PYTHAGOREAN
NUMBERS
Pythagorean
numbers, more typically referred to as Pythagorean Triples, are
those sets of three integers that define all integral primitive
and nonprimitive Pythagorean right triangles. General formulas
for deriving all integer sided rightangled Pythagorean
triangles, have been known since the days of Diophantus and the
early Greeks. For the right triangle with sides x, y, and z, z
being the hypotenuse, the lengths of the three sides of the
triangle can be derived as follows: x = k(m^2  n^2), y =
k(2mn), and z = k(m^2 + n^2) where k = 1 for primitive triangles
(x, y, and z having no common factor), m and n are arbitrarily
selected integers, one odd, one even, usually called generating
numbers, with m greater than n. Another set that was attributed
to Pythagoras took the form of x = 2n + 1, y = 2n^2 + 2n, and z
= 2n^2 + 2n + 1 where n is any integer. (It was ultimately
discovered that these formulas created only triangles where the
hypotenuse exceeded the larger leg by one.) Another set of
expressions that produce triples is x = n^2, y = (n^2  1)^2/2,
and z = (n^2 + 1)^2/2. For any positive integer m, 2m, m^2  1,
and m^2 + 1 are Pythagorean Triples.
Primitive
Pythagorean Triples always have one leg odd, one leg even, and
the hypotenuse odd (x and y cannot both be odd or even).
In every
primitive Pythagorean triple, either x or y is divisible by 3,
either x or y is divisible by 4, and either x, y, or z is
divisible by 5. The product of the two legs is always divisible
by 12, and the product of all three sides is always divisible by
60. The area is always divisible by 6.
Triples with
all sides even are nonprimitive but can be reduced to a
primitive triple by dividing through by a constant factor.
Triples with
all sides odd are impossible.
The sum and
difference of the hypotenuse and the even side of a primitive
Pythagorean triple are squares.
Only one side
of a Pythagorean triangle can be a square.
If m and n
have no factor in common but mn is a square, then both m and n
are squares.
All primitive
hypotenuses are primes of the form 4n + 1.
A number may
be a primitive hypotenuse only if all of its prime factors are
of the form 4n + 1. No primitive
Other unique
Pythagorean Triples can also be derived from m and n values
based on the triangular numbers T = n(n+1)/2, i.e., 1, 3, 6, 10,
15, 21, etc. Using consecutive Triangular numbers for m and n,
the triples that result have the smallest leg a perfect cube.
The product of
the two legs of a right triangle is equal to the product of the
hypotenuse and the altitude to the hypotenuse.
The area of a
Pythagorean triangle is never a square number or twice a
square.
There is a
much more in depth article on Pythagorean Triangles in the
Knowledge Database.
QUASIPERFECT NUMBERS
A quasiperfect
number N has been defined as one where the sum of all of its
factors/divisors is equal to one more than twice the number or
s(N) = 2N + 1. Up to now, I have not seen none or heard that any
exist which makes me wonder what led to the creation of the
definition in the first place.
(See perfect,
multiplyperfect, semiperfect, deficient, least deficient,
abundant, super abundant)
RANDOM
NUMBERS
Random numbers
are groups or sequences of numbers within which, eventually, all
of the numbers will occur equally often and where the occurrence
of any one number in any location within the group provides no
indication as to the occurrence of any other numbers in the
group. Random numbers are traditionally used in the design of
experiments and product sampling. The most basic method of
defining random numbers is the drawing of numbered cards,
tickets, balls, etc., from a container filled with the numbers
involved. There are computer programs that generate random
numbers.
RATIONAL
NUMBERS
Rational
numbers are any number that can be represented by an integer "a"
or the ratio of two integers, a/b, where 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 or b = 1, the ratio is an integer. If b is other than 1,
a/b is a fraction. 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.
3/5 is a proper fraction while 8/5 is an improper fraction
equaling 1 3/5. Any rational number can be expressed as a
decimal. All decimal equivalents of a/b, other than those with b
equal to 2 or 5, result in a repeating decimal. Decimals
resulting from fractions with the denominator being powers of 2,
5, or both, i.e., denominators of 2, 4, 5, 8, 10, 16, 20, 25,
32, 40, 50, 64, and 80, etc., are terminating decimals. The
group of numbers that are found to repeat in nonterminating
decimals are referred to as the period of the repeating decimal.
The number of digits in the repeating group is referred to as
the length of the period. You will note that when a is divided
by b, the remainders can only be 1, 2, 3,....(b  1), meaning
that after (b  1) steps of division, the possible remainders
must repeat themselves
REAL
NUMBERS
Real numbers
are those numbers that can be represented by the points on a
number line. Real numbers include both the rational and
irrational numbers. When using the term number, it is normally
assumed that the number is a real number.
RECTANGULAR
NUMBERS
Rectangular
numbers are numbers that represent the total number of dots
contained within a rectangular arrangement of dots with the
number of columns being one more than the number of rows.
