My Math Forum  

Go Back   My Math Forum > College Math Forum > Number Theory

Number Theory Number Theory Math Forum


Reply
 
LinkBack Thread Tools Display Modes
March 9th, 2015, 10:03 PM   #1
Senior Member
 
Joined: Sep 2013
From: Earth

Posts: 827
Thanks: 36

Square triangular number

The first two numbers that are both squares and triangles are 1 and 36. Find the
next one and, if possible, the one after that. Can you figure out an efficient way to find
triangular-square numbers? Do you think that there are infinitely many?

This question is from http://www.amazon.com/Friendly-Intro.../dp/0321816196.

I'm totally new to number theory. Of course, I google-ed this question and I got the answer as well. But it's useless for me because I don't know how to generate the formula to find the answer? Can anyone explain how to figure out an efficient way to find the numbers ? Thanks.
jiasyuen is offline  
 
March 10th, 2015, 04:58 AM   #2
Math Team
 
Joined: Apr 2010

Posts: 2,780
Thanks: 361

Have you seen this page: Square Triangular Number -- from Wolfram MathWorld?
Hoempa is offline  
March 10th, 2015, 05:03 AM   #3
Senior Member
 
Joined: Sep 2013
From: Earth

Posts: 827
Thanks: 36

Yes. I just learned the general formula. How to derive the general formula actually ?
jiasyuen is offline  
March 10th, 2015, 05:23 AM   #4
Global Moderator
 
CRGreathouse's Avatar
 
Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Quote:
Originally Posted by jiasyuen View Post
Yes. I just learned the general formula. How to derive the general formula actually ?
You can use Pell's equation, I believe.
CRGreathouse is offline  
March 10th, 2015, 06:30 AM   #5
Senior Member
 
MarkFL's Avatar
 
Joined: Jul 2010
From: St. Augustine, FL., U.S.A.'s oldest city

Posts: 12,211
Thanks: 521

Math Focus: Calculus/ODEs
Quote:
Originally Posted by jiasyuen View Post
Yes. I just learned the general formula. How to derive the general formula actually ?
If you take the 2nd order recurrence:

$\displaystyle u_{n+2}-6u_{n+1}+u_{n}=0$

You find the characteristic roots are:

$\displaystyle r=3\pm2\sqrt{2}$

And so the general form of the closed solution is:

$\displaystyle u_{n}=k_1\left(3+2\sqrt{2}\right)^n+k_2\left(3-2\sqrt{2}\right)^n$

Using the initial conditions, we find:

$\displaystyle u_0=k_1+k_2=0$

$\displaystyle u_1=k_1\left(3+2\sqrt{2}\right)+k_2\left(3-2\sqrt{2}\right)=1$

From this, we find:

$\displaystyle \left(k_1,k_2\right)=\left(\frac{1}{4\sqrt{2}},-\frac{1}{4\sqrt{2}}\right)$

Hence:

$\displaystyle u_{n}= \frac{1}{4\sqrt{2}}\left(\left(3+2\sqrt{2}\right)^ n- \left(3-2\sqrt{2}\right)^n\right)$
MarkFL is offline  
March 10th, 2015, 02:57 PM   #6
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,685
Thanks: 2665

Math Focus: Mainly analysis and algebra
The $n$th triangle number $T_n$ is given by$$T_n = \tfrac12 n (n-1) = \begin{cases} k(2k + 1) &\text{or} \\ k(2k - 1)\end{cases}$$ for some natural number $k$. I shall henceforth write $T_n = k(2k \pm 1)$ to represent this.

Now, if a natural number $r \gt 1$ divides $k$, then $r$ also divides $2k$. Thus $r$ does not divide $(2k \pm 1)$.

We want $T_n$ to be square, so $r \divides T_n \implies r^2 \divides T_n$ and thus $r^2$ also divides $k$. Hence $k$ is square.

Since both $T_n$ and $k$ we must have that $2k \pm 1)$ is also square.

Thus, to find square triangular numbers, examine $(2k \pm 1)$ for each square $k$. If either $(2k \pm 1)$ is also square then the triangular number $k(2k \pm 1)$ is a square triangle number.
v8archie is offline  
Reply

  My Math Forum > College Math Forum > Number Theory

Tags
number, square, triangular



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Why do we square a number in an equation? Not me Elementary Math 3 August 13th, 2014 07:33 PM
Square Number Theorem M_B_S Algebra 23 November 20th, 2013 12:01 AM
about sum of a number to square powers Dougy Number Theory 6 June 17th, 2012 07:00 PM
square number 144...4*1444...44 Cruella_de_Vil Number Theory 5 March 25th, 2012 06:03 PM
There are infinite primes p: p-1 is Square number mathcool Math Events 0 December 31st, 1969 04:00 PM





Copyright © 2019 My Math Forum. All rights reserved.