Introduction
Those ten
simple symbols, digits, or numbers that we all learn early in
life that influence our lives in far more ways than we could
ever imagine. Have you ever wondered what our lives would be
like without these 10 elegant digits and the infinite array of
other numbers that they can create? Birthdays, ages, height,
weight, dimensions, addresses, telephone numbers, license plate
numbers, credit card numbers, PIN numbers, bank account numbers,
radio/TV station numbers, time, dates, years, directions, wake
up times, sports scores, prices, accounting, sequences/series of
numbers, magic squares, polygonal numbers, factors, squares,
cubes, Fibonacci numbers, perfect, deficient, and abundant
numbers, and the list goes on ad infinitum. Engineers,
accountants, store clerks, manufacturers, cashiers, bankers,
stock brokers, carpenters, mathematicians, scientists, and so
on, could not survive without them. In a sense, it could easily
be concluded that we would not be able to live without them.
Surprisingly, there exists an almost immeasurable variety of
hidden wonders surrounding or emanating from these familiar
symbols that we use every day, the natural numbers.
Over time,
many of the infinite arrays, or patterns, of numbers derivable
from the basic ten digits have been categorized or classified
into a variety of number types according to some purpose that
they serve, fundamental rule that they follow, or property that
they possess. Many, if not all, are marvelously unique and serve
to illustrate the extreme natural beauty and wonder of our
numbers as used in both classical and recreational mathematics.
In the
interest of stimulating a broader interest in number theory and
recreational mathematics, this collection will endeavor to
present basic definitions and brief descriptions for several of
the number types so often encountered in the broad field of
recreational mathematics. The number type descriptions that
follow will not be exhaustive in detail as space is limited and
some would take volumes to cover in detail. A list of excellent
reading references is provided for those who wish to learn more
about any specific number type or explore others not included.
It is sincerely hoped that the material contained herein will
stimulate you to read and explore further. I also hope that
after reading, digesting, and understanding the material offered
herein, that you will have enjoyed the experience and that you
will never utter those terrible, unforgettable words, "I hate
math."
Some basic
definitions of terms normally encountered in the classroom are
given first.
Numbers 
The Basics
Integers
 Any of the positive and negative whole numbers,
..., ^{}3,
^{}2, ^{}1,
0, ^{+}1, ^{+}2, ^{+}3, ...
The positive integers, 1, 2, 3..., are called the
natural numbers or counting numbers. The set of all integers is
usually denoted by Z or Z+
Digits
 the 10 symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9, used to
create numbers in the base 10 decimal number system.
Numerals
 the symbols used to denote the natural numbers. The Arabic
numerals 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are those used in the
HinduArabic number system to define numbers.
Natural
Numbers
 the set of
numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,
17,....., that we see and use every day. The natural numbers are
often referred to as the counting numbers and the positive
integers.
Whole
Numbers
 the natural numbers plus the zero.
Rational
Numbers
 any number that is either an integer "a" or is
expressible as
the ratio of two integers, a/b. The numerator, "a", may be any
whole number, and the denominator, "b", may be any positive
whole number greater than zero. If the denominator happens to be
unity, b = 1, the ratio is an integer. If "b" is other than 1,
a/b is a fraction.
Fractional
Numbers
 any number expressible by the quotient of two numbers as in
a/b, "b" greater than 1, where "a" is called the numerator and
"b" is called the denominator. If "a" is smaller than "b" it is
a proper fraction. If "a" is greater than "b" it is an improper
fraction which can be broken up into an integer and a proper
fraction.
Irrational
Numbers
 any number that cannot be expressed by an integer or the ratio
of two integers. Irrational numbers are expressible only as
decimal fractions where the digits continue forever with no
repeating pattern. Some examples of irrational numbers are
.
Transcendental Numbers
 any number
that cannot be the root of a polynomial equation with rational
coefficients. They are a subset of irrational numbers examples
of which are Pi = 3.14159... and e = 2.7182818..., the base of
the natural logarithms.
Real
Numbers
 the set of real numbers including all the rational and
irrational numbers.
Irrational
numbers are numbers such as
Rational
numbers include the whole numbers (0, 1, 2, 3, ...), the
integers (...,  2,  1, 0, 1, 2, ...), fractions, and repeating
and terminating decimals.
Draw a line.
Put on it
all the whole numbers 1,2,3,4,5,6,7.....etc
then put
0
then put
all the negatives of the whole numbers to the left of 0
........10,9,8,7,6,5,4,3,2,1,0
Then put
in all of the fractions .
Then put
in all of the decimals [ some decimals aren't fractions ]
Now you
have what is called the "real number line"
The way to
get a number that is not "real" is to try to find the square
root of  1
can't be 1 because 1 squared is 1, not 1
can't be 1 because the square of 1 is 1, not 1
So there is
no number on your number line to be
and new numbers
would need to be put somewhere.
The number
types currently entered, and/or planned to be entered, are
listed below and will be updated as new entries are made in the
future. When appropriate, and time permitting, some of the
number definitions/descriptions will be expanded further to
provide additional information.
Abundant,
Algebraic, Amicable, Arrangement, Automorphic, Binary, Cardinal,
Catalan, Complex, Composite, Congruent, Counting, Cubic,
Decimal, Deficient, Even, Factor, Factorial, Fermat, Fibonacci,
Figurate, Fractional, Friendly, Generating, Gnomon, Golden,
Gyrating, Happy, HardyRamanujan, Heronian, Imaginary, Infinite,
Integers, Irrational, Mersenne, Monodigit, Narcissistic,
