Mathmatical Side Show - 5

Soroban · #1 $Old$ March 27th, 2009, 08:07 AM

Mathematical Side Show - 5

Pythagorean Triangles with Equal Area

Let the legs of the triangles be: . $(M,N),\:(P.Q),\:(T,U)$

The generating formulas are:

$\begin{array}{ccc} M &=& r^2+rs + s^2 \\ N &=& r^2-s^2\end{array}\quad\begin{array}{ccc}P &=& r^2+rs+s^2 \\ Q &=& 2rs + s^2 \end{array}\quad \begin{array}{ccc}T &=& r^2 + 2rs \\ U &=& r^2 + rs + r^2\end{array}$

Examples: . $(40,42,58),\;(24,70,74),\;(15,112,113)$
$(105,208,233),\;(120,182,218),\;(56,390,394)$

Three Squares in Arithmetic Progression

We want $(X,Y,Z)$ so that $X^2,Y^2,Z^2$ form an A.P.

The generating formulas are:

$\begin{array}{ccc}X &=& m^2-n^2 - 2mn \\ Y &=& m^2+n^2 \\ Z &=& m^2-n^2 + 2mn \end{array}$

Examples: . $(1,5,7),\;(2,10,14),\;(7,13,17),\;(7,17,23)$

An Amazing Bigrade

The equation: . $123789^n + 561945^n + 642864^n \:=\:242868^n + 323787^n + 761943^n$

holds true for $n = 1,2$ . . . and is called a bigrade.

It is abbreviated: . $123789,561945,642864 \:\begin{array}{c}_2\\^{=}\end{array}\:242868, 323787, 761943$

The left digits may be dropped, and the bigrade still holds.

$23789, 61945, 42864 \:\begin{array}{c}_2\\^{=}\end{array}\:42868,23787, 61943$

$3789, 1945, 2864 \:\begin{array}{c}_2\\^{=}\end{array}\:2868,3767,1943$

$789,945,864 \:\begin{array}{c}_2\\^{=}\end{array}\:868, 767, 943$

$89,45,64 \:\begin{array}{c}_2\\^{=}\end{array}\:68,67,43$

$9,5,4 \:\begin{array}{c}_2\\^{=}\end{array}\:8,7,3$

The right digits may be dropped, and the bigrade still holds.

$12378, 56194, 64286 \:\begin{array}{c}_2\\^{=}\end{array}\:24286, 32378, 76194$

$1237, 5619, 6428 \:\begin{array}{c}_2\\^{=}\end{array}\:2428, 3237, 7619$

$123, 561, 642 \:\begin{array}{c}_2\\^{=}\end{array} \:242,323, 761$

$12, 56, 64 \:\begin{array}{c}_2\\^{=} \end{array}\;24,32,76$

$1,5,6 \:\begin{array}{c}_2\\^{=}\end{array} 2,3,7$

The second and third digit may dropped (simultaneously):

$1789, 5945, 6864 \:\begin{array}{c}_2\\^{=}\end{array}\:2868, 3787, 7943$

$19, 55, 64 \:\begin{array}{c}_2\\^{=}\end{array}\:28, 37, 73$