A Two-Power Algebraic Identity

​
x
-1
y
1
3
z
-1
t
1
3
X
Y
Z
U
V
10
15
-6
-5
30
2
X
+
2
Y
+ (U + V - Z
2
)
2
U
+
2
V
+ (X + Y + Z
2
)
1286
1286
2
3
Z
+
3
X
+
3
Y
+ (U + V - Z
3
)
3
U
+
3
V
+ (X + Y + Z
3
)
33734
33734
Let
x
,
y
,
z
,
t
be four arbitrary complex numbers and set
X=(
2
t
+
2
z
)
2
t

4
x
+2
3
x
y+2
2
x
2
y
-2x
3
y
+
4
y
+tz
4
x
-6
2
x
2
y
+
4
y
-2xy
2
z

2
x
-
2
y

,
Y=-(
2
t
+
2
z
)2
2
t
xy
2
x
-
2
y
+tz
4
x
-6
2
x
2
y
+
4
y
-
2
z

4
x
+2
3
x
y+2
2
x
2
y
-2x
3
y
+
4
y

,
Z=(
2
t
-
2
z
)t
2
x
-t
2
y
-2xyz2txy+
2
x
z-
2
y
z
,
U=(
2
t
+
2
z
)
2
t

4
x
-2
3
x
y+2
2
x
2
y
+2x
3
y
+
4
y
-tz
4
x
-6
2
x
2
y
+
4
y
+2xy
2
z

2
x
-
2
y

,
V=(
2
t
+
2
z
)2
2
t
xy
2
x
-
2
y
+tz
4
x
-6
2
x
2
y
+
4
y
+
2
z

4
x
-2
3
x
y+2
2
x
2
y
+2x
3
y
+
4
y

.
Then we have
2
X
+
2
Y
+
2
(U+V-Z)
=
2
U
+
2
V
+
2
(X+Y+Z)
and
2
3
Z
+
3
X
+
3
Y
+
3
(U+V-Z)
=
3
U
+
3
V
+
3
(X+Y+Z)
.
In this Demonstration,
x
and
y
are integers.
For example:
2
10
+
2
15
+
2
((-5)+30-(-6))
=
2
(-5)
+
2
30
+
2
(10+15+(-6))
,
2
3
(-6)
+
3
10
+
3
15
+
3
((-5)+30-(-6))
=
3
(-5)
+
3
30
+
3
(10+15+(-6))
.

External Links

abc Conjecture
Coincidences in Powers of Integers
Diophantine Equation (Wolfram MathWorld)
Seven Points with Integral Distances
Simultaneous Diophantine Equations for Powers 1, 2, 4 and 6
A Four-Power Diophantine Equation

Permanent Citation

Minh Trinh Xuan
​
​"A Two-Power Algebraic Identity"​
​http://demonstrations.wolfram.com/ATwoPowerAlgebraicIdentity/​
​Wolfram Demonstrations Project​
​Published: January 4, 2023