Coincidentally, rectangular numbers are the sums of the first
"n" even integers as defined by the expression Rn = n(n + 1).
They are also twice the triangular numbers and sometimes
referred
to as oblong or pronic numbers.
RELATIVELY
PRIME NUMBERS
Relatively
prime numbers are pairs of numbers that have no common factor
other than one, or unity. The numbers 2 and 7, 3 and 8, 11 and
27, etc., are all relatively prime. Pairs of numbers satisfying
this criteria are also referred to as coprime numbers.
REPUNIT
NUMBERS
A repunit
number is one consisting of a continuous string of "n" ones in a
specific base. 11,111 and 111,111,111,111 are simple examples.
Repunits in base 10 with n = 2, 19, 23, 317, and 1031 are prime
numbers.
SEMIPERFECT NUMBERS
A semiperfect
number is a number that is equal to the sum of "some" of its
aliquot parts or proper divisors. The smallest semiperfect
number is 12 which is the sum of 2, 4, and 6. 18 is the next
with 18 = 3 + 6 + 9. 24 is another with 24 = 4 + 8 + 12 or 1 + 2
+ 3 + 4 + 6 + 8.
(See perfect,
multiplyperfect, quasiperfect, deficient, least deficient,
abundant, super abundant)
SEQUENCE
NUMBERS
Number
sequences of many varieties have been the focus of study and
enjoyment since the ten digits first appeared on the scene. Our
first exposure to number sequences was, in fact, the counting
numbers, or natural numbers, that we learned in our first year
of school. Without question, the sequence of counting numbers is
the most fundamental of number sequences and all other number
sequences derive from manipulations of these counting numbers. A
broad, in depth, discussion of sequences may be found elsewhere
in the Knowledge Database under the heading of Sequences. The
following is but a brief introduction to the topic. It is my
hope that the material that follows will be interesting and
entertaining enough to spur you to seek out further information,
and believe me, there is much more to be learned. Lets see where
these magical numbers take us.
The Natural
Numbers are the familiar set of whole numbers, 1, 2, 3, 4, 5, 6,
7, 8, 9, 10, 11,.....etc., that we see and use every day. (The 0
is sometimes included.) The set of natural numbers is often
referred to as the counting numbers and often denoted by N.
A number
sequence, or progression, is an orderly set of numbers arranged
in such a way that each
successive
number in the sequence is defined by a fixed rule or law related
to the position in the sequence or the previous number, or
numbers, in the sequence. The specific numbers in the sequence
are called the terms or elements of the sequence.
A number
series is the sum of the terms of a number sequence.
The sum of the
numbers in a sequence, or the series of numbers, can be finite
or infinite. Finite sequences and series have a well defined
first and last term while infinite sequences and series continue
forever. A finite set of numbers is typically portrayed as [1,
4, 7, 10, 13, 16] while an infinite set of numbers is usually
portrayed as [1, 3, 5, 7, 9, ......]. Very often, the brackets
are omitted.
A divergent
sequence/series is one having no finite sum.
A convergent
sequence/series is one having a finite sum or limit.
Sequences/Series come in many varieties, arithmetic, geometric,
harmonic, power, finite difference, binary, triad, to name a
few, plus many known primarily by the name of their discoverer
such as Fibonacci, Lucas, Pell, and Pascal. Others derive their
designations from specific geometric manipulations or
arrangements of the counting numbers such as figurate or
polygonal, pyramidal, tetrahedral, etc. Please refer to the
separate KD article titled Sequences. (Submitted in full for KD
entry on 02/11/02)
SOCIABLE
NUMBERS
You may
already be familiar with perfect numbers and amicable numbers.
Perfect numbers are those numbers whose aliquot divisors (all
the divisors except the number itself) add up to the number
itself. The numbers 6, 28 and 496 are examples of perfect
numbers. Amicable numbers are pairs of numbers, each of which
is the sum of the other numbers aliquot divisors. 220 and 284
are a pair of amicable numbers. This can be viewed in another
way. If the aliquot divisors of a number sum to the number, it
is called perfect. If the aliquot divisors sum to another number
whose aliquot divisors then sum to the first number, the two
numbers are called amicable. Strange as it might seem, there are
numbers whose successive sums of aliquot divisors sum to the
initial number after more than two summations. Over the years,
two examples of this have surfaced. The first, 12,496, has
aliquot divisor sums of 14,288, 15,472, 14,536, 14,264, and
12,496. The number 14,316 returns to itself after 28 additions
of the aliquot divisors. These numbers have been referred to as
sociable numbers and amicable chains.
SQUARE
NUMBERS +
A square
number, sometimes called a perfect square, is the result of
multiplying a number by itself as in N = nxn = n^2.
The perfect
squares are the squares of the counting numbers, 1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12, etc. which produce 1, 4, 9, 16, 25, 36,
49, 64, 81, 100, 121, 144, etc.
Believe it or
not, the perfect squares are simply the successive sums of the
odd numbers.