Natural, Oblong, Octahedral, Odd, Ordinal, Parasite, Pell,
Pentatope, Perfect , Persistent, Polygonal, Pronic, Pyramidal ,
Pythagorean, Quasiperfect, Random, Rational, Real, Rectangular,
Relatively Prime, Semiperfect, Sequence, Sociable, Square,
Superabundant, Tag, Tetrahedral, Transcendental, Triangular,
Unit Fraction, Whole.
Several of the
numbers form unique patterns that are often used in the solution
of mathematical problems. When distinct patterns are
applicable, the first ten numbers of the patterns will be given
along with specific relationships, or equations, that will
enable you to find any number in the pattern.
End of
Introduction
ABUNDANT
NUMBERS
A number n for which the sum of divisors σ(n)>2n,
or, equivalently, the sum of proper divisors (or
aliquot sum) s(n)>n.
An abundant number is a number n for
which the sum of divisors σ(n)>2n, or,
equivalently, the sum of proper divisors (or
aliquot sum) s(n)>n.
Abundant
numbers are part of the family of numbers that are either
deficient, perfect, or abundant.
Abundant
numbers are numbers where the sum, Sa(N), of its aliquot
parts/divisors is more than the number itself Sa(N) > N or S(N)
> 2N. (In the language of the Greek mathematicians, the divisors
of a number N were defined as any whole number smaller than N
that, when divided into N, produced whole numbers. The
factors/divisors of a number N, less the number itself, are
referred to as the aliquot parts, aliquot divisors, or proper
divisors, of the number.) Equivalently, N is also abundant if
the sum, S(N), of "all" its divisors is greater than 2N.
From the
following list
N>.......1..2..3..4..5...6....7...8....9..10..11..12..13..14..15..16..17..18..19...20..21..22..23..24
Sa(N)>1..1..1..3..1...6....1...7....4...8....1...16...1...10...9...15...1....21...1...22..11..12...1...36
S(N)>
1..3..4..7..6..12...8..15..13.18..12..28..14..22..24..31..18...39..20..42..32..36..24..60
..........................................12,18,20, and 24 are
abundant.
It can be
readily seen that using the aliquot parts summation, sa(24) =
1+2+3+4+6+8+12 = 36 > N = 24 while s(24) = 1+2+3+4+6+8+12+24 =
60 > 2N = 48, making 24 abundant using either definition.
Only 21 of the
numbers from 1 to 100 are abundant. The number 945 is the first
odd abundant number.
Every even
integer greater than 46 is expressible by the sum of two
abundant numbers.
Every integer
greater that 83,159 is expressible by the sum of two abundant
numbers.
Any multiple
of an abundant number is abundant.
A prime number
or any power of a prime number is deficient. The divisors of a
perfect or deficient number is deficient.
The abundant
numbers below 100 are 20, 24, 30, 36, 40, 42, 48, 54, 56, 60,
66, 70, 72, 78, 80, 84, 88, 90 and 96.
(See perfect,
semiperfect, multiplyperfect, quasiperfect, deficient, least
deficient, super abundant)
ALGEBRAIC
NUMBERS
Algebraic
numbers are the real or complex number solutions to polynomial
equations of the form:
The coefficients a, b, c, d, ....p, q,
are integers or fractions. All rational numbers are algebraic
while some irrational numbers are algebraic.
ALIQUOT
PART
An aliquot
part is any divisor of a number, not equal to the number itself.
The divisors are often referred to as proper divisors. The
aliquot parts of the number 24 are 1, 2, 3,4, 6, 8 and 12..
ALMOST
PERFECT NUMBER
An almost
perfect number is typically applied to the powers of 2 since the
sum of the aliquot parts is
, or just 1 short of being a
perfect number. It follows that any power of 2 is a deficient
number
ALPHAMETIC
NUMBERS
Alphametic
numbers form
cryptarithms where a set of numbers are assigned to
letters that usually spell out some meaningful thought. The
numbers can form an addition, subtraction, multiplication or
division problem. One of the first cryptarithms came into being
in 1924 in the form of an addition problem the words being
intended to represent a student's letter from college to the
parents. The puzzle read SEND + MORE = MONEY. The answer was
9567 += 1085 = 10,652. Of course, you have to use logic to
derive the numbers represented by each letter..
AMICABLE
NUMBERS
Amicable
numbers are pairs of numbers, each of which is the sum of the
others aliquot divisors. For example, 220 and 284 are amicable
numbers whereas all the aliquot divisors of 220, i.e., 110, 55,
44, 22, 10, 5, 4, 2, 1 add up to 284 and all the aliquot
divisors of 284, i.e., 142, 71, 4, 2, 1 add up to 220. Also true
for any two amicable numbers, N1 and N2, is the fact that the
sum of all the factors/divisors of both, Sf(N1 + N2) = N1 + N2.
Stated another way, Sf(220 + 284) = 220 + 284 = 504. Other
amicable numbers are:
There are
more than 1000 known amicable pairs. Amicable numbers are
sometimes referred to as friendly numbers.
Several
methods exist for deriving amicable numbers, though not all,
unfortunately. An Arabian mathematician devised one method. For
values of n greater than 1, amicable numbers take the form:
given that x, y, and z are prime numbers. n = 2
produces x = 11, y = 5 and z = 71 which are all prime and
therefore result in the amicable number pair of 220/284. n = 3
produces the x = 23, y = 11 and z = 287 but 287 is composite,
being 7x41. n = 4 produces x = 47, y = 23 and z = 1151, all
prime, and thereby resulting in the amicable pair of 17,296 and
18,416. In reviewing the few amicable pairs shown earlier, it is
obvious that this method does not produce all amicable pairs.
If the number
1 is not used in the addition of the aliquot divisors of two
numbers, and the remaining aliquot divisors of each number still
add up to the other number, the numbers are called
semiamicable. For example, the sum of the aliquot divisors of
48, excluding the 1, is 75 while the sum of the aliquot divisors
of 75, excluding the 1, is 48.
As a matter of
general information:
The sum of all
the factors/divisors of a number N is given by
..................................Sf(N) = [p^(a+1)  1] x
[q^(b+1)  1] x [r^(c+1)  1]
.....................................................(p 
1).............(q  1).............(r  1)
which when
applied to an example of 60 = 2^2x3x5 results in
..................................Sf(60) = (2^3  1) x
