

INTERESTING NUMBERS The interesting
number paradox is a paradox that arises from attempting to classify
natural numbers as "interesting" or "dull". The paradox states that all
natural numbers are interesting. The "proof" is by contradiction.
If there were uninteresting numbers, there would be a smallest
uninteresting number But the smallest uninteresting number is itself
interesting by being so, producing a contradiction.
Leopold Kronecker said: " God himself made the integers: everything else is
the work of man".
1729 is also known as the HardyRamanujan
number. Godfrey Hardy, an English mathematician was visiting Srinivasa
Ramanujan in a hospital when Ramanujan told him about this number.
Here are Hardy's words: I remember
once going to see him when he was ill at Putney. I had ridden in taxi cab
number 1729 and remarked that the number seemed to me rather a dull one, and
that I hoped it was not an unfavorable omen. "No," he replied, "it is a very
interesting number; it is the smallest number expressible as the sum of two
cubes in two different ways." Notice that 1729 = 1^3 + 12^3 =
9^3 + 10^3.
Twin Primes It is interesting that it cannot
be proven that there are or there are not an infinite number of "twin
primes"
Similarly the Goldbach Conjecture states:
Every
even
integer greater than 2 can be expressed as
the sum of two
primes. The conjecture has been shown to
hold for all integers less than 4 × 10^{18} but remains unproven
despite considerable effort.
The even integers from 4 to 28 as sums of two primes: Even integers
correspond to horizontal lines. For each prime, there are two oblique lines,
one red and one blue. The sums of two primes are the intersections of one
red and one blue line, marked by a circle. Thus the circles on a given
horizontal line give all partitions of the corresponding even integer into
the sum of two primes.
111,111,111 x 111,111,111 = 12345678987654321
Some of the following comes from David Wells book: "The Penguin Dictionary
of Curious and Interesting Numbers". Also examine Patrick de
Geest's website:
http://www.worldofnumbers.com/
See more on numbers at
https://www.mathgoodies.com/articles/numbers
1 This is e raised to i
times pi.
The Euler formula, sometimes also called the Euler identity and can be
derived from
The special case of the formula with gives
the beautiful identity


This equation connects the fundamental numbers i, pi, e,
1, and 0 (zero),
the fundamental operations , ,
and exponentiation, the most important relation ,
and nothing else. Gauss is reported to have commented that if this formula
was not immediately obvious, the reader would never be a firstclass
mathematician.
The Euler formula can also be demonstrated using a series expansion
It can also be demonstrated using a complex integral.
Let
so
A mathematical joke asks, How many mathematicians does it take to change a
light bulb?" and answers "
(which, of course, equals 1.
Also see
https://www.youtube.com/watch?v=yPl64xi_ZZA
0 (zero) It started its life as a space on a counting board. Medieval
mathematicians could not decide if it was a number. It originated in
India as an extra numeral.
0.01123595505... equals 1/89. This is related to the
Fibonacci series. 89 is a Fibonacci number and a Fibonacci prime.
The decimal expansion of its reciprocal has the first six Fibonacci numbers:
0,1,1,2,3,5
Think of the Fibonacci series as being a
sequence of decimal fractions, arranged so the right most digit of the
nth Fibonacci number is in the n+1th decimal place. Then
add: .01
.001
.0002
.00003
.000005
.0000008
.00000013
.000000021
.0000000034
.00000000055
.000000000089
.0000000000144
.
.
+ .

.01123595505...
Proof at http://www.geom.uiuc.edu/~rminer/1over89/1over89proof.html 
0.1234567891011121314... The
Champernowne constant C_{10} is
a
transcendental
real
constant whose decimal expansion has
important properties. It is named after
mathematician
D. G. Champernowne..
For
base 10, the number is defined by
concatenating representations of successive integers:
 C_{10} =
0.12345678910111213141516… Champernowne constants can also
be constructed in other bases, similarly, for example:
C_{2} = 0.11011100101110111… _{2
}C_{3} = 0.12101112202122… _{3}.
The Champernowne constant can be expressed exactly as an
infinite series:

and this series generalizes to arbitrary bases b
by replacing 10 and 9 with
b and
b − 1 respectively.
0.20788... is i to the ith power. If you are familiar
with
complex numbers, the "imaginary" number i
has the property that the square of i is 1. It is a rather curious fact
that i raised to the ith power is actually a real number.
From
Euler's formula, we know that exp(i*x) =
cos(x) + i*sin(x), where "exp(z)" is the exponential function e^{z}.
Then
exp(i*Pi/2) = cos(Pi/2) + i*sin(Pi/2) = i.
Raising both sides to ith power, we see that the right side is the desired
quantity i^{i}, while the left side becomes exp(i*i*Pi/2), or
exp(Pi/2), which is approximately .20788.
This is one of many possible values for i to the i, because,
for instance, exp(5i*Pi/2) = i. In
complex analysis, one learns that
exponentiation with respect to i is a multivalued function.
0.57721566... The Euler–Mascheroni constant (also
called Euler's constant) is a
mathematical constant recurring in
analysis and
number theory, usually denoted by the
lowercase Greek letter gamma
().
It is defined as the
limiting difference between the
harmonic series and the
natural logarithm:

Here,
represents the
floor function.
The numerical value of the Euler–Mascheroni constant, to 50 decimal
places, is
 0.57721566490153286060651209008240243104215933593992
1 (One) The Greeks did not consider 1 to be a number. It is not
defined as a prime number.
The Collatz conjecture is a
conjecture in
mathematics named after
Lothar Collatz, who first proposed it in
1937. The conjecture is also known as the 3n + 1 conjecture,
the Ulam conjecture (after
Stanisław Ulam), Kakutani's problem
(after
Shizuo Kakutani), the Thwaites
conjecture (after Sir Bryan Thwaites), Hasse's algorithm (after
Helmut Hasse), or the Syracuse problem;
the sequence of numbers involved is referred to as hailstone
numbers (because the values are usually subject to multiple descents and
ascents like hailstones in a cloud), or as wondrous numbers.
Take any
natural number n. If n is
even, divide it by 2 to get n / 2. If n is odd, multiply it by
3 and add 1 to obtain 3n + 1. Repeat the process indefinitely.
The conjecture is that no matter what number you start with, you will always
eventually reach 1. The property has also been called oneness.
Paul Erdős said, allegedly, about the
Collatz conjecture: "Mathematics is not yet ripe for such problems." He also
offered $500 for its solution. Use a calculator at
http://plus.maths.org/content/mathematicalmysterieshailstonesequences
The onewoman town of
Monowi, Nebraska is the only
officially incorporated municipality with a population of 1. The sole,
83yearold resident is the city's mayor, librarian, and bartender.
1.41421356237309504880168872420969807856967187537694807317667973799...
= Square root of 2.
The square root of 2 is equal to the length of the
hypotenuse of a
right triangle with legs of
length 1It was probably the first number known to be
irrational.
The numerical value for the square root of two, truncated
to 65
decimal places, is:
 1.41421356237309504880168872420969807856967187537694807317667973799,
See
https://en.wikipedia.org/wiki/Square_root_of_2 and
https://www.youtube.com/watch?v=6ymTZEeTjI8
1.5 Is the average of 0 and 1, See
https://www.youtube.com/watch?v=0Oazb7IWzbA
1.618030339887... This is the Golden Ratio. Its value
is:
The Fibonacci numbers are the sequence of numbers
defined by the
linear recurrence equation
with
.
It is conventional to define
.
The Fibonacci numbers for
,
2, ... are 1, 1, 2, 3, 5, 8, 13, 21, ...
Johannes Kepler discovered that as n
increases, the ratio of the successive terms of the Fibonacci sequence {F_{n}}
approaches the
golden ratio.
. In 1765,
Leonhard Euler published an explicit
formula, known today as the
Binet formula,

