top ten coolest numbers

top ten coolest numbers

This is an attempt to give a count-down of the

top ten coolest
numbers

. Let’s first admit that this is a highly subjective
ordering–one person’s 14.38 is another’s

π

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
“canonicity”. 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

⁢

π

+

1

=

0

. This was a tough decision. Perhaps
these five (

e

,

i

,

π

, 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 a comparatively uninteresting number. Similarly not included in the list were
876-5309, 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 Champoleon’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 odd
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.

•

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

,

ϕ

This 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

:

1

1

+

1

1

+

1

1

+

1

⋱

and

1

+

1

+

1

+

⋯

Also, this number plays an important role in the hstory of

algebraic
number theory

. The field it generates is the first known example of a
field in which unique factorization fails. Trying to come to grips with
this fact led to the invention of ideal theory, class nubers, etc.

#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

⁢

(

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

k

⁢

2

n

+

1

is

composite

for

every

n

≥

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

⋅

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)

π

2

6

Perhaps the first striking
this about this number is that it is the sum of the reciprocals of the
positive integer squares:

1

+

1

4

+

1

9

+

⋯

+

1

n

2

+

⋯

=

π

2

6

.

Though the choice of

2

here for the

exponent

is somewhat
non-canonical (i.e. we’ve just noted that

ζ

⁢

(

2

)

=

π

2

6

,
where

ζ

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

ζ

⁢

(

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

ζ

⁢

(

n

)

is known to be a

rational

multiple

of

π

n

) or odd (in which case extremely little is known, even for

ζ

⁢

(

3

)

.

Further, there’s something slightly more canonical about the fact that
its reciprocal,

6

π

2

, gives the “probability” (in a
suitably-defined sense) that two randomly chosen positive integers are
relatively prime.

#6) Feigenbaum’s constant

- 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) 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

ℝ

has index 2 in its

algebraic
closure

ℂ

. The factor

2

⁢

π

⁢

i

is prevalent enough in
complex and

Fourier analysis

that I’ve heard people lament that

π

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) 808017424794512875886459904961710757005754368

×

10

9

The above

integer

is the size of the

monster group

, 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)

Euler-Mascheroni Constant

,

γ

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:

lim

n

→

∞

⁡

1

+

1

2

+

1

3

+

⋯

+

1

n

-

ln

⁡

(

n

)

=

γ

.

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

Finally, perhaps due to importance inherited from the crucially
important harmonic series, the Euler-Mascheroni constant appears
magically all over mathematics.

#2) Khinchin’s constant,

K

≈

2.685452

⁢

…

For a

real number

x

, we define a

geometric mean

function

f

⁢

(

x

)

=

lim

n

→

∞

⁡

(

a

1

⁢

⋯

⁢

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, which is called Khinchin’s constant and is 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

,

…

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 an
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

ℚ

⁢

(

-

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
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

π

⁢

163

is within

10

-

12

of an integer.

•

The

polynomial

f

⁢

(

x

)

=

x

2

+

x

+

41

has the property that for integers

1

≤

x

≤

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

ℚ

⁢

(

-

d

)

is the

last

such field, the two listed
properties are in some sense the best possible.

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

ℚ

⁢

(

-

d

)

has class number
1, it is the

first

value of

p

such that

ℚ

⁢

(

ζ

p

+

ζ

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

=

11

⁢

x

2

+

14

⁢

x

+

1

. This field has

discriminant

163

2

. (See Shanks’

The Simplest Cubic Fields

).

•

The maximal

conductor

if an

imaginary

abelian number field

of
class number 1 corresponds to the field

ℚ

⁢

(

-

67

,

-

163

)

,
which has conductor

10921

=

67

*

163

.

It is unclear whether or not these additional arithmetical properties
reflect 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/ mcleman

Cam’s homepage

Title

top ten coolest numbers

Canonical name

TopTenCoolestNumbers

Date of creation

2013-03-22 15:38:03

Last modified on

2013-03-22 15:38:03

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

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Author

rspuzio (6075)

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