(3^2  1) x (5^2  1) = 7 x 4 x 6 = 168.
......................................................1...............2..............4
Stated another
way, the sum of the factors of a number N is given by
..................................Sf(N) =
(1+p+p^2+....p^a)(1+q+q^2+....q^b)(1+r+r^2+....r^c)
which when
applied to an example of
results in
.................................Sf(60) = (1+2+4)(1+3)(1+5) =
168.
As noted
earlier, the sum of the aliquot factors/divisors is the sum of
all the factors/divisors minus the number itself.
APOCALYPSE NUMBER
The apocalypse number, 666, often referred to as the beast
number, is referred to in the bible, Revelations 13:18.
While the actual meaning or relevance of the number remain
unclear, the number itself has some surprisingly interesting
characteristics.
The sum of the first 36 positive numbers is 666 which makes it
the 36th triangular number.
The sum of the squares of the first seven prime numbers is 666.
Multiplying the sides of the primitive right triangle 123537
by 18 yields nonprimitive sides of 216630666.
Even more surprising is the fact that these sides can be written
in the Pythagorean Theorem form:
ARRANGEMENT
NUMBERS
Arrangement
numbers, more commonly called permutation numbers, or simply
permutations, are the number of ways that a number of things can
be ordered or arranged. They typically evolve from the question
how many arrangements of "n" objects are possible using all "n"
objects or "r" objects at a time. We designate the permutations
of "n" things taken "n" at a time as _{n}P_{n} and the permutations of
"n" things taken "r" at a time as _{n}P_{r} where P stands for
permutations, "n" stands for the number of things involved, and
"r" is less than "n". To find the number of permutations of "n"
dissimilar things taken "n" at a time, the formula is _{n}P_{n} =
n!
which is "n" factorial which means:
Example: How
many ways can you arrange the letters A & B. Clearly 2 which is
2 x 1 = 2, namely AB and BA.
How many ways
can you arrange the letters A, B & C in sets of three? Clearly
_{3}P_{3} = 3 x 2 x 1 = 6, namely ABC, CBA, BAC, CAB, ACB, and BCA.
How many ways
can you arrange A, B, C & D in sets of four? Clearly
_{4}P_{4} = 4 x 3
x 2 x 1 = 24.
To find the
number of permutations of "n" dissimilar things taken "r" at a
time, the formula is:
Example: How
many ways can you arrange the letters A, B, C, and D using 2 at
a time? We have
_{4}P_{4} = 4 x (42+1) = 4 x 3 = 12 namely AB, BA, AC,
CA, AD, DA, BC, CB, BD, DB, CD, and DC.
How many
3place numbers can be formed from the digits 1, 2, 3, 4, 5, and
6, with no repeating digit? Then we have _{
6}P_{3} = 6 x 5 x (63+1) =
6 x 5 x 4 = 120.
How many
3letter arrangements can be made from the entire 26 letter
alphabet with no repeating letters? We now have
_{26}P_{3} =
26 x 25 x (263+1) = 26 x 25 x 24 = 15,600.
Lastly, four
persons enter a car in which there are six seats. In how many
ways can they seat themselves?
_{6}P_{4} = 6 x 5 x 4 x (64+1) =
6 x 5 x 4 x 3 = 360.
Another
permutation scenario is one where you wish to find the
permutations of "n" things, taken all at a time, when "p" things
are of one kind, "q" things of another kind, "r' things of a
third kind, and the rest are all different. Without getting into
the derivation,
Example, how
many different permutations are possible from the letters of the
word committee taken all together? There are 9 letters of which
2 are m, 2 are t, 2 are e, and 1 c, 1 o, and 1 i. Therefore, the
number of possible permutations of these 9 letters is:
AUTOMORPHIC
NUMBERS
Automorphic
numbers are numbers of "n" digits whose squares end in the
number itself. Such numbers must end in 1, 5, or 6 as these are
the only numbers whose products produce 1, 5, or 6 in the units
place. For instance, the square of 1 is 1; the square of 5 is
25; the square of 6 is 36.
What about 2
digit numbers ending in 1, 5, or 6? It is well known that all 2
digit numbers ending in 5 result in a number ending in 25 making
25 a 2 digit automorphic number with a square of 625. No other 2
digit numbers ending in 5 will produce an automorphic number.
Is there a 2
digit automorphic number ending in 1? We know that the product
of 10A + 1 and 10A + 1 is 100A^{2} + 20A + 1. "A" must be a number
such that 20A produces a number whose tens digit is equal to
"A". For "A" = 2, 2 x 20 = 40 and 4 is not 2. For "A" = 3, 3 x 20 =
60 and 6 is not 3. Continuing in this fashion, we find no 2
digit automorphic number ending in 1.
Is there a 2
digit automorphic number ending in 6? Again, we know that the
product of 10A + 6 and 10A + 6 is 100A^{2} + 120A + 36. "A" must
be a number such that 120A produces a number whose tens digit
added to 3 equals "A". For "A" = 2, 2 x 120 = 240 and 4 + 3 = 7
which is not 2. For "A" = 3, 3 x 120 = 360 and 6 + 3 = 9 which is
not 3. Continuing in this manner through A = 9, for "A" = 7, we
obtain 7 x 120 = 840 and 4 + 3 = 7 = "A" making 76 the only other
2 digit automorphic number whose square is 5776.
By the same
process, it can be shown that the squares of every number ending
in 625 or 376 will end in 625 or 376.
The sequence
of squares ending in 25 are 25, 225, 625, 1225, 2025, 3025, etc.
The nth square number ending in 25 can be derived directly from
N(n)^{2} = 100n(n  1) + 25. (This expression derives from the
Finite Difference Series of the squares.)
BEAST
NUMBER
See apocalypse
number
BINARY
NUMBERS
Binary numbers
are the natural numbers written in base 2 rather than base 10.
While the base 10 system uses 10 digits, the binary system uses
only 2 digits, namely 0 and 1, to express the natural numbers in
binary notation. The binary digits 0 and 1 are the only numbers
used in computers and calculators to represent any base 10
number. This derives from the fact that the numbers of the
familiar binary sequence, 1, 2, 4, 8, 16, 32, 64, 128, etc., can
be combined to represent every number. To illustrate, 1 = 1, 2 =
2, 3 = 1 + 2, 4 = 4, 5 = 1 + 4, 6 = 2 + 4, 7 = 1 + 2 + 4, 8 = 8,
9 = 1 + 8, 10 = 2 + 8, 11 = 1 + 2 + 8, 12 = 4 + 8, and so on. In
this manner, the counting numbers can be represented in a
computer using only the binary digits of 0 and 1 as follows.
Number 
B I N A R Y S E Q U E N C E 