It demonstrates that the Fibonacci numbers grow at an exponential rate
equal to the golden ratio φ.
The random Fibonacci sequence is a stochastic analogue of the
Fibonacci sequence defined by the
recurrence relation f_{n}
= f_{n−1} ± f_{n−2}, where the
signs + or − are chosen
at random with equal probability 1/2,
independently for different n. In
1960,
Hillel Furstenberg and
Harry Kesten showed that for a general
class of random
matrix products, the
norm grows as λ^{n},
where n is the number of factors. Their results apply to a broad
class of random sequence generating processes that includes the random
Fibonacci sequence
In 1999,
Divakar Viswanath showed that the growth
rate of the random Fibonacci sequence is equal to 1.13198824…, a
mathematical constant that was later named
Viswanath's constant. As a consequence, the nth root of f_{n}
converges to a constant value
almost surely, or with probability one:

2 (Two) raised to the 5th power times 9 raised to the 2nd power = 2592
"A man is a person who will pay two
dollars for a onedollar item he wants. A woman will pay one dollar
for a twodollar item she doesn't want..."  William Binger
2.0663615...
This is
the square root of one half the product of
π and e. Ramanujan showed this equals:
1 + 1/3 + 1/3*5 + 1/3*5*7 + 1/3*5*7*9 + ...
 It is not known if this is transcendental.
2.71828.... is approximately
equal to e is
an important
mathematical constant that is the base of
the
natural logarithm. It is the
limit of (1 + 1/n)^{n}
as n
approaches infinity, an expression that arises in the study of
compound interest. It can also be
calculated as the sum of the infinite
series:

In 1737 Euler proved that e is irrational. In 1873
Hermite proved that e was transcendental.
It is not known if
or
is
irrational.
The number e
occurs naturally in connection with many problems involving
asymptotics. A prominent example is
Stirling's formula for the
asymptotics of the
factorial function, in which both the
numbers e and
π enter:

A particular consequence of this is

.

The letter e was first publicly used to represent the number
2.71828... in a letter from Euler to Goldbach in 1731.
See
http://mathworld.wolfram.com/e.html for many formulas.
The constant is
base of the natural
logarithm. is
sometimes known as Napier's constant, although its symbol ()
honors Euler.
is
the unique number
with the property that the area of the region bounded by the hyperbola ,
the xaxis,
and the vertical lines and is
1. In other words,

(1)

With the possible exception of , is
the most important constant in mathematics since it appears in myriad
mathematical contexts involving limits and derivatives.
The numerical value of is

(2)

can
be defined by the limit

(3)

(illustrated above), or by the infinite series

(4)

as first published by Newton (1669; reprinted in Whiteside 1968, p. 225).
is
given by the unusual limit

(5)

(Brothers and Knox 1998).
Euler (1737; Sandifer 2006) proved that is irrational by
proving that has
an infinite simple continued fraction (;
Nagell 1951), and Liouville proved in 1844 that does
not satisfy any quadratic
equation with integral coefficients (i.e.,
if it is algebraic,
it must be algebraic of degree greater
than 2). Hermite subsequently settled the issue, proving to
be transcendental in
1873. However, is
the "least" transcendental possible, with irrationality
measure .
Sondow (2006) proved that is
irrational using a construction for as
the intersection of a nested sequence of closed intervals. This method also
provides a measure of irrationality in terms of the Smarandache
function (denoted here as instead
of the conventional in
order to avoid confusion with the irrationality
measure) by showing that if and are
any integers with ,
then

(6)

It is not known if or is irrational.
It is known that and do
not satisfy any polynomial equation
of degree with integer coefficients of
average size (Bailey
1988, Borwein et al. 1989), but it is not known if either of these is
transcendental.
It is not known if is
normal to any base (Stoneham 1970).
has
the series representation

(7)

as well as
The special case of the Euler
formula

(15)

with gives
the beautiful identity

(16)

an equation connecting the fundamental numbers i, pi, ,
1, and 0 (zero)
and involving the fundamental operations of equality (), addition (), multiplication (),
and exponentiation.
A nested series for can
be obtained by rewriting the series (2) for as
which gives a pretty nested
radical result when is
taken to the power of both sides.
An unexpected Wallislike
formula for is
given by the Pippenger
product

(20)

(OEIS A084148 and A084149;
Pippenger 1980). Another product for given
by

(21)

due to Guillera (Sondow 2006). This is analogous to the products

(22)

and

(23)

(Guillera and Sondow 2005, Sondow 2006).
Using the recurrence
relation

(24)

with ,
compute
2.9951... The Gelfond–Schneider constant or
Hilbert number is

which was proved to be a
transcendental number by
Rodion Kuzmin in 1930. In 1934,
Aleksandr Gelfond proved the more general
Gelfond–Schneider theorem, which solved
the part of
Hilbert's seventh problem described below.
Part of the seventh of
Hilbert's twenty three problems posed in
1900 was to prove (or find a counterexample to the claim) that a^{b}
is always transcendental for algebraic a ≠ 0, 1 and irrational
algebraic b. In the address he gave two explicit examples, one of
them being the Gelfond–Schneider constant 2^{√2}.
In 1919, he gave a lecture on
number theory and spoke of three
conjectures: the
Riemann hypothesis,
Fermat's Last Theorem, and the
transcendence of 2^{√2}. He mentioned to the audience that he didn't
expect anyone in the hall to live long enough to see a proof of this final
result. But the proof of this number's transcendence was published by Kuzmin
in 1930,^{
}well within
Hilbert's own lifetime. Namely, Kuzmin
proved the case where the exponent b is a real
quadratic irrational, which was later
extended to an arbitrary algebraic irrational b by Gelfond.
3 (Three)
The
only three consecutive integers whose cubes sum to a cube are given by the
Diophantine equation:
3 cubed + 4 cubed + 5 cubed = 6 cubed.
"Cubes": 153, 370, 371 and 407 are all the "sum of
the cubes of their digits". In other words 153=1^{3}+5^{3}+3^{3}
A circle can be drawn through three points that are not colinear.
3.14159... called
π, is the
ratio of a
circle's
circumference to its
diameter, approximately equal to 3.14159.
It is an irrational number. Like all irrational numbers,
π cannot be represented as a
common fraction.
Every irrational
number, including π, can be represented by an
infinite series of nested fractions, called a
continued fraction:

When Euler solved the
Basel problem (See
https://en.wikipedia.org/wiki/Basel_problem) in 1735, finding the exact
value of the sum of the reciprocal squares, he established a connection
between π and the
prime numbers that later contributed to
the development and study of the
Riemann zeta function:

Swiss scientist
Johann Heinrich Lambert in 1761
proved that π is
irrational, meaning it is not
equal to the quotient of any two whole number
Lambert's proof exploited a
continuedfraction representation of the tangent function. French
mathematician
AdrienMarie Legendre proved in 1794 that
π^{2} is also irrational. In 1882,
German mathematician
Ferdinand von Lindemann proved
that π is
transcendental, confirming a
conjecture made by both
Legendre and Euler.
Other series involving
π are found at:
http://mathworld.wolfram.com/PiFormulas.html Here a
very few:
=
=
discovered by Leibnitz in 1673.
=
=
=
=
=
=
=
=
=
was determined by Isaac Newton in 1666.
=
=
=
where
is the
golden ratio.
The first 16 million digits of pi have passed all the tests of being random.
Also look at
http://numbers.computation.free.fr/Constants/Pi/piSeries.html and
https://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80
Bellard's formula, as used by
PiHex, the nowcompleted
distributed computing project, is used
to calculate the nth digit of
π in
base 2. It is a faster version (about
43% faster^{[1]})
of the
Bailey–Borwein–Plouffe formula.
It was discovered by
Fabrice Bellard in 1997.
See
http://en.wikipedia.org/wiki/Category:Pi
Pi is also a
transcendental number, which means it cannot be the root of an algebraic
equation.
The probability that an
integer does not contain any squared factors, and the probability that two
numbers chosen at random are relatively prime is π^{2}/6
is about 61%.
If ζ refers to the
Riemann zeta function, the identity
relating the product over primes to ζ(2) is an example of an
Euler product, and the evaluation of ζ(2)
as π^{2}/6 is the
Basel problem, solved by
Leonhard Euler in 1735.
More generally, the probability of k randomly chosen integers
being coprime is 1/ζ(k).
Relatively Prime
Two integers are relatively prime if they share no common
positive factors (divisors) except 1. Using the notation
to denote the
greatest common divisor, two
integers
and
are relatively prime if
.
Relatively prime integers are sometimes also called
strangers or coprime and are
denoted
.
The plot above plots
and
along the two axes and colors a square black if
and white otherwise (left figure) and simply colored according
to
(right figure).
Two numbers can be tested to see if they are relatively prime
in the
Wolfram Language using
CoprimeQ[m, n].
Two distinct primes
and
are always relatively prime,
,
as are any positive integer powers of distinct primes
and
,
.
Relative primality is not
transitive. For example,
and
,
but
.
The probability that two
integers
and
picked at random are relatively prime is

(1)

(OEIS
A059956; Cesàro and Sylvester
1883; Lehmer 1900; Sylvester 1909; Nymann 1972; Wells 1986,
p. 28; Borwein and Bailey 2003, p. 139; Havil 2003, pp. 40 and
65; Moree 2005), where
is the
Riemann zeta function. This
result is related to the fact that the
greatest common divisor of
and
,
,
can be interpreted as the number of
lattice points in the
plane which lie on the straight
line connecting the
vectors
and
(excluding
itself). In fact,
is the fractional number of
lattice points
visible from the
origin (Castellanos 1988,
pp. 155156).
Given three
integers
chosen at random, the probability that no common factor will
divide them all is

(2)

(OEIS
A088453; Wells 1986, p. 29),
where
is
Apéry's constant (Wells 1986,
p. 29). In general, the probability that
random numbers lack a
th
power common divisor is
(Cohen 1959, Salamin 1972, Nymann 1975, Schoenfeld 1976,
Porubský 1981, Chidambaraswamy and Sitaramachandra Rao 1987,
Hafner et al. 1993).
Interestingly, the probability that two
Gaussian integers
and
are relatively prime is

(3)

(OEIS
A088454), where
is
Catalan's constant (Pegg;
Collins and Johnson 1989; Finch 2003, p. 601).
Similarly, the probability that two random
Eisenstein integers are
relatively prime is

(4)

(OEIS
A088467), where

(5)

(Finch 2003, p. 601), which can be written analytically as
where
is the
trigamma function
Amazingly, the probabilities for random pairs of integers and
Gaussian integers being relatively prime are the same as the
asymptotic densities of
squarefree integers of these
types.

See
https://www.youtube.com/watch?v=XfGesUai7Fs4 and
https://www.youtube.com/watch?v=vbl5NWt19E
The Feynman point is a sequence of six 9s that begins at the 762nd
decimal place of the
decimal representation of
π. It is named after physicist
Richard Feynman, who once stated during a
lecture he would like to memorize the digits of π
until that point, so he could recite them and quip "nine nine nine nine nine
nine and so on", suggesting, in a
tongueincheek manner, that
π is
rational.
Which is a better fit: a round peg in a square hole or a square peg in a
round hole? The ratios are:
π/2 and 2/π .As π squared is greater than
8, the round peg is a better fit and fills more of the square hole.
"Pi Day" is celebrated on March 14. "Feigenbaum
Constant Day" is celebrated on April 6 (or April 7 if you round up)
Buffon's Needle Problem involves pi. . See
http://mste.illinois.edu/reese/buffon/buffon.html and
http://www.mathsisfun.com/activity/buffonsneedle.html
http://pi314.net/eng/euler.php
additional formulas by Euler and note the famous formula
for
/4 equals (3x5x7x11x13x17x19x ...)/(4x4x8x12x12x16 ,,,)
where the numerator consists of the odd prime numbers and the
numbers in the denominator are multiples of 4 that are closest
to the corresponding number in the numerator. 
It can shown that Pi is irrational. Also see
https://en.wikipedia.org/wiki/Proof_that_%cf%80_is_irrational
However, John Arioni's
simple proof of the infinitude of primes uses the fact that Pi/4 can be expressed
as a product of two series each involving primes. John observed that the
identity implies infinitude of primes, because had the number of primes been
finite, the righthand side would have been rational, while the left side
Pi/4 is known to be irrational.
John Wallis (1655) took what can now be expressed as
and without using the binomial theorem or integration (not
invented yet) painstakingly came up with a formula for
to be
.
William Brouncker (ca. 1660's) rewrote Wallis' formula as a
continued fraction, which Wallis and later Euler (1775) proved to be
equivalent. It is unknown how Brouncker himself came up with the
continued fraction,
.
James Gregory (1671) & Gottfried Leibniz (1674) used the series
expansion of the arctangent function,
,
and the fact that arctan(1) =
/4
to obtain the series
.
Unfortunately, this series converges to slowly to be useful, as
it takes over 300 terms to obtain a 2 decimal place precision. To
obtain 100 decimal places of
,
one would need to use at least 10^50 terms of this expansion!
History books credit Sir Isaac Newton (ca. 1730's) with using
the series expansion of the arcsine function,
,
and the fact that arctan(1/2) =
/6
to obtain the series
.
This arcsine series converges much faster than using the
arctangent. (Actually, Newton used a slightly different expansion in
his original text.) This expansion only needed 22 terms to obtain 16
decimal places for
.
Leonard Euler (1748) proved the following equivalent relations
for the square of
,
Ko Hayashi (1989) found another
infinite expression for
in terms of the Fibonacci numbers,
.