0 + 1 = 1
1 + 3 = 4
1 + 3 + 5 = 9
1 + 3 + 5 + 7
= 16
1 + 3 + 5 + 7
+ 9 = 25, etc.
The last digit
in the square of a number must be one of the following: 0, 1, 4,
5, 6, or 9. Note that there are numbers that end in these digits
that are not squares but to be a square, they must end in one of
these digits.
The last two
digits of a 3 or more digit square number must be one of the
following: 00, 01, 04, 09, 16, 21, 24, 25, 29, 36, 41, 44, 49,
56, 61, 64, 69, 76, 81, 84, 89, or 96.
To be a
square, the sum of the digits in a square must add up to 1, 4,
7, or 9.
The nth square
is equal to n^2.
n....1....2....3....4....5....6....7....8....9....10
Sq..1...4....9...16..25..36..49..64...81..100
1 = 1 = 1^2
1 + 2 + 1 =
4 = 2^2
1 + 2 + 3 +
2 + 1 = 9 = 3^2
1 + 2 + 3 +
4 + 3 + 2 + 1 = 16 = 4^2
1 + 2 + 3 +
4 + 5 + 4 + 3 + 2 + 1 = 25 = 5^2
The nth
nonsquare is n + r.o.[sqrt(n)] (r.o. = rounded off)
n....................1....2....3....4....5....6....7....8....9....10....11....12....13....14....15
Nonsquare.....2....3....4....6....7....8...10..11...12...13....14....15....17....18....19
6th nonsquare
= 6 + 2[.449] = 8; 12th nonsquare = 12 + 3[.464] = 15.
The perfect
squares are also the successive sum of the triangular numbers,
Tn = n(n + 1)/2 = 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, etc.
1^2 = 0 + 1 =
1
2^2 = 1 + 3 =
1 + 3
3^2 = 1 + 3 +
5 = 3 + 6
4^2 = 1 + 3 +
5 + 7 = 6 + 10
5^2 = 1 + 3 +
5 + 7 + 9 = 10 + 15  1, 3, 6, 10, 15, etc., being the
sequential triangular numbers.
The square of
a number is either divisible by 4 or leaves a remainder of 1
when divided by 4.
The square of
an odd number is always of the form 8n + 1.
The square,
when divided by 8, leaves a remainder of 0, 1, or 4.
A perfect
square always has an odd number of factors/divisors. The total
number of factors of a number derives from f(N) = (a + 1)(b +
1)(c + 1)......(n + 1) where a, b, c,.....n are the exponents of
the prime factors of the number. Since all the exponents of a
perfect square have to be even, a, b, c, ....n being even, (a +
1)(b + 1)(c + 1)....(n + 1 becomes the product of n odd numbers
resulting in another odd number.
Examples:
25 = 5^2 and
f(25) = (2 + 1) = 3 the factors being 1, 5, and 25.
36 = 6^2, the
prime factors being 2^2(3^2) and f(36) =(2 + 1)(2 + 1) = 9, the
factors being , 1, 2, 3, 4, 6, 9, 12, 18, and 36, 9 in all.
81 = 9^2, the
prime factors being 3^4 and f(81) = (4 + 1) = 5, the factors
being 1, 3, 9, 27, and 81 or 5 in all.
144^2 = 20,736
= the prime factors being 2^6(3^4)2^2 and f(20,736) = (6 + 1)(4
+ 1)(2 + 1) = 105 factors in all.
The square of
any prime number has only 3 factors.
The sum of the
first n squares, i.e., 1^2 + 2^2 + 3^2 + 4^2 + .....+ n^2 = n(n
+ 1)(2n + 1)/6.
It follows
that every odd number is the difference of two squares.
1 = 1^2  0^1,
3 = 2^2  1^2, 5 = 3^2  2^2, 7 = 4^2  3^2, 9 = 5^2  4^2, etc.
All even
numbers with a prime factor of 2 and exponent of 2 or more is
the difference of two squares in one or more ways..
4 = 2^2 = 2^2
 0^2, 12 = 2^2(3) = 4^2  2^2, 16 = 4^2 = 4^2  0^2, 20 =
2^2(5) = 6^2  4^2, 24 = 2^3(3) = 5^2  1^2, etc.
Every
octagonal number is the difference of two squares.
n........................................1...........2.............3.............4.............5.............6
Noct = n(3n 
2)................1............8............21...........40...........65...........96.........etc.
(2n1)^2 
(n1)^2.........1^20^2...3^21^2...5^22^2...7^25^2...9^24^2...11^25^2
SQUAREFULL
NUMBERS
A squarefull number is a positive whole number N having the
property that for very prime number, P, that evenly divides into
N, P^2 also evenly divides N. All squarefull numbers are of the
form a^2(b^3), a and b being positive whole numbers also. The
first few known squarefull numbers are 1, 4, 8, 9, 16, 25, 27,
32, 36 and 49.
SQUBES
There are
infinitely many squbes derived from n^6 = n^3xn^3 = (n^3)^2 =
n^2xn^2xn^2 = (n^2)^3.
