# Types of Numbers, Part V

__PYTHAGOREAN NUMBERS__

Pythagorean numbers, more typically referred to as Pythagorean Triples, are those sets of three integers that define all integral primitive and non-primitive Pythagorean right triangles. General formulas for deriving all integer sided right-angled 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 non-primitive 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, multiply-perfect, semi-perfect, 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 non-terminating 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 co-prime 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.

__SEMI-PERFECT NUMBERS__

A semi-perfect number is a number that is equal to the sum of “some” of its aliquot parts or proper divisors. The smallest semi-perfect 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, multiply-perfect, quasi-perfect, 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 non-square 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

Non-square…..2….3….4….6….7….8…10..11…12…13….14….15….17….18….19

6th non-square = 6 + 2[.449] = 8; 12th non-square = 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.

(2n-1)^2 – (n-1)^2………1^2-0^2…3^2-1^2…5^2-2^2…7^2-5^2…9^2-4^2…11^2-5^2

__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 Hardy-Ramanujan 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 semi-annually, 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 non-repeating 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/(2n-1)^2 = 1/1^2 + 1/3^2 + 1/5^2 +…..as n—>inf. = Pi^2/8.

The limit of Sn = 1/(2n-1)^4 = 1/1^4 + 1/3^4 + 1/5^4 +…..as n—>inf. = Pi^4/96.

The limit of Sn = [(-1)^(n+1)]/(2n-1) = 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)]/(2n-1)^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 1-2-3-4-…..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 1-2-3-4-…..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,

101-103, 107-109, 137-139, 149-151, 179-181, 191-193, and 197-199.

__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.

1–Mathematical Recreations by Maurice Kraichik–Dover Publications, Inc.–1942

2–Arithmetical Excursions–by H. Bowers and J.E. Bowers–Dover Publications, Inc.–1961

3–The Joy of Mathematics–by Theoni Pappas–Wide World Publishing/Tetra–1993

4–Mathematics for the General Reader–by E.C. Titchmarsh–Dover Publications, Inc.–1981

5–More Joy of Mathematics–by Theoni Pappas–Wide World Publishing/Tetra–1994

6–Recreations in the Theory of Numbers–by Albert H. Beiler–Dover Publications, Inc.–1964

7–Excursions in Number Theory–by C.S. Ogilvy & J.T. Anderson–Dover Publications, Inc.–1988

8–The Magic of Mathematics–by Theoni Pappas–Wide World Publishing/Tetra–1994

9–Mathematics for the Million–by Lancelot Hogben–WW Norton & Co.–1985

10–The Enjoyment of Mathematics–by Hans Rademacher & Otto Toeplitz–Dover Publ. Inc.–1990

11–Mathematics Appreciation–by Theoni pappas–Wide World Publishing/Tetra–1992

12–Mathematics for the NonMathemetician–by Morris Kline–Dover Publications, Inc.–1985

13–Journey Through Genius–by William Dunham–Penguin Books–1990

14–The Master Book of Mathematical Recreations–by Fred Schuh–Dover Publications, Inc.–1968

15–Mathematical Recreations and Essays–by W.W. Rouse Ball & H.S.M Coxeter–Dover Pub., 1987

16–Madachy’s Mathematical Recreations–by Joseph S. Madachy–Dover Publications, Inc.–1979

17–Number Theory and its History–by Oystein Ore–Dover Publications, Inc.–1976

18–The Divine Proportion–by H.E. Huntley–Dover Publications, Inc.–1970

19–More Mathematical Morsels by Ross Honsberger–Math. Assoc. of America–1991

20–Mathematical Gems II by Ross Honsberger–Math. Assoc. of America–1976

21–Elementary Theory of Numbers by W.J. LeVeque–Dover Publications–1990

22–An Adventurer’s Guide to Number Theory by R. Friedberg–Dover Publications–1994

23–The Rhind Mathematical Papyrus by G. Robins & C. Shute–Dover Publications–1987

24–Number Theory by George E. Andrew, Dover Publications, Inc., 1994.