128 
64 
32 
16 
8 
4 
2 
1 


1 







1 

B 
2 






1 
0 

I 
3 






1 
1 

N 
4 





1 
0 
0 

A 
5 





1 
0 
1 

R 
6 





1 
1 
0 

Y 
7 





1 
1 
1 


8 




1 
0 
0 
0 

N 
9 




1 
0 
0 
1 

O 
10 




1 
0 
1 
0 

T 
11 




1 
0 
1 
1 

A 
12 




1 
1 
0 
0 

T 
13 




1 
1 
0 
1 

I 
14 




1 
1 
1 
0 

O 
15 




1 
1 
1 
1 

N 
76 

1 
0 
0 
1 
1 
0 
0 


157 
1 
0 
0 
1 
1 
0 
1 
1 

As you can
see, the location of the ones digit in the binary representation
indicates the numbers of the binary sequence that are to be
added together to yield the base 10 number of interest.
CARDINAL
NUMBERS
A cardinal
number is a number that defines how many items there are in a
group or collection of items. Typically, an entire group of
items is referred to as the "set" of items and the items within
the set are referred to as the "elements" of the set. For
example, the number, group, or set, of players on a baseball
team is defined by the cardinal number 9. The set of 200 high
school students in the graduating class is defined by the
cardinal number 200. (See ordinal numbers and tag numbers.)
CATALAN
NUMBERS
Catalan
numbers are one of many special sequences of numbers that derive
from combinatorics problems in recreational mathematics.
Combinatorics deals with the selection of elements from a set of
elements typically encountered under the topics of probability,
combinations, permutations, and sampling. The specific Catalan
numbers are 1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16,796 and
so on deriving from
This
particular set of numbers derive from several combinatoric
problems, one of which is the following.
Given "2n"
people gathered at a round table. How many person to person,
noncrossing, handshakes can be made, i.e., no pairs of arms
crossing one another across the table? A few quick sketches of
circles with even sets of dots and lines will lead you to the
first three answers easily. Two people, one handshake. Four
people, two handshakes. Six people, 5 handshakes. With a little
patience and perseverance, eight people will lead you to 14
handshakes. Beyond that, it is probably best to rely on the
given expression.
CHOICE
NUMBERS
Choice
numbers, more commonly called combination numbers, or simply
combinations, are the number of ways that a number of things can
be selected, chosen, or grouped. Combinations concern only the
grouping of items and not the arrangement of those items. They
typically evolve from the question how many combinations of "n"
objects are possible using all "n" objects or "r" objects at a
time? To find the number of combinations of "n" dissimilar
things taken "r" at a time, the formula is:
which can be stated as "n" factorial divided by the product of
"r" factorial times (n  r) factorial.
Example: How
many different ways can you combine the letters A, B, C, and D
in sets of three? Clearly,
namely ABC, ABD, ACD, and BCD. (Note that ACB, BAC, BCA, CAB and
CBA are all the same combination just arranged differently. In how many
ways can a committee of three people be selected from a group of
12 people? We have:
How many
handshakes will take place between six people in a room when
they each shakes hands with all the other people in the room one
time? Here,
Notice that
no consideration is given to the order or arrangement of the
items but simply the combinations.
Another way of
viewing combinations is as follows. Consider the number of
combinations of 5 letters taken 3 at a time. This produces:
Now assume you permute (arrange)
the r = 3 letters in each of the 10 combinations in all possible
ways. Each group would produce r! permutations. Letting x = _{5}C_{3}
for the moment, we would therefore have a total of x(r!)
different permutations. This total, however, represents all the
possible permutations (arrangements) of n things taken r at a
time, which is shown under arrangement numbers and defined as
_{n}P_{r}. Therefore,
which results in
Using the committee of 3 out of 12 people example from
above,
Consider the
following: How many different ways can you enter a 4 door car?
It is clear that there are 4 different ways of entering the car.
Another way of expressing this is:
If we ignore
the presence of the front seats for the purpose of this example,
how many different ways can you exit the car assuming that you
do not exit through the door you entered? Clearly you have 3
choices. This too can be expressed as:
Carrying
this one step further, how many different ways can you enter the
car by one door and exit through another? Entering through door
#1 leaves you with 3 other doors to exit through. The same
