Polygonal, Prime and Perfect Numbers

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Presentation transcript:

Polygonal, Prime and Perfect Numbers Greeks tried to transfer geometric ideas to number theory. One of such attempts led to the appearance of polygonal numbers 10 6 3 triangular 1 16 9 4 square 1 22 12 5 pentagonal 1

Results about polygonal numbers General formula: Let X n,m denote mth n-agonal number. Then X n,m = m[1+ (n-2)(m-1)/2] Every positive integer is the sum of four integer squares (Lagrange’s Four-Square Theorem, 1770) Generalization (conjectured by Fermat in 1670): every positive integer is the sum of n n-agonal numbers (proved by Cauchy in 1813) Euler’s pentagonal theorem (1750):

Prime numbers An (integer) number is called prime if it has no rectangular representation Equivalently, a number p is called prime if it has no divisors distinct from 1 and itself There are infinitely many primes. Proof (Euclid, “Elements”)

Perfect numbers Examples: 6=1+2+3, 28=1+2+4+7+14 Definition (Pythagoreans): A number is called perfect if it is equal to the sum of its divisors (including 1 but not including itself) Examples: 6=1+2+3, 28=1+2+4+7+14

Triangular numbers 1; 1+2=3 1+2+3=6 1+2+3+4=10 1+2+3+4+5=15

Let’s Build the 9th Triangular Number 1 Let’s Build the 9th Triangular Number +2 +3 +4 +5 +6 +7 +8 +9

Q: Is there some easy way to get these numbers? A: Yes, take two copies of any triangular number and put them together…..with multi-link cubes.

9x10 = 90 Take half. Each Triangle has 45. 9 9+1=10

Each Triangle has n(n+1)/2 Take half. Each Triangle has n(n+1)/2 n n+1

Each Triangle has n(n+1)/2 Take half. Each Triangle has n(n+1)/2 n n+1

Another Cool Thing about Triangular Numbers Put any triangular number together with the next bigger (or next smaller). And you get a Square!

Another Cool Thing about Triangular Numbers First + Second  1+3 =4=22 Second +Third  3+6 = 9 = 32 Third + Forth  6 +10 = 16 = 42

Interesting facts about Triangular Numbers Number of People in the Room Number of Handshakes 2 1 3 4 6 5 10 15 The Triangular Numbers are the Handshake Numbers Which are the number of sides and diagonals of an n-gon.

A B E D C Number of Handshakes = Number of sides and diagonals of an n-gon. A B E D C

Why are the handshake numbers Triangular? Let’s say we have 5 people: A, B, C, D, E. Here are the handshakes: A-B A-C A-D A-E B-C B-D B-E C-D C-E D-E It’s a Triangle !

PASCAL TRIANGLE The first diagonal are the “stick” numbers. …boring, but a lead-in to…

Triangle Numbers in Pascal Triangle The second diagonal are the triangular numbers. Why? Because we use the Hockey Stick Principle to sum up stick numbers.

TETRAHEDRAL NUMBERS The third diagonal are the tetrahedral numbers. Why? Because we use the Hockey Stick Principle to sum up triangular numbers.

Triangular and Hexagonal Numbers Relationships between Triangular and Hexagonal Numbers….decompose a hexagonal number into 4 triangular numbers. Notation Tn = nth Triangular number Hn = nth Hexagonal number

Decompose a hexagonal number into 4 triangular numbers.

A Neat Method to Find Any Figurate Number Number example: Let’s find the 6th pentagonal number.

The 6th Pentagonal Number is: Polygonal numbers always begin with 1. Now look at the “Sticks.” There are 4 sticks and they are 5 long. Now look at the triangles! There are 3 triangles. and they are 4 high. 1 + 5x4 + T4x3 1+20+30 = 51

The kth n-gonal Number is: Polygonal numbers always begin with 1. Now look at the “Sticks.” There are n-1 sticks and they are k-1 long. Now look at the triangles! There are n-2 triangles. and they are k-2 high. 1 + (k-1)x(n-1) + Tk-2x(n-2)