Examples:
1^6 = 1 = 1^2
= 2^3, 2^6 = 64 = 8^2 = 4^3, 3^6 = 729 = 27^2 = 9^3, 4^6 = 4096
= 64^2 = 16^3, 5^6 = 15,625 = 125^2 = 25^3, 6^6 = 46,656 = 216^2
= 36^3, 7^6 = 117,649 = 343^2 = 49^3, etc.
SUPERABUNDANT
An extension
of abundant numbers, a superabundant number has been defined as
a number, N where the sum of all of its factors/divisors divided
by N results in a ratio that is greater than the sum of all the
factors/divisors of the number K divided by K for all values of
K less than N. This is expressed by s(N)/N > s(K)/K for all K <
N.
Of the first
10 numbers, the s(N) are
N..........1.....2.....3.....4.....5.....6.....7.....8......9......10
s(N).....1.....3.....4.....7.....6....12....8....15....13.....18
s(N)/N..1...3/2..4/3..7/4..6/5.12/6.8/7.15/8.13/9.18/10.
By inspection,
2, 4, and 6 are superabundant. There are supposedly an infinite
number of superabundant numbers
TAG NUMBERS
Tag numbers
are simply other names for objects in our every day use of
numbers. A telephone number, a license number, an airline flight
number, a bus number, etc., are examples of tag numbers. Tag
numbers represent the third application of numbers in our lives
after cardinal and ordinal numbers.
TAXICAB
NUMBERS
Numbers that
can be expressed as the sum of two cubes in two different ways
are referred to as Taxicab numbers or HardyRamanujan numbers.
They were so named as a result of a conversation between the
mathematicians G.H. Hardy and Srinivasa Ramanujan. Hardy had
driven to a hospital to visit Ramanujan in a taxi numbered 1729.
Upon arrival, Hardy mentioned this to Ramanujan, saying how
uninteresting the number 1729 was. Ramanujan's immediate
response was that it was quite the contrary explaining that 1729
was the smallest positive number that was the sum of two cubes
in two different ways, 1729 = 12^3 + 1^3 = 10^3 + 9^3.
4104 = 16^3 +
2^3 = 15^3 + 9^3
20,682 = 19^3
+ 24^3 = 10^3 + 27^3
39,312 = 15^3
+ 33^3 = 2^3 + 34^3
40,033 = 16^3
+ 33^3 = 9^3 + 34^3
87,539,319 =
167^3 + 436^3 = 228^3 + 423^3 = 255^3 + 414^3.
are some other
examples of which there are an infinite number.
TETRAHEDRAL
NUMBERS
Tetrahedral
numbers are the consecutive sums of the triangular numbers. With
the triangular numbers being, 1, 3, 6, 10, 15, 21, 28, 36, etc.,
the tetrahedral numbers become 1, 4, 10, 20, 35, 56, 84, 120,
etc. and are derived from Tet(n) = n(n + 1)(n + 2)/6. Note that
the stacked triangular numbers form a three sided pyramid which
leads to their often being called triangular pyramidal numbers.
If there 50 women in the chorus line, and the 8th women's
employee number is 28945, the 50 is a cardinal number, the 8 is
an ordinal number, and the 28945 is a tag number.
TRANSCENDENTAL NUMBERS
Transcendental
numbers are numbers that cannot be the roots of polynomial
equations with rational coefficients. They are a sub set of
irrational numbers. Trigonometric functions as well as numbers
like Pi and "e" are transcendental.
e
The number
referred to as "e" is one of the most important numbers in
mathematics. Its first reference was by John Napier in 1618 and
first approximated by Jacob Bernouli. It was named by Euler in
1727, a date relatively new in the broad history of mathematics.
It subsequently was referred to as the Euler number. The
definition of "e" is the ultimate limiting sum of an infinite
series of numbers. The initial awareness of the number came from
Bernouli while working through a series of computations
involving compound interest.
Consider the
growth of an investment of $1.00 at 100% interest. If the
interest is compounded annually, the $1.00 grows to $2.00 at the
end of the first year. If the interest is compounded
semiannually, the $1.00 grows to $2.25 at the end of the first
year. If the interest is compounded quarterly, the $1.00 grows
to $2.4414. If the interest is compounded monthly,, the $1.00
grows to $2.613; weekly  $2.6925; daily  2.71456; continuously
 2.7182. Depending on the nature of continuously, the limiting
number becomes 2.718281828... The sums can also be expressed by
S = (n + 1/n)^n where S = the compounded growth of the initial
$1.00 investment and n = the number of compounded periods. Thus
for n = 1, S = 2; for n = 2, S = $2.25; for n = 4, S = $2.4414;
for n = 12, S = $2.613; for n = 52, S = 2.6925; for n = 365, S =
$2.71456; for n = 31,536,000, S = 2.718281828.
e can also be
derived from e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6!
+...
e^2 = 1 +
2^1/1! + 2^2/2! + 2^3/3! + 2^4/4! + 2^5/5! + 2^6/6! +...