result exists if you enter through either of the other 3 doors.
Therefore, the total number of ways of entering and exiting
under the specified conditions is:
Another example of this type of situation is how many ways
can a committee of 4 girls and 3 boys be selected from a class
of 10 girls and 8 boys? This results in:
See
ARRANGEMENT NUMBERS for determining the number of possible
arrangements between items.
CIRCULAR PRIMES
A circular prime number is one that remains a prime number after
repeatedly relocating the first digit of the number to the end
of the number. For example, 197, 971 and 719 are all prime
numbers. Similarly, 1193, 1931, 9311 and 3119 are all prime
numbers. Other numbers that satisfy the definition are 11, 13,
37, 79, 113, 199 and 337.
Primes of two or more digits can only contain the digits 1, 3, 7
because, If 0, 2, 4, 5,6, or 8 were part of the number, in the
units place, the number would be divisible by 2 or 5.
It is thought that there are an infinite number of circular
primes but has not yet been proven.
COMPLEX
NUMBERS
Complex
numbers are formed by the addition of a real number and an
imaginary number, the general form of which is a + bi where i =
= the imaginary number and a and b are real numbers.
The "a" is said to be the real part of the complex number and b
the imaginary part.
COMPOSITE
NUMBERS
Probably the
easiest number to define after prime numbers.
The
Fundamental Theorem of Arithmetic states that every positive
integer greater than 1 is either a prime number or a composite
number. As we know, a prime number "p" is any positive number
the only divisors of which are 1 and p (or 1 and p). Thus, by
definition, any number that is not a prime number must be a
composite number.
A composite
number is any number having 3 or more factors/divisors and is
the result of multiplying prime numbers together. Most of the
positive integers are the product of smaller prime numbers.
Examples: 4,
6, 8, 10, 12, 14, 15, 16, 18, 20, 22, 24, 25, 26, 27, 28, 30,
32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 49, 50,
etc., are all composite numbers, each being divisible by lower
prime numbers. Every number divisible by 2, the only even prime,
is composite.
Every
composite number can be broken down to a single unique set of
prime factors and their exponents.
Examples: 210
= 2 x 3 x 5 x 7; 495 = 3^{2 }x 5^{1 }x 11^{1} or 4500 = 2 x 2 x 3 x 3 x 5 x 5 x 5 =
2^{2} x 3^{2}
x 5^{3}. This is the one and only possible factorization of
the number 210.
If a positive
number N is evenly divisible by any prime number less than
, the number N is composite.
Wilson's
Theorem states that for every prime number "p", [(p + 1)! + 1]
is evenly divisible by "p". The converse was shown to also be
true in that every integer "N" that evenly divides [(N + 1)! +
1] is prime. Combining these leads to the famous general theorem
that a necessary and sufficient condition that an integer "N" be
prime is that "N" evenly divide [(n + 1)! + 1]. Conversely, if
"N" does not divide [(N + 1)! + 1], "N" is composite.
Unfortunately,
the practical use of this method is minimal due to the large
numbers encountered with high N's.
CONGRUENT
NUMBERS
A number N is
said to be congruent if there are two integers, x and y, that
result in the expressions x^{2} + Ny^{2}
and x^{2}  Ny^{2} being
perfect squares. The smallest known congruent number is 5 which
satisfies 41^{2} + 5(12^{2}) = 49^{2}
and 41^{2}  5(12^{2}) = 31^{2}.
The use of the square of a negative number results in another
solution of 2^{2} + 5(1^{2})
= 3^{2}
and 2^{2}  5(1^{2}) = (1)^{2}.
While there are many congruent numbers, finding them is an
arduous task. The expressions x^{2} + Ny^{2}
and x^{2}  Ny^{2} are often useful
in solving many problems in recreational mathematics.
COUNTING
NUMBERS
The counting
numbers are the familiar set of whole numbers, 1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11,....., that we see and use every day. (The 0 is
sometimes included.) The set of counting numbers is often
referred to as the natural numbers.
CUBIC
NUMBERS or CUBES
A cubic number
is the third power of a number as in a x a x a = a^{3}. Those
familiar with the evolution of the squares from adding
successive odd numbers might not be too surprised to discover
how the cubes evolve from summing odd numbers also. Clearly, the
nth cube is simply n^{3}. Cubes can be derived in other ways also:
The cube of
any integer, "n", is the sum of the series of odd numbers
beginning with (n^{2}  n + 1) and ending with (n^{2}
+ n  1). Example: For n = 6, (n^{2}
 n + 1) = 31 and 31 + 33 + 35 + 37 + 39 + 41 = 216 = 6^{3}.
n^{3} 
(n)^{3} 

n^{3} 
(n^{2}
 n + 1) 

