Mathematical Side Show - 3

Soroban · #1 $Old$ March 25th, 2009, 11:52 AM

Mathematical Side Show - 3

Fascinating sums of squares

We have: . $4^2 + 5^2 + 6^2 \:=\:8^2 + 3^2 + 2^2$

Form two-digit numbers by chossing a digit
from the left side and from the right side.
For example: . $48,53,62.$
Consider their reversals: . $84,35,36$

We find that: . $48^2 + 53^2 + 62^2 \:=\:84^2 + 35^2 + 26^2$

This is true for all $3! = 6$ choices for the two-digit numbers.

$48^2+53^2+62^2\:=\:84^2+35^2+26^2$
$48^2+52^2+63^2\:=\:84^2+25^2+36^2$
$43^2+58^2+62^2\:=\:34^2+85^2+26^2$
$43^2+52^2+68^2\:=\:34^2+25^2+86^2$
$42^2+58^2+63^2\:=\:24^2+85^2+36^2$
$42^2+53^2+68^2\:=\:24^2+35^2+86^2$

Another Sum-of-Squares

It is possible to have three squares $(X^2,Y^2,Z^2)$
such that their sum in pairs is also a square.

Example: . $(X,Y,Z) \:=\:(44,117,240)$

$\begin{array}{ccc}44^2 + 117^2 &=& 240^2 \\ 117^2 + 240^2 &=& 267^2 \\ 44^2 + 240^2 &=& 244^2 \end{array}$

This is a box with dimensions: $44 \times 117 \times 240$
whose faces have integral diagonals.

Example: . $(828, 2035, 3120)$

$\begin{array}{ccc}828^2 + 2035^2 &=& 2197^2 \\ 2035^2 + 3120^2 &=& 3725^2 \\ 828^2 + 3120^2 &=& 3228^2 \end{array}$

The general equations are:

$\begin{array}{c}X \;=\;2mn(3m^2-n^2)(3n^2-m^2) \qquad\qquad\qquad\qquad \\ Y \;=\;8mn(m^4-n^4)\qquad\qquad\qquad\qquad\qquad\qquad\quad \\ Z \;=\;(m^2-n^2)(m^2+n^2+4mn)(m^2+n^2-4mn) \end{array}$