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Figurate Numbers


Date: 12/31/97 at 09:36:11
From: JIMMY
Subject: Figurate Numbers

Hi!

Here are figurate numbers.

Part 1: Triangular Numbers.

Triangular numbers are of the form n(n+1)/2. The first few are 1, 3, 
6, 10, 15, 21, 28, 36, 45, 55, 66, 78, and 91. All numbers of such a 
type end in 0, 1, 3, 5, 6, or 8. Every perfect number has the form 
(2^n-1)2^(n-1,) so they are triangular numbers with the base number 
(2^n-1.)

Part 2: Square Numbers.

Square numbers are of the form n*n. The first few are 1, 4, 9, 16, 25, 
36, 49, 64, and 81. All numbers of such a type end in 0, 1, 4, 5, 6, 
or 9.

Part 3: Pentagonal Numbers.

Pentagonal numbers are of the form n(3n-1)/2. The first few are 1, 5, 
12, 22, 35, 51, 70, and 92. All numbers of such a type end in 0, 1, 2, 
5, 6, or 7. 

Part 4: Heptagonal Numbers.

Heptagonal numbers can be gotten by taking a triangular number, 
multiplying it by 6 and adding 1. The first few are 1, 7, 19, 37, 
61, and 91. All numbers of such a type end in 1, 7, or 9.

Part 5: Questions.

1. Why can a triangular number be gotten by adding up all the numbers 
from 1 to a given number??

2. I can prove easily that there are no square numbers that end in 2 
or 8 as follows: these are the digits 2 to an odd power will end in. 
But, how can you prove that there are none that end in 3 or 7?

3. How do we know every pentagonal number is 1/3 of a triangular 
number??

4. 7, 19, 37, and 61 are prime numbers. 91, however, is composite. Are 
there an infinite number of prime heptagonal numbers?

5. Sometimes, at the bottom of reply messages, it says Doctor Rob, 
sometimes Doctor Wilkinson, and sometimes Doctor Pete. What will it 
say this time?


Date: 12/31/97 at 10:40:45
From: Doctor Rob
Subject: Re: Figurate Numbers

1. Look at the rows in a triangle of objects:

         o
        o o
       o o o
      o o o o

   This should give you a clue as to "why".

2. Divide the squares by 5, and look at the remainders.  They are

   0, 1, 4, 4, 1, 0, 1, 4, 4, 1, 0, ...

   They repeat with period 5.  (Prove this!)  A number ending in 2, 3, 
   7, or 8, when divided by 5, will leave remainder 2 or 3, so can 
   never be a square.

3. Triple the n-th pentagonal number,

   3*[n*(3*n-1)/2] = (3*n)*[(3*n)-1]/2

                   = (3*n-1)(3*n)/2

                   = (3*n-1)(3*n-1+1)/2

                   = (3*n-1)((3*n-1)+1)/2

                   = the (3*n-1)-th triangular number.

4. These kinds of questions are unsolved. No one knows whether 
   any given irreducible polynomial of degree 2 or more (like 
   3*n^2 - 3*n + 1) represents infinitely many prime numbers.  
   A classical unsolved problem is to answer this question for 
   n^2 + 1.

5. It will say Doctor Rob this time, because that is who is answering
   this question.  There are dozens of Math Doctors who answer the 
   questions submitted. The choice of who answers which questions is 
   determined by who chooses to answer the question first.

-Doctor Rob,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
Middle School Number Sense/About Numbers

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