(n^{2}
+ n  1) 
1 
1^{3} 
= 
1 
1 
= 
1(1) 
2 
2^{3} 
= 
8 
3 + 5 
= 
2(1 + 2 + 1) 
3 
3^{3} 
= 
27 
7 + 9 + 11 
= 
3(1 + 2 + 3 +
2 + 1) 
4 
4^{3} 
= 
64 
13 + 15 +
17 + 19 
= 
4(1 + 2 + 3 +
4 + 3 + 2 + 1) 
5 
5^{3} 
= 
125 
21 + 23 +
25 + 27 + 29 
= 
5(1 + 2 + 3 +
4 + 5 + 4 + 3 + 2 + 1) 
Starting with
1....2....3....4....5....6....7....8....9....10....11....12....13....14....15....16....17....18....19....20....21
Cross out
every third number giving us
1....2..........4....5.........7.....8..........10....11............13....14............16....17............19....20
Summing
1...3...........7...12.......19...27.........37....48............61....75...........91...108...........127..147
Cross out
every second number giving us
1................7..............19................37....................61...................91....................127
Summing
1................8..............27................64..................125.................216....................343
The last line
of numbers are the perfect cubes.
The first ten
cubes are 1, 8, 27, 64, 125, 216, 343, 512, 729 and 1000.
The sum of the
first n cubes starting with 1, is 1^{3}
+ 2^{3}
+ 3^{3} +........+
n^{3}.
which, quite
surprisingly, is the square of the nth triangular number,
defined by Tn = n(n+1)/2.
The sum of the
cubes of the first n odd numbers is
2n^{4}  n^{2} = n^{2}(2n^{2} 
1).
The sum of the
cubes of the first n even numbers is
2n^{4} + 4n^{3 }+^{ }2n^{2} = 2n^{2}(n
+ 1)^{2}.
The sum of the
first n cubes,
1^{3}
+ 2^{3}
+ 3^{3} + 4^{3}
+........+ n^{3} is equal to
the square of the sum of the first n integers. Thus,
1^{3}
+ 2^{3}
+ 3^{3} + 4^{3}
+........+ n^{3} = (1 + 2 + 3 + 4
+........+
n)^{2}.
The cube of
any integer is the difference of the squares of two other
integers.
Every cube is
either a multiple of 9 or right next to one.
A perfect cube
can end in any of the digits 0 through 9.
Three digit
numbers that are the sum of the cubes of their digits: 153, 370,
371, 407.
The smallest
number that is the sum of 2 cubes in two different ways. 1729 =
1^{3} + 12^{3} = 10^{3} + 9^{3}.
What are the
dimensions of two cubes with integral sides that have their
combined volume equal to the combined length of their edges.
What are the dimensions of the cubes? x = 2 and y =
4.
The sum of any
two cubes can never be a cube.
The sum of a
series of three or more cubes can equal a cube.
Examples:
3^{3}
+ 4^{3}
+ 5^{3} = 6^{3}
11^{3}
+ 12^{3}
+ 13^{3}
+ 14^{3} = 20^{3}
1134^{3}
+ 1135^{3}
+ .........2133^{3} = 16,830^{3}
The cube root
of a number N is the number "a" which, when multiplied by itself
twice, results in the number N or N = axaxa. There is no formula
for extracting the cube root of a number. It can be obtained by
means of a long division method or a simple estimation method.
The sum of "n"
terms of an arithmetic progression with the first term equal to
the sum of the first "n" natural numbers and a common difference
of "n" is n^{3}.
First, a
method for approximating the cube root of a number to several
decimal places which is usually sufficient for everyday use.
Estimation
Method
1Make an
estimate of the cube root of N = n lying between successive
integers a and b.
2Compute A =
N  a^{3}
and B = b^{3}  N
2n = a +
(bA)/[bA + aB]
Example: Find
the cube root of 146
1With 146
lying between 125 and 216, let a = 5 and b = 6.
2A = 21 and
B = 70
3n = 5 +
(6 x 21)/[6 x 21 + 5 x 70] = 5 + 126/476 = 5 + .264 = 5.2647
4The cube
root of 146 is 5.2656
A more exact
method for determining integer cube roots.
Note that each
cube of the numbers 1 through 10 ends in a different digit:
n^{3}....1......2......3......4......5......6.......7......8.......9......10
n^{3}....1......8....27....64...125...216...343...512...729...1000
Also note that
the last digit is the cube root for all cases except 2, 3, 7 and
8. A quick review of these exceptions leads to the fact that
these four digits are the difference between 10 and the cube
root, i.e., 8 = 10  2, 7 = 10  3, 3 = 10  7 and 2 = 10  8.
How can this information be used to determine the cube root of a
number?
Given the cube
of a number between 1 and 100, say 300,763.
The last digit
tells us that the last digit of the cube root is 10  3 = 7.
Eliminating
the last 3 digits of the cube leaves the number 300.
The number 300
lies between the cubes of 6 and 7 in our listing above.
The first
digit of our cube root will be the lowest of these two numbers,
in this case 6.
Therefore, the
cube root of 30,763 becomes 67.
Given the cube
592,704.
The last digit
of 4 is the last digit of the cube root.
The number 592
lies between 512 and 729, the cubes of 8 and 9.
Therefore, the
cube root of 592,704 becomes 84.
Of course,
this is only useful if you know ahead of time that the cube is a
perfect cube, i.e., having an integral cube root.
CYCLIC NUMBERS
A cyclic
number is a number of "n" digits that when multiplied by 1, 2,
3,...n, results in the same digits but in a different order. For
example, the number 142,857 is a cyclic number since 142,857 x 2 =