Pi  π
The number Pi
was known for ages before anyone attempted to develop a proof of
its value.
Pi  the 16th
letter of the Greek alphabet
π  the symbol
for the ratio of the circumference of a circle to its diameter,
3.141592...
The symbol π,
and the number it represents, has been the focus of
mathematicians for close to four thousand years. Many hours,
years, decades and centuries have been spent searching for an
exact value of this extraordinary number with no success. There
are numerous methods that were developed over the years aimed at
deriving π to as many digits of the number as possible. History
records that the earliest recording of the number was by an
Egyptian scribe on the Rhind Papyrus in 1650 B.C.E.
Archimedes was
the first to apply a very analytical, and realistic, approach to
calculating π to a high degree of accuracy. His method was to
calculate the perimeters of a pair of inscribed, and
circumscribed, hexagons of a given circle. Given the diameter of
the circle, he was then able to calculate the lengths of the
inscribed and circumscribed, sides of the hexagons. The sums of
the respective dides produced the perimeters of the two
hexagons. Clearly, the perimeter of the inscribed hexagon was
less that the circumference of the circle and the perimeter of
the circumscribed hexagon was greater than the circumference of
the circle.
Perhaps, you
can see where Archimedes was going with this approach. Having
the ratio of each hexagon to the diameter, he proceeded to
double the number of sides to inscribed and circumscribed
polygons and derive a new set of perimeter to diameter ratios.
He continued this process up to 96 sided polygons. His final set
of calculations resulted in the perimeter of the 96 sided
inscribed polygon having a perimeter to diameter ratio of
3.14103 and that of the similarly circumscribed polygon having
the ratio of 3.14271, leading him to conclude that π lied
somewhere between 3 10/71 and 3 10/70.
π is an
infinite nonrepeating decimal with no repeating pattern.
Some unusual
sequences that produce π are:
The limit
of Sn = 1/n^2 = 1/1^2 + 1/2^2 + 1/3^2 +......as n>inf. =
Pi^2/6.
The sum of
Sn = 1/n^4 = 1/1^4 + 1/2^4 + 1/3^4 +......as n>inf. =
Pi^4/90.
The limit
of Sn = 1/(2n1)^2 = 1/1^2 + 1/3^2 + 1/5^2 +.....as n>inf. =
Pi^2/8.
The limit
of Sn = 1/(2n1)^4 = 1/1^4 + 1/3^4 + 1/5^4 +.....as n>inf. =
Pi^4/96.
The limit
of Sn = [(1)^(n+1)]/(2n1) = 1  1/3 + 1/5  1/7 + 1/9 
.........as n>inf. = Pi/4.
The limit
of Sn = [(1)^(n+1)]/n^2 = 1/1^2  1/2^2 + 1/3^2  1/4^2
+......as n>inf. = Pi^2/12.
The limit
of Sn = [(1)^(n+1)]/(2n1)^3 = 1/1^3  1/3^3 + 1/5^3  1/7^3
+......as n>inf. = Pi^3/32.
Needless to
say, π to more than 10 decimal places is ever required.
π =
3.1415926535 Use as many digits as you think you need.
TRAPEZOIDAL NUMBERS
Trapezoidal numbers are all positive
integers which can be written as a sum of at least 2 consecutive
positive integers.
Examples:…….1………….11………….***********
………………2….3……..12..13………*************
Sum……..……6…..……...36………..***************
TRIANGULAR
NUMBERS
Triangular
numbers, members of the class of figurate, or polygonal,
numbers, are numbers that can be represented by geometrically
arranged groups of dots. They derive their names from the
numerous geometric arrangements of dots, circles, etc., that
they form. The number of dots, circles, etc., that can be
arranged in a line is called a linear number. The number of
dots, circles, etc., that can be arranged in an equilateral or
right triangular pattern is called a triangular number. The 10
bowling pins form a triangular number as do the 15 balls racked
up on a pool table. Upon further inspection, it becomes
immediately clear that the triangular numbers, T1, T2, T3,
T4,.....Tn, are simply the sum of the consecutive integers
1234.....n or Tn = n(n + 1)/2, namely, 1, 3, 6, 10, 15, 21,
28, 36, 45, 55, 66,78, 91, etc.
The number of
dots, circles, spheres, etc., that can be arranged in an
equilateral or right triangular pattern is called a triangular
number. The 10 bowling pins form a triangular number as do the
15 balls racked up on a pool table. Upon further inspection, it
becomes immediately clear that the triangular numbers, T1, T2,
T3, T4, etc., are simply the sum of the consecutive integers
1234.....n or Tn = n(n + 1)/2, namely, 1, 3, 6, 10, 15, 21,
28, 36, 45, 55, 66,78, 91, etc.
Triangular
numbers are the sum of the balls in the triangle as defined by
Tn = n(n + 1)/2.
