeed each continued fraction gives an infinite collection of that form. That the partial geometric means of these sequences is almost everywhere constant tells us a great deal about the distribution of sequences showing up as continued fraction sequences, in turn revealing something very fundamental about the structure of real numbers.

#1) 163

Well, we've come down to it, this author's humble opinion of the coolest number in existence. Though a seemingly unlikely candidate, I hope to show you that 163 satisfies so many eerily related properties as to earn this title.

I'll begin with something that most number theorists already know about this number - it is the largest value of $ d$ such that the number field $ \mathbb{Q}(\sqrt{-d})$ has class number 1, meaning that its ring of integers is a unique factorization domain. The issue of factorization in quadratic fields, and of number fields in general, is one of the principal driving forces of algebraic number theory, and to be able to pinpoint the end of perfect factorization in the quadratic imaginary case like this seems at least arguably fundamental.

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

  • $ e^{\pi\sqrt{163}}$ is within $ 10^{-13}$ of an integer.
  • 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 of the $ j$ -function, the latter using relatively simple arguments concerning the splitting of primes in number fields) to the above quadratic imaginary number field having class number 1. Further, since $ \mathbb{Q}(\sqrt{-d})$ is the last such field, the two listed properties are in some sense the best possible. Along a similar vein, $ p=163$ is the largest prime such that there exists an elliptic curve $ E$ over $ \mathbb{Q}$ with an isogeny of degree $ p$ , which in turn makes $ N=163$ the last $ N$ such that the ``modular curve" $ Y_0(N)$ has $ Y_0(N)(\mathbb{Q})$ being non-empty.

Most striking to me, however, is the amazing frequency with which 163 shows up in a wide variety of class number problems. In addition to being the last value of $ d$ such that $ \mathbb{Q}(\sqrt{-d})$ has class number 1, it is the first value of $ p$ such that $ \mathbb{Q}(\zeta_p+\zeta_p^{-1})$ (the maximal real subfield of the $ p$ -th cyclotomic field) has class number greater than 1. That 163 appears as the last instance of a quadratic field having unique factorization, and the first instance of a real cyclotomic field not having unique factorization, seems too remarkable to be coincidental. This is (maybe) further substantiated by a couple of other factoids:

  • Hasse asked for an example of a prime and an extension such that the prime splits completely into divisors which do not lie in a cyclic subgroup of the class group. The first such example is any prime less than 163 which splits completely in the cubic field generated by the polynomial $ x^3=11x^2+14x+1$ . This field has discriminant $ 163^2$ . (See Shanks' The Simplest Cubic Fields).
  • The maximal conductor of an imaginary abelian number field of class number 1 corresponds to the field $ \mathbb{Q}(\sqrt{-67},\sqrt{-163})$ , which has conductor $ 10921=67*163$ .
It is unclear the extent to which these additional arithmetical properties reflect deeper properties of the $ j$ -function or other modular forms, and remains a wide open field of study.

Filed under having  

0 comments:

Post a Comment

Subscribe to: Post Comments (Atom)

Ad

Share