285,714, 142,857 x 3 = 428,571, 142,857 x 4 = 571,428, and so on. It
is not known just how many cyclic numbers exist.
DECIMAL
NUMBERS
Decimal
numbers are numbers expressed through the decimal, or base 10,
number system where each digit represents a multiple of some
power of 10. The term applies primarily to numbers that have
fractional parts so indicated by a decimal point. A number less
than 1 is called a decimal fraction, e.g., .673. A mixed decimal
is one consisting of an integer and a decimal fraction, e.g.,
37.937.
.
Rational
numbers can be expressed in the form of a fraction, 1/2, or as a
decimal, .50, 1/8, or as a decimal, .125. From experience, we
know that a fraction expressed in decimal form will either
terminate without a remainder such as 3/8 = 0.375 or 7/8 =
0.875, repeat the same digit endlessly such as 1/3 =
.3333333..... or 2/3 = .6666666....., repeat a series of
different digits repeatedly such as 1/27 = .037037037... or 1/7
= .142857142857...., or repeat a series of digits after some non
repeating digits such as 1/12 = .0833333.....
All prime
denominators produce repeating decimals. Fractions with the same
denominator often produce decimals with the same period and
period length but with the digits starting with a different
number in the period. For instance, 1/7 =
.142857142857142857...., 2/7 = .285714285714285714...., 3/7 =
.428571428571428571...., 4/7 = .571428571428571428...., 5/7 =
.714285714285714285...., and 6/7 = .857142857142857142.... Other
denominators produce two or more repeating periods in different
orders. Investigate those with prime denominators of 11, 13, 17,
19, etc. and then 9, 12, 14, 16, 18, etc.and see what results.
What do you notice?
Another
interesting property of repeating decimals of even period length
is illustrated by the following. Take the decimal equivalent of
2/7 = .285714285714...... The repeating period is 285714. Split
the period into two groups of three digits and add them
together. The result is 999. Do the same with any other
repeating decimal period and the result will always be a series
of nines. Take 1/17 = .0588235294117647........ Adding 05882352
and 94117647 gives you 99999999.
All repeating
decimals, regardless of the period and length, are rational
numbers. This simply means that it can be expressed as the
quotient of two integers. A question that frequently arises is
how to convert a repeating decimal, which we know to be
rational, back to a fraction.
Rational
decimal fractions may be converted to fractions as follows:
Given the
decimal number N = 0.078078078...
Multiply N by
1000 or 1000N = 78.078078078...
Subtracting N = .078078078...
.................................999N = 78 making N = 78/999 =
26/333.
Given the
decimal number N = .076923076923...
Muliply N by
1,000,000 or 1,000,000N = 76,923.0769230769230...
Subtracting N =
.0769231769230...
.........................................999,999N = 76,923
making N = 76,923/999,999 = 1/13
An easier way
to derive the fraction is to simply place the repeating digits
over the same number of 9's. For example, the repeating decimal
of .729729729729 converts to the fraction of 729/999 = 27/37.
Another trick
where zeros are involved is to place the repeating digits over
the same number of 9's with as many zeros following the 9's as
there are zeros in the repeating decimal. For example, .00757575
leads to 075/9900 = 1/132.
DEFICIENT
NUMBERS
Deficient
numbers are part of the family of numbers that are either
deficient, perfect, or abundant.
Deficient
numbers, dN, are numbers where the sum of its aliquot parts
(proper divisors), sa(N), is less than the number itself sa(N) <
N. (In the language of the Greek mathematicians, the divisors of
a number N were defined as any whole numbers smaller than N
that, when divided into N, produced whole numbers. The
factors/divisors of a number N, less the number itself, are
referred to as the aliquot parts, or aliquot divisors, of the
number.) Equivalently, N is also deficient if the sum, s(N), of
all its divisors is less than 2N. It can be readily seen that
using the aliquot parts summation, sa(16) = 1+2+4+8 = 14 < N =
16 while using all of the divisors, s(16) = 1+2+4+8+16 = 31 < 2N
= 32, making the number 16 deficient under either definition.
From the
following list
N>.......1..2..3..4..5...6....7...8....9..10..11..12..13..14..15..16..17..18..19...20..21..22..23..24
Sa(N)>1..1..1..3..1...6....1...7....4...8....1...16...1...10...9...15...1....21...1...22..11..12...1...36
S(N)>
1..3..4..7..6..12...8..15..13.18..12..28..14..22..24..31..18...39..20..42..32..36..24..60
1,2,3,4,5,7,8,9,10,11,13,14,15,16,17,19,21,22,23 are all
deficient.
A prime number
or any power of a prime number is deficient. The divisors of a
pefect or deficient number is deficient.
DIGITAL
ROOT
The digital
root of a number is the single digit that results from the
continuous summation of the digits of the number and the numbers
resulting from each summation. For example, consider the number