Order.n...1........2...............3.....................4.............................5....................6.....7.....8.....9
.............O.......O...............O....................O............................O
....................O...O.........O...O...............O...O.......................O...O
..................................O...O...O.........O....O....O...............O....O....O
......................................................O....O...O....O.........O.....O...O....O
.................................................................................O....O....O....O....O
Total......1........3.................6....................10..........................15..................21...28...36...45...etc.
The sum of a
series of triangular numbers from 1 through Tn is given by S =
(n^3 + 3n^2 + 2n)/6.
After staring
at several triangular and square polygonal number arrangements,
one can quickly see that the 1st and 2nd triangular numbers
actually form the 2nd square number 4. Similarly, the 2nd and
3rd triangular numbers form the 3rd square number 9, and so on.
By inspection, one can see that the nth square number, Sn, is
equal to Tn + T(n  1) = n^2. This can best be visualized from
the following:
.........Tn

1...3...6...10...15...21...28...36...45...55...66...78...91
.........T(n 
1)........1...3....6....10...15...21...28...36...45...55...66...78
.........Sn.........1...4...9....16...25...36...49...64...81..100.121.144.169
A number
cannot be triangular if its digital root is 2, 4, 5, 7 or 8.
Some
interesting characteristics of Triangular numbers:
The numbers 1
and 36 are both square and triangular. Some other triangular
squares are 1225, 41,616, 1,413,721, 48,024,900 and
1,631,432,881. Triangular squares can be derived from the series
0, 1, 6, 35, 204, 1189............Un where Un = 6U(n  1)  U(n
 2) where each term is six times the previous term, diminished
by the one before that. The squares of these numbers are
simultaneously square and triangular.
The difference
between the squares of two consecutive rank triangular numbers
is equal to the cube of the larger numbers rank.
Thus, (Tn)^2 
(T(n  1))^2 = n^3. For example, T6^2  T5^2 = 441  225 = 216 =
6^3.
The summation
of varying sets of consecutive triangular numbers offers some
strange results.
T1 + T2 + T3 =
1 + 3 + 6 = 10 = T4.
T5 + T6 + T7 +
T8 = 15 + 21 + 28 + 36 = 100 = 45 + 55 = T9 + T10.
The pattern
continues with the next 5 Tn's summing to the next 3 Tn's
followed by the next6 Tn's summing to the next 4 Tn's, etc.
The sum of the
first "n" cubes is equal to the square of the nth triangular
number. For instance:
n............1.....2.....3.....4.......5
Tn..........1.....3.....6....10.....15
n^3.........1
+ 8 + 27 + 64 + 100 = 225 = 15^2
Every number
can be expressed by the sum of three or less triangular numbers,
not necessarily different.
1 = 1, 2 = 1 +
1, 3 = 3, 4 = 3 + 1, 5 = 3 + 1 + 1, 6 = 6, 7 = 6 + 1, 8 = 6 + 1
+ 1, 9 = 6 + 3, 10 = 10, etc.
Alternate ways
of finding triangular squares.
From Tn = n(n
+ 1)/2 and Sn = m^2, we get m^2 = n(n + 1)/2 or 4n^2 + 4n =
8m^2.
Adding one to
both sides, we obtain 4n^2 + 4n + 1 = 8m^2 + 1.
Factoring, we
find (2n + 1)^2 = 8m^2 + 1.
If we allow
(2n + 1) to equal "x" and "y" to equal 2m, we come upon x^2 
2y^2 = 1, the famous Pell Equation.
We now know
that the positive integer solutions to the Pell equation, x^2 
2y^2 = +1 lead to triangular squares. But how?
Without
getting into the theoretical aspect of the subject, sufficeth to
say that the Pell equation is closely connected with early
methods of approximating the square root of a number. The
solutions to Pell's equation, i.e., (x,y), often written as
(x/y) are approximations of the square root of D in x^2  2y^2 =
+1. Numerous methods have evolved over the centuries for
estimating the square root of a number.
Diophantus'
method leads to the minimum solutions to x^2  Dy^2 = +1, D a
non square, by setting x = my + 1 which leads to y = 2m/(D 
m^2).
From values of
m = 1.......n, many rational solutions evolve.
Eventually, an
integer solution will be reached.
For instance,
the smallest solution to x^2  2y^2 = +1 derives from m = 1
resulting in x = 3 and y = 2 or sqrt(2) ~= 3/2..
Newton's
method leads to the minimum solution sqrt(D) = sqrt(a^2 + r) =
(a + D/a)/2 ("a" = the nearest square) = (3/2).
Heron/Archimedes/El Hassar/Aryabhatta obtained the minimum
solution sqrt(D) = sqrt(a^2 +r) = a +r/2a = (x/y) = (3/2).
Other methods
exist that produce values of x/y but end up being solutions to
x^2  Dy^2 = +/C.
Having the
minimal solutions of x1 and y1 for x^2  Dy^2 = +1, others are
derivable from the following:
(x + ysqrtD) =
(x1 + y1sqrtD)^n, n = 1, 2, 3, etc.