4 One half of five is four,
as one half of five is iv. Write any number from 1 to
100 using four 4's: (See
http://mathforum.org/ruth/four4s.puzzle.html) A hyp erbola can be drawn
though four points, three of which are not colinear. In 1878 Francis Guthrie
said a map could be colored with at most four colors so no abutting
country had the same color. This 4 color conjecture has been "proven"
correct nontraditionally using a computer. Every integer can be represented
as the sum of at most four squares.
4.11325037878... This is
e
raised to the square root of 2.
4.6692... Some mathematical
mappings involving a single linear parameter exhibit the apparently random
behavior known as chaos when the parameter lies within certain ranges. As
the parameter is increased towards this region, the mapping undergoes
bifurcations at precise values of the
parameter. At first there is one stable point, then bifurcating to an
oscillation between two values, then bifurcating again to oscillate between
four values and so on. In 1975, Dr. Feigenbaum, using the small calculator
he had been issued, discovered that the ratio of the difference between the
values at which such successive
perioddoubling bifurcations occur tends to
a constant of around 4.6692... He was able to provide a mathematical proof
of that fact, and he then showed that the same behavior, with the same
mathematical constant, would occur within a wide class of mathematical
functions, prior to the onset of chaos. For the first time, this universal
result enabled mathematicians to take their first steps to unraveling the
apparently intractable "random" behavior of chaotic systems. This "ratio of
convergence" is now known as the first
Feigenbaum constant which equals
4.6692...
And the ratio between the values where the period doubles ends up
approaching the Feigenbaum constant, approximately 4.669.
So, should we celebrate Feigenbaum Constant Day on
April 6th or 7th? Due to rounding, it's probably the 7th, but feel free to celebrate both.
Bifurcation diagram of the logistic map below. Feigenbaum noticed in 1975
that the quotient of successive distances between bifurcation events tends
to 4.6692... This is the first Feigenbaum constant.
The second is 2.5029.... described later. See
http://en.wikipedia.org/wiki/Feigenbaum_constants,
http://en.wikipedia.org/wiki/Chaos_theory,
http://en.wikipedia.org/wiki/Nonlinear_dynamics, and especially
http://en.wikipedia.org/wiki/Mandelbrot_set
The first Feigenbaum constant is the limiting
ratio of each bifurcation interval
to the next between every
period doubling, of a oneparameter
map

where f(x) is a function parameterized by the
bifurcation parameter a.
It is given by the
limit:

where a_{n} are discrete values of a at the nth
period doubling.
We can make some extraordinary observations from the bifurcation diagram.
The ratio between each bifurcation converges to the first Feigenbaum’s constant,
a universal constant for functions that have a periodicdoubling route to
chaos, a fact discovered by Mitchell Feigenbaum in 1975. This constant is
approximately 4.669... .. We can interpret from this constant that as
we approach chaos each periodic region is smaller than the previous region
by a factor of 4.669 (the first Feigenbaum constant).
The extraordinary thing about this constant is that its the same for all
unimodal functions (functions having a quadratic maximum and approach chaos
via period doubling). If we call
the value of r for which the
period becomes unstable then
.
2.5029... is the second Feigenbaum constant.
The following functional equation also arises in the study of
onedimensional maps that, as a function of a parameter, go through a
perioddoubling cascade. The functional equation is the mathematical
expression of the
universality of period doubling. The
equation is used to specify a function g and a parameter λ by
the relation

with the boundary conditions
 g(0) = 1, 
 g′(0) = 0, and 
 g′′(0) < 0 
For a particular form of solution with a quadratic dependence of the
solution near x=0, the inverse 1/λ=2.5029... is the
second
Feigenbaum constant. In
bifurcation theory, the Feigenbaum
constants are two
mathematical constants which both express
ratios in a
bifurcation diagram for a nonlinear
map. They are named after the mathematician
Mitchell Feigenbaum.
The second Feigenbaum constant
2.502907875095892822283902873218..., is the ratio between the width of
a
tine and the width of one of its two
subtines (except the tine closest to the fold). A negative sign applied to
when the ratio between the lower subtine and the width of the tine is
measured. (In
chaos theory (
nonlinear dynamics), the branches of a
bifurcation diagram are called tines
and subtines.)
The Feigenbaum constants apply to a large class of
dynamical systems. Both constants are
believed to be
transcendental.
5 Five is the first prime of the form 6n 
1. In fact all primes, except 2 and 3, are one more or less than a
multipe of 6. Pappus showed how to construct a conic through any five
points in the plane, no three of which are colinear.
Is also a pentagonal number. A pentagonal number is a figurate
number that extends the concept of triangular and square
numbers to the pentagon,
but, unlike the first two, the patterns involved
in the construction of pentagonal numbers are not rotationally
symmetrical. The nth pentagonal number p_{n} is
the number of distinct dots in a pattern of dots consisting of the outlines of
regular pentagons with sides up to n dots, when the pentagons are overlaid
so that they share one vertex.
For instance, the third one is formed from outlines comprising 1, 5 and 10
dots, but the 1, and 3 of the 5, coincide with
3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2
inside.
p_{n} is
given by the formula: (n)(3n  1)/2
for n ≥ 1. The first few pentagonal numbers are:
1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247,
287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001,
1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147,
2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725,
3876, 4030, 4187..
6
The problem of finding two rational numbers whose cubes sum to
six was "proved" impossible by Legendre. However, Dudeney found
the simple solutions 17/21 and 37/21. Did you know that snowflakes have six
sides. Six equals the square root of the sum of the cubes of 1 and 2
and 3. There are 6 regular polytopes. They are analogous
in four dimensions of the regular polyhedra in three dimensions and the
regular polygons in two dimensions.
7 Did you know the 10 millionth digit of
π is a 7? So what!
Seven is a prime. See the following table by Martin Gardner, which
contains all primes. Start with the top one and peel off right hand
digits.
73939133 
7393913 
739391 
73939 
7393 
739 
73 
7 
The "Flower of Life"is a name for a
geometrical figure composed of multiple
evenlyspaced, overlapping
circles. This figure, used as a
decorative motif since ancient times, forms a
flowerlike pattern with the
symmetrical structure of a
hexagon. A "Flower of Life"
figure consists of seven overlapping circles, in which the center
of each circle is on the circumference of up to six surrounding circles
of the same diameter. However, the surrounding circles need not be
clearly or completely drawn; in fact, some ancient symbols that are
claimed as examples of the Flower of Life contain only a single circle
or hexagon.
The "Seed of Life" is formed from seven circles being
placed with sixfold symmetry, forming a pattern of circles and lenses,
which act as a basic component of the Flower of Life's design. 