7935. The summation of its digits is 24. The summation of 2 and
4 is 6, the digital root of 7935. Digital roots are used to
check addition and multiplication by means of a method called
casting out nines. For example, check the summation of 378 and
942. The DR of 378 is 9, 3+7+8=18, 1+8=9.. The DR of 942 is 6,
9+4+2=15, 1+5=6. Adding 9 and 6 produces 15, The DR of 15 is 6,
1+5=6. The summation of 378 and 942 is 1320. The DR of 1320 is
6. With the two final DR's are equal, the addition is correct.
EGYPTIAN
FRACTIONS
Egyptian fractions are the reciprocals of the positive integers
where the numerator is always one. They are often referred to as
unit fractions. They were used exclusively by the Egyptians to
represent all forms of fractions. The two fractions that they
used that did not have a unit fraction was 2/3 and 3/4. The only
other fractions that they seemed to have a strong interest in
were those of the form 2/n where n was any positive odd number.
The Rhind papyrus contains a list of unit fractions representing
a series of 2/n for odd n's from 5 to 501. It is unclear as to
why they found these 2/n fractions so important. These and other
2/n fractions may be derived from 2/n = 1/[n(n+1)/2] +
1/[n+1)/2].
In the year 1202, Leonardo Fibonacci proved that any ordinary
fraction could be expressed as the sum of a series of unit
fractions in an infinite number of ways. He used the then named
greedy method for deriving basic unit fraction expansions. He
described the greedy method in his Liber Abaci as simply
subtracting the largest unit fraction less than the given non
unit fraction and repeating the process until only unit
fractions remained. It was later shown that the greedy method,
when applied to any fraction m/n, results in a series of no more
than "m" unit fractions. An example will best illustrate the
process.
Reduce the fraction 13/17 to a sum of unit fractions. Dividing
the fraction yields .7647. Of the unit fractions 1/2, 1/3, 1/4,
1/5, etc., 1/2 is the largest that is smaller than 13/17 so we
compute 13/17  1/2 = 9/34 making 13/17 = 1/2 + 9/34. Repeating
with 9/34, we have 9/34  1/4 = 2/136 = 1/68 making 13/17 = 1/2
+ 1/4 + 1/68. Alternatively, divide the numerator into the
denominator and use the next highest integer as the new
denominator. 17/13 = 1.307 making 2 the denominator of the
fraction to be subtracted from 13/17.
Reduce the fraction 23/37 to a sum of unit fractions. 23/37 
1/2 = 9/74  1/16 = 70/1184  1/17 = 6/20,128  1/3355 =
1/33,764,720 making 23/37 = 1/2 + 1/16 + 1/17 + 1/3355 +
1/33,764,720.
There are many methods or algorithms that derive the unit
fractions for any fraction m/n. Much more information regarding
unit fractions can be found at
Unit Fractions Search at
http://mathpages.com/cgilocal/ATmathpakbsearch.cgi
Egyptian Fractions at
http://www1.ics.uci.edu/~eppstein/numth/egypt/
Unit Fraction from Math World at
http://mathworld.wolfram.com/UnitFraction.html
Algorithms for Egyptian Fractions at
http://www1.ics.uci.edu/~eppstein/numth/egypt/force.html
Creating Unit Fractions at
http://www.mathpages.com/home/kmath150.html
The unit fractions derived by means of the method shown, or any
other method, can be further broken down into other unit
fractions by means of the identity 1/a = 1/(a+1) + 1/a(a+1),
also known to Fibonacci. For example, 1/2 = 1/(2+1) + 1/2(2+1) =
1/3 + 1/6. Further still, 1/3 = 1/(3+1) + 1/3(3+1) = 1/4 + 1/12
and 1/6 = 1/(6+1) + 1/6(6+1) = 1/7 + 1/42 yielding 1/2 = 1/4 +
1/7 + 1/12 + 1/42. While the number of unit fractions derivable
for any given fraction is therefore infinite, there is
apparently no known procedure for deriving a series with the
least number of unit fractions or the smallest largest
denominator.
.
EQUABLE TRIANGLES
A shape is
called
equable if its area equals its perimeter. There are exactly
five equable Heronian triangles: the ones with side lengths
(5,12,13), (6,8,10), (6,25,29), (7,15,20), and (9,10,17).
EQUIVALENT NUMBERS
Equivalent numbers are numbers where the aliquot parts (proper
divisors other than the number itself) are identical. For
instance, 159, 559 and 703 are equivalent numbers since their
aliquot parts sum to 57.
f(159) = 1, 3, 53 and 159 where 1 + 3 + 53 = 57.
f(559) = 1, 13, 43 and 559 where 1 + 13 + 43 = 57.
f(703) = 1, 19, 37 and 703 where 1 + 19 + 37 = 57.
EVEN
NUMBERS
Probably the
easiest number to define, an even number is any number that is
evenly divisible by 2.
The nth even
number is given by Ne = 2n.
The sum of the
set of "n" consecutive even numbers beginning with 2 is given by
Se = n(n + 1).
The sum of the
set of m consecutive even numbers starting with n1 and ending
with n2 is given by Se(n1n2) = n2^2  n1^2 + (n1 + n2) or (n1 +
n2)(1 + n1  n2).
The sum of the
squares of the even numbers starting with 2^2 is given by Se^2 =
(4n^3 + 6n^2 + 2n)/3.
An even number
multiplied by any number, or raised to any power, results in
another even number.
The number 2
is the only even prime number.
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