Alternative approach
Given x = p
and y = q satisfying x^2  2y^2 = +1, we can write (x + sqrtD)(x
 sqrtD) = 1.
x = [(p +
qsqrt(2))^n + (p  qsqrt(2))^n]/2
y = [(p +
qsqrt(2))^n  (p  qsqrt(2))^n]/(2sqrt(2))
Having the
minimum solution of x = 3 and y = 2, the next few solutions
derive from n = 2 and 3 where x = 17, y = 12, x = 99 and y = 70
respectively.
Alternative approach
Subsequent
solutions can also be obtained by means of the following:
x^2  2y^2 =
+1 can be rewritten as x^2  2y^2 = (x + yqrt(2)(x  ysqrt(2)) =
+1.
Using the
minimum solution of x = 3 and y = 2, we can now write
.................(3 + sqrt(2))^2(3  sqrt(2))^2 = 1^2 = 1
.................(17 + 12sqrt(2))(17  12sqrt(2)) = 1
.................289  2(144) = 17^2  2(12)^2 = 1
the next smallest solution.
The next
smallest solution is derivable from
.................(3 + sqrt(2))^3(3  sqrt(2))^3 = 1^2 = 1 which
works out to
.................(99 + 70sqrt(2))(99  70sqrt(2)) = 1 or
.................99^2  2(70)^2 = 1.
Similarly, (3
+ sqrt(2))^4(3  sqrt(2))^4 = 1^2 = 1 leads to
.................(577 + 408sqrt(2))(577  408sqrt(2)) = 1 and
.................577^2  2(408)^2 = 1.
Regardless of
the method, we ultimately end up with the starting list of
triangular squares.
..x........y........n.......m........Tn = Sm^2
..3........2........1........1..............1
.17......12.......8........6..............36
.99......70......49......35...........1225
577....408....288.....204.........41,616 etc.
TWIN PRIME
NUMBERS
Two prime
numbers differing by 2 are called twin primes. Examples are 3 
5, 5  7, 11  13, 17  19, 659  661, 2687  2689, and
1,000,000,009,649  1,000,000,009,651, to name a few. A. de
Polignac (Nouvelles Annales de Math., 1849, vol. VIII, p. 428)
proposed that every even number is the difference of two
consecutive prime numbers in infinitely many ways. This leads to
the conclusion that there are an infinite number of pairs of
primes that are consecutive odd numbers. Examples are 5 and 7,
11 and 13, 17 and 19, 29 and 31, and so on ad infinitum. As for
primes greater than 100,
101103,
107109, 137139, 149151, 179181, 191193, and 197199.
TRIPLE
PRIME NUMBERS
Triple prime
numbers are three prime numbers in succession, each successive
pair differing by 2. Not surprisingly, there is only one set of
triple primes, 3  5  7.
UNIT
FRACTION NUMBERS
Unit fractions
are the reciprocals of the positive integers where the numerator
is always one. They are also called Egyptian fractions, and 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]. There is a
more in depth article in the Knowledge Database under Unit
Fractions.
REFERENCES
If you are
inspired to dig deeper into the history of any specific number
type, I offer you the following list of reading references,
which will exponentially increase your knowledge and enjoyment
of the subject.
1Mathematical Recreations by Maurice KraichikDover
Publications, Inc.1942
2Arithmetical Excursionsby H. Bowers and J.E. BowersDover
Publications, Inc.1961
3The Joy of
Mathematicsby Theoni PappasWide World Publishing/Tetra1993
4Mathematics
for the General Readerby E.C. TitchmarshDover Publications,
Inc.1981
5More Joy of
Mathematicsby Theoni PappasWide World Publishing/Tetra1994
6Recreations
in the Theory of Numbersby Albert H. BeilerDover
Publications, Inc.1964
7Excursions
in Number Theoryby C.S. Ogilvy & J.T. AndersonDover
Publications, Inc.1988
8The Magic
of Mathematicsby Theoni PappasWide World
Publishing/Tetra1994
9Mathematics
for the Millionby Lancelot HogbenWW Norton & Co.1985
10The
Enjoyment of Mathematicsby Hans Rademacher & Otto
ToeplitzDover Publ. Inc.1990
11Mathematics Appreciationby Theoni pappasWide World
Publishing/Tetra1992
12Mathematics for the NonMathemeticianby Morris KlineDover
Publications, Inc.1985
13Journey
Through Geniusby William DunhamPenguin Books1990
14The Master
Book of Mathematical Recreationsby Fred SchuhDover
Publications, Inc.1968
15Mathematical Recreations and Essaysby W.W. Rouse Ball &
H.S.M CoxeterDover Pub., 1987
16Madachy's
Mathematical Recreationsby Joseph S. MadachyDover
Publications, Inc.1979
17Number
Theory and its Historyby Oystein OreDover Publications,
Inc.1976
18The Divine
Proportionby H.E. HuntleyDover Publications, Inc.1970
19More
Mathematical Morsels by Ross HonsbergerMath. Assoc. of
America1991
20Mathematical Gems II by Ross HonsbergerMath. Assoc. of
America1976
21Elementary
Theory of Numbers by W.J. LeVequeDover Publications1990
22An
Adventurer's Guide to Number Theory by R. FriedbergDover
Publications1994
23The Rhind
Mathematical Papyrus by G. Robins & C. ShuteDover
Publications1987
24Number
Theory by George E. Andrew, Dover Publications, Inc., 1994.