8.53973422267... This is
π times
e. Most sums, products, powers of the number π and
the
number e are not known to be rational, algebraic irrational or
transcendental ( π + e, π − e, πe, π/e, π^{π},
e^{e}, π^{e}, π^{√2},
e^{π2} ). However, π + e^{π},
πe^{π} and e^{π√n}
,for any positive integer n, have been proven to be transcendental.
9
Dudeney found two rational numbers other than 1 and 2 whose
cubes sum to nine:
[415280564497 / 348671682660] and
[676702467503 / 348671682660]
Nine is the maximum number of cubes that are needed to sum to any
positive integer. Nineteen is the maximum number of fourth
powers needed to sum to any positive integer.
A number is divisible by 9 if, and only if, if it divides the sum of the
number's digits.
9.8960440189358 ... This is
π
squared. In 1794, Legendre proved this was irrational.
10 The base of the decimal system. 10! =6! x 7! These are the only
consecutive integers, 6 and 7, that solve the equation N! = A! x (A+1)!
11 British mathematician J J Sylvester said:
"Mathematics is the music of reason." In 1884 at age 70, he proved
that the highest number that cannot be created from using two numbers x
and y equals xy  x  7. In Rugby, where drop goals are
scored as 3 and converted tries are scored as 7, there cannot ever be a
score of 11.
That is 3x7  3  7 = 11.
11 is a palindrome, a number that can be read forward and backward.
But so are some of their squares.
Dattaraya Ramchandra Kaprekar (1905–1986) was an
Indian
recreational mathematician who described
several
classes of natural numbers including the
Kaprekar,
Harshad and
Self numbers and discovered the
Kaprekar constant, named after him. Despite
having no formal postgraduate training and working as a schoolteacher, he
published extensively and became well known in recreational mathematics
circles. .International fame arrived when
Martin Gardner wrote about Kaprekar in his
March 1975 column of Mathematical Games for
Scientific American. Today his name is
wellknown and many other mathematicians have pursued the study of the
properties he discovered.
Kaprekar also studied the
Demlo numbers, named after a train
station 30 miles from Bombay on the then
G. I. P. Railway where he had the idea of
studying them.^{
}These are the numbers 1, 121, 12321, …, which are the
squares of the
repunits 1, 11, 111,
11 is the second repunit, a number whose digits are all 1s.
This works for n =2 and for some higher n's:
n 
11 to the nth 
111 to the nth 
1111 to the nth 
11111 to the nth 
111111 to the nth 
1 
11 
111 
1111 
11111 
111111 
2 
121 
12321 
1234321 
123454321 
12345654321 
3 
1331 
1367631 



4 
14641 




12 The base of the duodecimal system. The Romans
used only duodecimal fractions. The dodecahedron has 12 faces.
13 The next number in the sequence 345 is 1. The next number in
the sequence 543 is 1. The next number in the sequence 222 is 7.
The next number in the sequence 123 is 7. The next number in the
sequence 333 is 4. These are all bridge hand distributions.
"Triskaidekaphobia" is the fear of Friday the 13th. Gauss developed a
calendar formula to determine the day of the week upon which any day of the
week falls. In a 400 year cycle, there are 12 months that have a 13th,
or 4800 days. The average you might expect on any day of the week is 4800/7
= 685.7. But Friday is the most frequent at 688, followed by Wed and
Sun at 687, Mon and Tue at 685, and Thur and Sat at 684.
13 is supposedly unlucky: 13 present at Last Supper, 13 at a banquet in Bahalla when Balder (don of Odin) was slain. Friday is not a good day: Eve
tempted Adam to eat the forbidden fruit, the Great Flood occurred, the start
of linguistic confusion at the Tower of Babel, the destruction of Solomon's
Temple, the death of Jesus Christ, British sailors fear of sailing on a
Friday.
14.134725 ... The first zero in the Reiman zeta function is at
1/2 + i(14.134725 ... )
17 A 17 sided regular polygon is a Heptadecagon.In 1796, when
Gauss was only 19 he announced he could construct this with straight edge
and compass. He went further and indicated which ngons could be
constructed using Euclidean tools and which could not. 17 is equal to the um
of the digits of its cube, 4913. It is the only prime number to have
this property. Also, the formula, n squared + n + 17, generates prime
numbers for n = 0 to 15.
19 Every single positive integer can be written as the sum of
at most 19 powers.
22, 23, and 24 are the only positive integers
(other than 1) for which n! has precisely n digits.
22.4591577183 ... This is pi raised to the
e power, π^{e}. It is
not known if this is irrational or transcendental.
23 Its digits are consecutive prime numbers. It is
the smallest odd prime that is not a twin prime. 23 is the smallest prime
for which the sum of the squares of its digits is also a prime.
There were 23 problems on
David Hilbert's
famous
list of unsolved mathematical problems, Beckham chose the number 23 on his shirt to play for Real
Madrid. Michael Jordan wore the number 23. We have 23 pairs of
chromosomes. Caesar was stabbed 23 times. 23 mph is the maximum
speed of an American crow.
23 equals 3x5 + 3 +5
23 factorial contains 23 digits, the only prime number with
this property.
23.1406926328... is e raised to the power
pi. In
mathematics, Gelfond's constant,
named after
Aleksandr Gelfond, is e^{π},
that is,
e to the
power of
π. Like both e and
π, this constant is a
transcendental number. This was first
established by Gelfond and may now be considered as an application of the
Gelfond–Schneider theorem, noting the fact
that

where i is the
imaginary unit. Since −i is
algebraic but not rational, e^{π}
is transcendental. The constant was mentioned in
Hilbert's seventh problem.^{
} A related constant is
,
known as the
Gelfond–Schneider constant. The related
value π + e^{π}
is also irrational.^{
}
^{24 is a factor of the
product of 4 consecutive integers. The product of n consecutive
integers is divisible by n factorial !}
25 is a Friedman Number: A positive integer which can be
written in some nontrivial way using its own digits, together with the
symbols +  x / ^ ( ) and concatenation. See:
http://www2.stetson.edu/~efriedma/mathmagic/0800.html
Here, 25 = 5^{2 }
31 is a Mersenne prime number. That is, it is a prime
number that can be written in the form M(n) = 2ⁿ − 1 for some integer n.
They are named after Marin Mersenne, a French Minim friar, who studied them
in the early 17th century. The first four Mersenne primes are 3, 7, 31,
and 127
^{31 is the number of regions a circle can be
divided into by connecting 6 points on the edge of a circle. See
http://www.mast.queensu.ca/~peter/inprocess/circleregions.pdf and
http://blogimages.bloggen.be/gnomon/attach/218796.pdf
}
^{The surprising result is
not 32 as the pattern of the number of regions seems
to be following a power of 2.}
Points 
1 
2 
3 
4 
5 
6 
7 
8 
Regions 
1 
2 
4 
8 
16 
31 
57 
99 
If you have a circle with n points on the circle. You can
connect lines connecting these points, where the number of lines =
L(n)= (n)(n1)/2
These number of intersections of these lines = I(n) =
(n)(n1)(n2)(n3)/24
The number of regions formed by these lines and their
intersections = R(n) = 1 + L(n) + I(n)
R(n) = 1 + (n)(n1)/2 + (n)(n1)(n2)(n3)/24 
31 is the number of
regular polygons with an odd number of
sides that are known to be
constructible with compass and straightedge.
37 (666)/(6 + 6 + 6) = 111/3
Forty is the only number whose letters are in
alphabetical order.
45 Lake Chaubunagungamaug,
also known as Webster Lake, is a lake in the
town of
Webster,
Massachusetts,
United States. It is located near the
Connecticut border and has a surface area
of 1442 acres. Since 1921, the lake has also been known by a much longer
name having 45 letters comprising fourteen syllables: Lake
Chargoggagoggmanchauggagoggchaubunagungamaugg.
53 If there are 53 people in a room,
the probability is approximately 1/53 none of them will share a birthday.
This is .019... , about 2%.
55 is the largest two digit number used in the
NBA. This makes it easy for referees to communicate a player's number
using hand signals using the fingers on each hand. Each digit is 0 thru 5.
41 Euler discovered the formula n
squared + n + 41 generates prime numbers for n equal to 0 to 39.
60 The base of the sexagesimal system
used by the Sumerians as early as 3500 BC. The Babylonians used this
system for mathematical and astronomimal work. The division of the circle
into 360 degrees and the degrees into 60 parts originated with the
Babylonians. We still do this with time, measured in 60 minutes and 60
seconds.
61 The smallest solution of the
Pellian equation x squared minus 61 times y squared = 1, is x =
1,766,319,049 and y = 226,153,980. Wow!
71 71 squared = 7 factorial + 1
factorial. That is, the sum of the factorials of its digits.
88 The number of keys on a piano, 52 white
keys and 36 black keys. . There are 7 white keys and 5 black keys to an
octave. 88 The number that is called "two fat
ladies" in Bingo. 88 The number of feet per second, when
driving 60 miles per hour.
93 You can chop a
big lump of cheese into a maximum of 93 bits with 8 straight cuts.
100 The square of 10, the base of the
decimal system, but also the square of the base in any other base.
113 The smallest three digit prime
such that all other arrangements of its digits are also prime (131 and 311).
Other three digit numbers are 199 (919 and 991) and 337 (373 and 733).
Two digit primes are 12, 13, 17, 37 and 79. The next two primes with
this property are 1,111,111,111,111, 111,111 and a similar prime consisting
of 23 instead of 19 1s.
128 A cord of wood is 4 feet by 4 feet by 8
feet or 128 cubic feet. A tennis tournament with 2^{n}
players will have n rounds. A Grand slam
Singles Tournament has 128 entrants and 7 rounds. 128 is the largest number
that cannot be expressed as the sum of three distinct squares. Like 25, 128
is another Friedman number: 128 = 2^{81}
129 It can be expressed as the sum of three distinct squares in
two different ways. 129 = 100 + 25 +4. Also 129 = 64 + 49 + 16.
144 It is the largest Fibonacci square.
153 It is the sum of the first
17
integers. .
It is also the sum of the first five positive
factorials:.
It is also the smallest threedigit number which can be expressed as the sum
of cubes of its digits.
All such numbers below 10 to the tenth
power are: 0, 1, 153, 370, 371, 407, 165033, 221859, 336700, 336701, 340067,
341067, 407000, 407001, 444664, 487215, 982827, 983221, 166500333,
296584415, 333667000,
333667001, 334000667, 710656413, 828538472.
216 It is the smallest cube that is the sum of three
cubes:
365 It is the smallest number which has more than
one expression as a sum of consecutive
squares.