25–Fundementals of Number Theory by William J. LeVeque, Dover Publications, Inc., 1996

26–Advanced Number Theory by Harvey Cohn, Dover Publications, Inc., 1980

27–Numbers-Their History and Meaning by Graham Flegg, Barnes & Noble Books, 1993.

28–Mathematical Scandals by theoni pappas, Wide World Publishing/Tetra, 1997.

29–The Man Who Loved Only Numbers by Paul Hoffman, Hyperion, NY,1998.

30–Mathematical Magic by William Simon, Dover Publicatins, Inc., 1993.

31–Mathematical Fallacies and Paradoxes by Bryan Bunch, Dover Publications, Inc., 1997.

32–Mathematical Recreations – by D. A. Klarner, Dover Publ., Inc., 1998.

33–The Penguin Dictionary of Curious and Interesting Numbers by David Wells, Penguin, 1987

34–Numbers – The Universal Language byDenis Guedj, Harry N Abrams, Inc., 1996.

35–You Are a Mathematician by David Wells, John Wiley & Sons, 1997.

36–The Penguin Book of Curious and Interesting Mathematics by D. Wells, Penguin Books, 1997

37–Magic Squares and Cubes by W.S. Andrews, Dover Publications, Inc., 1960.

38–The Magic of Numbers by Eric T. Bell, Dover Publications, Inc., 1991.

39–Mathematics Magic and Mystery by Martin Gardner, Dover Publications, Inc., 1956.

40–Mathematics Basic Facts by John Cullerne, Harper Collins Publishers, 1998.

41–Once Upon a Number by John Allen Paulos, Basic Books, 1998.

42–Mathematical Gems 1 by Ross Honsberger, Mathematical Association of America, 1973.

43–Mathematical Gems 2 by Ross Honsberger, Mathematical Association of America, 1976.

44–The Theory of Numbers and Diophantine Analysis by R. Carmichael, Dover Publ., Inc., 1915.

45–The Pleasures of Counting by T.W. Korner,Cambridge University Press, 1998.

46–Great Moments in Mathematics – Before 1650 by Howard Eves, MAA, 1983.

47–Great Moments in Mathematics – After 1650 by Howard Eves, MAA, 1983.

48–Mathematical Gems 3 by Ross Honsberger, MAA, 1985.

49–Diophantus and Diophantine Equations by I.G. Bashmakova, MAA, 1997.

50–Mathematical Morsels by Ross Honsberger, MAA,

51–Strength in Numbers by Sherman K. Stein, John Wiley & Sons, 1996

52–Lure of the Integers by Joe Roberts, MAA, 1992

53–Ingenuity in Mathematics by Ross Honsberger, MAA, 1970

54–The Book of Numbers by John H. Conway % Richard K. Guy, Springer Verlag, 1995

55–The Theory of Numbers and Diophantine Analysis by R. Carmichael, Dover Publ., Inc., 1959

56–New Mathematical Diversions–by Martin Gardner, MAA, 1995.

57–A Mathematical Mosaic–by Ravi Vakil, Brendan Kelly Publishing, 1996.

58–Mathamazement by Ronn Yablun, Contemporary Books, 1996.

59–Mathematical Mysteries–by Calvin C. Clawson, Perseus Books, 1996.

60–Mathematical Fallacies, Flaws, & Flimflam–by E.J. Barbeau, MAA, 1999.

61–Mathematics from the Birth of Numbers–by Jan Gullberg, W.W. Norton & Co., 1996.

62–Mathematical Sourcery, Calvin C. Clawson, Plenum Trade, 1999.

63–The Lore of Large Numbers, Philip J. Davis, MAA, 1961.

64–The Number Sense by Stanislas Debaene, Oxford University Press, New York, NY, 1997

65–Wonders of Numbers by Clifford A. Pickover, Oxford University Press, New York, NY, 2001

66–The Colossal Book of Mathematics by Martin Gardner, W.W. Norton & Co., 2001.

67–Mathematical Treks by Ivars Peterson, MAA, 2002

68–The Historical Roots of Elementary Mathematics by L.N.H. Bunt et al, Dover Publ., Inc., 1988.

69–Makers of Mathematics by Stuart Hollongdale, Penguin Books, 1994.

70–NUMBER 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.