25Fundementals of Number Theory by William J. LeVeque, Dover
Publications, Inc., 1996
26Advanced
Number Theory by Harvey Cohn, Dover Publications, Inc., 1980
27NumbersTheir History and Meaning by Graham Flegg, Barnes &
Noble Books, 1993.
28Mathematical Scandals by theoni pappas, Wide World
Publishing/Tetra, 1997.
29The Man
Who Loved Only Numbers by Paul Hoffman, Hyperion, NY,1998.
30Mathematical Magic by William Simon, Dover Publicatins,
Inc., 1993.
31Mathematical Fallacies and Paradoxes by Bryan Bunch, Dover
Publications, Inc., 1997.
32Mathematical Recreations  by D. A. Klarner, Dover Publ.,
Inc., 1998.
33The
Penguin Dictionary of Curious and Interesting Numbers by David
Wells, Penguin, 1987
34Numbers 
The Universal Language byDenis Guedj, Harry N Abrams, Inc.,
1996.
35You Are a
Mathematician by David Wells, John Wiley & Sons, 1997.
36The
Penguin Book of Curious and Interesting Mathematics by D. Wells,
Penguin Books, 1997
37Magic
Squares and Cubes by W.S. Andrews, Dover Publications, Inc.,
1960.
38The Magic
of Numbers by Eric T. Bell, Dover Publications, Inc., 1991.
39Mathematics Magic and Mystery by Martin Gardner, Dover
Publications, Inc., 1956.
40Mathematics Basic Facts by John Cullerne, Harper Collins
Publishers, 1998.
41Once Upon
a Number by John Allen Paulos, Basic Books, 1998.
42Mathematical Gems 1 by Ross Honsberger, Mathematical
Association of America, 1973.
43Mathematical Gems 2 by Ross Honsberger, Mathematical
Association of America, 1976.
44The Theory
of Numbers and Diophantine Analysis by R. Carmichael, Dover
Publ., Inc., 1915.
45The
Pleasures of Counting by T.W. Korner,Cambridge University Press,
1998.
46Great
Moments in Mathematics  Before 1650 by Howard Eves, MAA, 1983.
47Great
Moments in Mathematics  After 1650 by Howard Eves, MAA, 1983.
48Mathematical Gems 3 by Ross Honsberger, MAA, 1985.
49Diophantus
and Diophantine Equations by I.G. Bashmakova, MAA, 1997.
50Mathematical Morsels by Ross Honsberger, MAA,
51Strength
in Numbers by Sherman K. Stein, John Wiley & Sons, 1996
52Lure of
the Integers by Joe Roberts, MAA, 1992
53Ingenuity
in Mathematics by Ross Honsberger, MAA, 1970
54The Book
of Numbers by John H. Conway % Richard K. Guy, Springer Verlag,
1995
55The Theory
of Numbers and Diophantine Analysis by R. Carmichael, Dover
Publ., Inc., 1959
56New
Mathematical Diversionsby Martin Gardner, MAA, 1995.
57A
Mathematical Mosaicby Ravi Vakil, Brendan Kelly Publishing,
1996.
58Mathamazement by Ronn Yablun, Contemporary Books, 1996.
59Mathematical Mysteriesby Calvin C. Clawson, Perseus Books,
1996.
60Mathematical Fallacies, Flaws, & Flimflamby E.J. Barbeau,
MAA, 1999.
61Mathematics from the Birth of Numbersby Jan Gullberg, W.W.
Norton & Co., 1996.
62Mathematical Sourcery, Calvin C. Clawson, Plenum Trade,
1999.
63The Lore
of Large Numbers, Philip J. Davis, MAA, 1961.
64The Number
Sense by Stanislas Debaene, Oxford University Press, New York,
NY, 1997
65Wonders of
Numbers by Clifford A. Pickover, Oxford University Press, New
York, NY, 2001
66The
Colossal Book of Mathematics by Martin Gardner, W.W. Norton &
Co., 2001.
67Mathematical Treks by Ivars Peterson, MAA, 2002
68The
Historical Roots of Elementary Mathematics by L.N.H. Bunt et al,
Dover Publ., Inc., 1988.
69Makers of
Mathematics by Stuart Hollongdale, Penguin Books, 1994.
70NUMBER
from Ahmes to Cantor by Midhat Gazale, Princeton University
Press, 2000.
It is hoped
that others, if so inclined, will find some time to contribute
their favorite number types to the collection.
The
information in this article was compiled by
William Tappe over many years. He is a retired aeronautical
engineer from Long Island, NY.
Part I 
Part II 
Part III 
Part IV 