 538 is the number of electors in the U S Electoral
College. It is also the name of Nate Silver's website, purchased
by ESPN in July 2013, that focuses on opinion poll analysis, politics,
economics, and sports blogging. The website's logo is a fox, in
reference to a phrase attributed to Archilochus: "The fox knows many
things, but the hedgehog knows one big thing."

 Scholars have differed about the correct interpretation of these
dark words, which may mean no more than that the fox, for all his
cunning, is defeated by the hedgehog's one defense. But, taken
figuratively, the words can be made to yield a sense in which they mark
one of the deepest differences which divide writers and thinkers, and,
it may be, human beings in general. For there exists a great chasm
between those, on one side, who relate everything to a single central
vision which has significanceand, on the other side, those who pursue
many ends, often unrelated and even contradictory, connected, if at all,
only in some de facto way, for some psychological or
physiological cause, related by no moral or aesthetic principle.

 In The Hedgehog and the Fox, an essay by philosopher
Isaiah Berlin, Berlin expands upon an
idea to divide writers and thinkers into two categories: hedgehogs, who
view the world through the lens of a single defining idea (examples
given include
Plato,
Lucretius,
Dante Alighieri,
Blaise Pascal,
Georg Wilhelm Friedrich Hegel,
Fyodor Dostoyevsky,
Friedrich Nietzsche,
Henrik Ibsen,
Marcel Proust and
Fernand Braudel), and foxes, who draw
on a wide variety of experiences and for whom the world cannot be boiled
down to a single idea (examples given include
Herodotus,
Aristotle,
Desiderius Erasmus,
William Shakespeare,
Michel de Montaigne,
Molière,
Johann Wolfgang Goethe,
Aleksandr Pushkin,
Honoré de Balzac,
James Joyce and
Philip Warren Anderson).

