# Divisibility theory in the integers

• Apr 28th 2007, 05:51 AM
harrietchemist
Divisibility theory in the integers
Prove the following concerning triangular numbers:
a) A number is triangular if and only if it is of the form n(n+1)/2 for some n≧1
b) the integer n is a triangular number if and only if 8n+1is a perfect square
c) the sum of any two consecutive triangular numbers is a perfect square
d) if n is a triangular, then so are 9n+1, 25n+3,and 49n+6
• Apr 28th 2007, 07:25 AM
Soroban
Hello, harrietchemist!

I can help with the last two . . .

Quote:

Prove the following concerning triangular numbers:

c) the sum of any two consecutive triangular numbers is a perfect square

Let the first triangular number be: .T1 .= .k(k+1)/2
then the next triangular number is: .T
2 .= .(k+1)(k+2)/2

Then: .T
1 + T2 .= .k(k+1)/2 + (k+1)(k+2)/2 .= .(k² + k + k² + 3k + 2)/2

. . = .(2k² + 4k + 2)/2 .= .k² + 2k + 1 .= .(k + 1)² . . . a square

Quote:

d) if n is a triangular, then so are: 9n + 1, 25n + 3, and 49n + 6
Since n = k(k+1)/2, then: .9n + 1 .= .9·k(k+1)/2 + 1

. . = .(9k² + 9k + 2)/2 .= .(3k + 1)(3k + 2)/2 . . . a triangular number

Do the same for the other two expressions.

• Apr 28th 2007, 05:15 PM
ThePerfectHacker
Quote:

Originally Posted by harrietchemist
Prove the following concerning triangular numbers:
a) A number is triangular if and only if it is of the form n(n+1)/2 for some n≧1
b) the integer n is a triangular number if and only if 8n+1is a perfect square

Hmm, these problems seem to be taken from:
Burton, David. Elementary Number Theory 5/e. Page 15

Anyway.

a)Triangle means, t_n = 1+2+...+n
Use the summation formula 1+2+...+n=n(n+1)/2

b)If t_n is trianglular then it has form n(n+1)/2
Then, 8*t_n+1=8n(n+1)/2 + 1 = 4n(n+1)+1 = 4n^2+4n+1=(2n+1)^2

You try to do the converse.
• Apr 28th 2007, 08:10 PM
harrietchemist
thank you very much ~
I study number theory by myself...so I really happy that you both help me...