 563 In
number theory, Wilson's theorem
states that a natural number n > 1 is a
prime number
if and only if
.
Occasionally it is divisible by p squared. In 1953, using a
computer, Goldberg found that the only value for of n = p were 5,
13, and 563.
640 are the number of acres in one square mile.
666 is the occult "number of the beast," also
called the "sign of the devil", associated in the Bible with the
Antichrist. It has figured in many numerological studies. It is mentioned
in Revelation 13:18: "Here is wisdom. Let him that hath understanding
count the number of the beast: for it is the number of a man; and his
number is 666."
The beast number has several interesting properties which numerologists
may find particularly interesting (Keith 198283). In particular, the
beast number is equal to the sum of the squares of the first 7
primes
Emanouilidis (1998) also gives additional more obscure connections
between 666 and the numbers on a roulette wheel.
The number 666 is a sum and difference of the first three 6th
powers,
666 is the sum of numbers on a roulette wheel, which 38 pockets
labeled 0, 00, and 136.
666 is the sum of the first 144 decimal digits in pi.
714 The record number of home runs hit by
Babe Ruth, and surpassed by Hank Aaron's 756 and Barry Bonds' 762.
The product of 714 and 715 equals 2x3x5x7x11x13x17.
720 720 = 10x9x8 = 6x5x4x3x2.
1000 The smallest number spelled out that has an "a" in it.
1001 450 factorial has 1001 digits
and is appropriately known as the Arabian Nights Factorial.
In The Thousand and One Arabian Nights, a king found
out that his first wife was unfaithful to him. He then married a new
virgin each day who
would be beheaded the following morning. Scheherazade, the
vizier's daughter, volunteered to be his
next wife. She had perused the books, annals and legends of preceding
Kings, and the stories, examples and instances of bygone men and things;
indeed it was said that she had collected a thousand books of histories
relating to antique races and departed rulers. She had perused the works of
the poets and knew them by heart; she had studied philosophy and the
sciences, arts and accomplishments; and she was pleasant and polite, wise
and witty, well read and well bred. Once in the king's chambers,
Scheherazade began a telling a story. The king lay awake and listened with
awe as Scheherazade told her first story. The night passed by, and
Scheherazade stopped in the middle of the story. The king asked her to
finish, but Scheherazade said there was not time, as dawn was breaking. So,
the king spared her life for one day to finish the story the next night. So
the next night, Scheherazade finished the story and then began a second,
even more exciting tale which she again stopped halfway through at dawn. So
the king again spared her life for one day to finish the second story.
And so the King kept Scheherazade alive day by day, as he eagerly
anticipated the finishing of last night's story. At the end of 1,001 nights,
and 1,000 stories, Scheherazade told the king that she had no more tales to
tell him. During these 1,001 nights, the king had fallen in love with
Scheherazade, and so he spared her life, and made her his queen.
1233 Bhavaraju Sarveswara Rao (1915 –
2010) was an
Indian
economist
and a social scientist, who found that 1233 =
12² + 33²
. Also 8833 =
88²
+ 33² . These have four digits.
The number with six digits is: 990100 =
990² + 100²
.
Numbers with ten digits are:
1765038125 = 17650² + 38125² , 2584043776 =
25840² + 43776² , 7416043776 = 74160² + 43776² ,
8235038125 = 82350² + 38125²
1477
is a
factorial prime: a
prime number that is one less or one more
than a
factorial.. The first few factorial primes
are: 2
(0! + 1 or 1! + 1),
3 (2! + 1),
5 (3! − 1),
7 (3! + 1),
23 (4! − 1), 719 (6! − 1), 5039 (7! − 1),
39916801 (11! + 1), 479001599 (12! − 1), 87178291199 (14! − 1),
n! − 1 is prime for: n = 3, 4, 6, 7, 12, 14, 30, 32,
33, 38, 94, 166, 324, 379, 469, 546, 974, 1963, 3507, 3610, 6917, 21480,
34790, 94550, 103040, 147855, ...
n! + 1 is prime for: n = 0, 1, 2, 3, 11, 27, 37, 41,
73, 77, 116, 154, 320, 340, 399, 427, 872, 1477, 6380, 26951, 110059,
150209, ...
No other factorial primes are known as of September 2013^{.}
1729 is known as the Hardy–Ramanujan number
after a famous anecdote of the British mathematician
G. H.
Hardy regarding a visit to the hospital to see the Indian mathematician
Srinivasa Ramanujan. In Hardy's
words: I remember once going to see him when he was ill at
Putney. I had ridden in taxi cab number
1729 and remarked that the number seemed to me rather a
dull one, and that I hoped it was not an
unfavorable
omen. "No," he replied, "it is a very
interesting number; it is the smallest number expressible as the sum of two
cubes in two different ways." The two different ways are these:
1729 = 1^{3} + 12^{3} = 9^{3} + 10^{3 }
^{
https://www.youtube.com/watch?v=_o0cIpLQApk}
1729 is a Harshad number (or Niven number): an
integer that is divisible by the
sum of its digits. Harshad numbers were
defined by
D. R. Kaprekar, a
mathematician from
India. The word "Harshad" comes from the
Sanskrit
harṣa (joy) +
da (give), meaning joygiver. The term “Niven number” arose from
a paper delivered by
Ivan M. Niven at a conference on
number theory in 1977.
3320 The Gregorian calendar has 365 days, except those years
divisible by 400, which have 366 days. After 3320 days, this will
require a downward adjustment of 1.
4830 is the number of square yards in an acre.
8902 are the number of ways to play the first four moves in
chess.
26,861 There are as many primes of the form 4n + 1 as of
the form 4n + 3 below 26861. 26861 is a prime of the form 4n + 1.
Littlewood proved the lead keeps changing an infinite number of times.
43,470 is the number of square feet in an acre.
150, 209 factorial plus 1. This form of a factorial plus
or minus one, that equals a prime number, is shown below.
rank 
prime 
digits 
when 
1 
150209! + 1 
712355 
Oct 2011 
2 
147855!  1 
700177 
Sep 2013 
3 
110059! + 1 
507082 
Jun 2011 
4 
103040!  1 
471794 
Dec 2010 
5 
94550!  1 
429390 
Oct 2010 
6 
34790!  1 
142891 
May 2002 
7 
26951! + 1 
107707 
May 2002 
8 
21480!  1 
83727 
Sep 2001 
9 
6917!  1 
23560 
Oct 1998 
10 
6380! + 1 
21507 
Oct 1998 
11 
3610!  1 
11277 
Oct 1993 
12 
3507!  1 
10912 
Oct 1992 
13 
1963!  1 
5614 
Oct 1992 
14 
1477! + 1 
4042 
Dec 1984 
15 
974!  1 
2490 
Oct 1992 
16 
872! + 1 
2188 
Dec 1983 
17 
546!  1 
1260 
Oct 1992 
18 
469!  1 
1051 
Dec 1981 
23,456,789 is a prime with consecutive digits. All that are
known are: 23, 67, 89, 4567, 78901, 23456789, 45678901,
9012345678901, 789012345678901.
The symmetrical decomposition of its digits yields three primes: 23, 4567,
and 89.
73,939,133 is the largest prime, which when successive
digits are removed from the right the results are all prime. Another
example is 33,333,331.
73939133 7393913
739391 73939
7393 739
73
7 
272,400, 600 are the number of terms it takes for the partial
sum of the slowly diverging Harmonic Series to reach 20.
635,318, 657 See
https://www.youtube.com/watch?v=_o0cIpLQApk and the Indian mathematician
Srinivasa Ramanujan
1,000,000,000 One billion. A billion is 10^{9
(2 + 1)x3 = 9 zeros}
1,234,567,891 is one of three primes whose digits are in
ascending order. The other two are 12,345,678,901,234,567,891 and
1,234,567,891,234,567,891,234,567,891.
1,480,028,171 is the central prime in a magic square.
Harry Nelson won $100 from Martin Gardner for discovering a 3x3 magic square
of primes the other 8 primes are more or less than it by 12, 18, 30, and 42.
Harry Lewis Nelson (born January 8, 1932) is an American mathematician and
computer programmer. He was a member of the team that won the
World Computer Chess Championship in 1983
and 1986. He was a codiscoverer of the 27th
Mersenne prime in 1979 (at the time, the
largest known prime number). He also served as editor of the
Journal of Recreational Mathematics for
five years.
1,979,339,339 and its tail cuttings are all primes:
1979339339
197933933
19793393
1979339
197933
19793
1979
197
19 
158,753,389,900:1 The odds against a person being dealt a
single suit in bridge. 158 billion to 1. A billion is 10^{9
(2 + 1)x3 = 9 zeros}
608,981,813,029 is the number where primes of the form 3n + 1
become the majority.
Prime numbers of the form 3n + 1 are more
numerous than those of the form 3n + 2 at 608981813029, even though the
"race" initially favors 3n + 2. The lead then keeps changing hands,
but for an infinite number of primes, this 3n + 1 form has the majority.
This is over 608 billion. A billion is 10^{9 (2 + 1)x3 = 9
zeros}
635,013,559,600 are the number of possible bridge hands a
player can receive. This is over 635 billion. A billion is 10^{9
(2 + 1)x3 = 9 zeros}
52!
13! x 39!
18,446,744,073,709,551,615 This number 2^{64} 
1 exceeds 18 quintillion. A quintillion is 10^{18 (5 +1)x3 =
18 zeros }It appears at least twice in mathematical folklore.
Chessboard There are different stories about
the invention of chess. One of them includes the geometric progression
problem. Its earliest written record is contained in the
Shahnameh,
an epic poem written by the Persian poet
Ferdowsi
between c. 977 and 1010 CE. When the creator of the game of chess, an
ancient Indian
Brahmin
mathematician named
Sessa or Sissa,
showed his invention to the ruler of the country, the ruler was so pleased
that he gave the inventor the right to name his prize for the invention. The
man, who was very clever, asked the king that for the first square of the
chess board, he would receive one grain of wheat, two for the second one,
four on the third one, and so forth, doubling the amount each time. The
ruler quickly accepted the inventor's offer, even offended by his notion
that the inventor was asking for such a low price, and ordered the treasurer
to count and hand over the wheat to the inventor. However, when the
treasurer took more than a week to calculate the amount of wheat, the ruler
asked him for a reason for his tardiness. The treasurer then gave him the
result of the calculation, and explained that it would take more than all
the assets of the kingdom to give the inventor the reward. The story ends
with the inventor becoming the new king.
Tower of Hanoi In the great temple at Benares
beneath the dome that marks the centre of the world, rests a brass plate in
which are fixed three diamond needles, each a cubit high and as thick as the
body of a bee. On one of these needles, at the creation, God placed
sixtyfour discs of pure gold, the largest disk resting on the brass plate,
and the others getting smaller and smaller up to the top one. This is the
tower of Bramah. Day and night unceasingly the priest transfer the discs
from one diamond needle to another according to the fixed and immutable laws
of Bramah, which require that the priest on duty must not move more than one
disc at a time and that he must place this disc on a needle so that there is
no smaller disc below it. When the sixtyfour discs shall have been thus
transferred from the needle which at creation God placed them, to one of the
other needles, tower, temple, and Brahmins alike will crumble into dust and
with a thunderclap the world will vanish. The number of separate
transfers of single discs which the Brahmins must make to effect the
transfer of the tower is two raised to the sixtyfourth power minus 1 or
18,446,744,073,709,551,615 moves. Even if the priests move one disk every
second, it would take more than 500 billion years to relocate the initial
tower of 64 disks.
42,252,003,274,489,856,000 the
total possible positions that can be reached on the original 3x3x3 Rubic's
Cube. It can be solved in just 20 moves.
There are 6 colors on a Rubik's Cube:
yellow, blue, green, red, white, and orange. This number
exceeds 42 quintillion. A quintillion is 10^{18 (5 +1)x3 = 18
zeros}
147,573,952,589,676,412,927 was proven to not be a
Mersenne prime. This prime exceeds 257 quintillion. A quintillion is 10^{18
(5 +1)x3 = 18 zeros}
On one cold rainy October afternoon in 1903, Frank Cole was
scheduled to give a talk at a meeting of the American Mathematical Society
with the unassuming title, "On the Factorization of Large Numbers." When his
time came, Cole strode confidently to the blackboard and carefully wrote
out:
2^{67}1 = 147,573,952,589,676,412,927
Then he moved to another section of the board and began the long process
of multiplying two large numbers together: 193707721 and 761838257287.
Cole never spoke a word during the tedious multiplication process. But
when he finally laid down his chalk and the product agreed with the original
decimal expansion of 2^{67}1:
2^{67}1 = 147573952589676412927 = 193707721 *
761838257287 = 147573952589676412927
The crowd recognized his accomplishment and broke out in applause. At the
meeting, no one asked Cole a single question about his "talk," but later he
stated that it had required 20 years of continuous Sunday afternoons to find
the factorization of 2^{67}  1. (It takes my computer with a
Pentium 4 processor exactly 125 milliseconds.)
Frank Nelson Cole's factorization accomplishment is now so wellknown
that an account of it usually makes an appearance in every modern number
theory book written for nonmathematicians.
A list of Mersenne Primes is at
http://en.wikipedia.org/wiki/Mersenne_prime
357,686,312,646,216,567,629,137 is a prime which when
successively "beheaded" is also a prime. This prime is
over 357 sextillion. A sextillion is 10^{21 (6 + 1)x3 = 21 zeros}
357686312646216567629137
57686312646216567629137
7686312646216567629137
686312646216567629137
86312646216567629137
6312646216567629137
312646216567629137
12646216567629137
2646216567629137
646216567629137
46216567629137
6216567629137
216567629137
16567629137
6567629137
567629137
67629137
7629137
629137
29137
9137
137
37
7 
2,235,197,406,895,366,368,301,560,000 to 1 are the odds
against all four players in bridge being dealt a single suit, over 2.25
octillion to 1. An octillion is 10^{27 (8 +1)x3 =
27 zeros}
2^{257 } 1 is a large Mersenne number, but
is not a prime. The Mersenne conjecture, was a statement by
Marin Mersenne in his Cogitata
PhysicaMathematica (1644; see e.g. Dickson 1919) that the numbers
were prime for n = 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257, and
were
composite for all other positive integers
n < 258. (He missed n = 61) Due to the size of these numbers, Mersenne did not and could
not test all of them, nor could his peers in the 17th century. By 1947
Mersenne's range, n < 258, had been completely checked and it
was determined that the correct list is n = 2, 3, 5, 7, 13, 17, 19,
31, 61, 89, 107 and 127. In 1903, Frank Cole showed when n =67,
it can be factored into two huge factors. In 1922, Krait showed when n
=257, it can be factored. It has four huge factors. For Mersenne primes, see
http://mathworld.wolfram.com/MersennePrime.html
Large Numbers
NAME 
^{Short Scale} 
^{Long Scale} 
Million 
10^{6} 
10^{6} 
Milliard 

10^{9} 
Billion 
10^{9} 
10^{12} 
Trillion 
10^{12} 
10^{18} 
Quadrillion 
10^{15} 
10^{24} 
Quintillion 
10^{18} 
10^{30} 
Sextillion 
10^{21} 
10^{36} 
Septillion 
10^{24} 
10^{42} 
Octillion 
10^{27} 
10^{48} 
Nonillion 
10^{30} 
10^{54} 
Decillion 
10^{33} 
10^{60} 
Undecillion 
10^{36} 
10^{66} 
Duodecillion 
10^{39} 
10^{72} 
Tredecillion 
10^{42} 
10^{78} 
Quattuordecillion 
10^{45} 
10^{84} 
Quindecillion 
10^{48} 
10^{90} 
Sexdecillion 
10^{51} 
10^{96} 
Septendecillion 
10^{54} 
10^{102} 
Octodecillion 
10^{57} 
10^{108} 
Novendecillion 
10^{60} 
10^{114} 
Vigintillion 
10^{63} 
10^{120} 
Centillion 
10^{303} 
10^{600} 
Googol 
10^{100} 
Googolplex 
10^{Googol} 
* ^{Short Scale = North America & Modern British
} 
^{
** Long Scale = Europe & Older British?} 
When a number such as 10^{45} needs
to be referred to in words, it is simply read out: "ten to the fortyfifth".
This is just as easy to say, easier to understand, and less ambiguous than "quattuordecillion",
which means something different in the long scale and the short scale.
An easy way to find the value of the above numbers in the short
scale (as well as the number of zeroes needed to write them) is to take the
number indicated by the prefix (such as 2 in billion,
4 in quadrillion, 18 in
octodecillion, etc.), add one to it, and multiply that
result by 3. For example, in a trillion, the prefix is tri,
meaning 3. Adding 1 to it gives 4. Now multiplying 4 by 3 gives us 12, which
is the power to which 10 is to be raised to express a shortscale trillion
in scientific notation: one trillion = 10^{12}.
In the long scale, this is done simply by multiplying the number
from the prefix by 6. For example, in a billion, the prefix is
bi, meaning 2. Multiplying 2 by 6 gives us 12, which
is the power to which 10 is to be raised to express a longscale billion in
scientific notation: one billion = 10^{12}.
The intermediate values (billiard, trilliard, etc.) can be converted in a
similar fashion, by adding ½ to the number from the prefix and then
multiplying by six. For example, in a septilliard, the prefix is
sept, meaning 7. Multiplying 7½ by 6 yields 45, and
one septilliard equals 10^{45}. Doubling
the prefix and adding one then multiplying the result by three would give
the same result.
Easy??








