A Four-Power Algebraic Identity

​
k
1
2
3
4
x
2
y
3
z
5
x
1
x
2
x
3
x
4
x
5
x
6
23
28
21
15
14
22
y
1
y
2
y
3
y
4
y
5
y
6
26
27
19
20
18
13
4
x
1
+
4
x
2
+
4
x
3
+
4
x
4
+
4
x
5
+
4
x
6
4
y
1
+
4
y
2
+
4
y
3
+
4
y
4
+
4
y
5
+
4
y
6
1412275
1412275
Let
x
,
y
,
z
be three arbitrary complex numbers.
Set
x
1
=xy+yz+x
,
x
2
=yz+zx+y
,
x
3
=zx+xy+z
,
x
4
=x+y+zx
,
x
5
=y+z+xy
,
x
6
=z+x+yz
,
and
y
1
=xy+yz+z
,
y
2
=yz+zx+x
,
y
3
=zx+xy+y
y
4
=x+y+yz
,
y
5
=y+z+zx
,
y
6
=z+x+xy
.
Then for
k=0,1,2,3,4
,
"k"
"x"
1
+
"k"
"x"
2
+
"k"
"x"
3
+
"k"
"x"
4
+
"k"
"x"
5
+
"k"
"x"
6
=
"k"
"y"
1
+
"k"
"y"
2
+
"k"
"y"
3
+
"k"
"y"
4
+
"k"
"y"
5
+
"k"
"y"
6
.
In this Demonstration, the input variables are integers.
For example,
1
(-40)
+
1
(-26)
+
1
3
+
1
6
+
1
(-13)
+
1
(-23)
=
1
(-37)
+
1
(-18)
+
1
(-8)
+
1
(-34)
+
1
(9)
+
1
(-5)
,
2
(-40)
+
2
(-26)
+
2
3
+
2
6
+
2
(-13)
+
2
(-23)
=
2
(-37)
+
2
(-18)
+
2
(-8)
+
2
(-34)
+
2
(9)
+
2
(-5)
,
3
(-40)
+
3
(-26)
+
3
3
+
3
6
+
3
(-13)
+
3
(-23)
=
3
(-37)
+
3
(-18)
+
3
(-8)
+
3
(-34)
+
3
(9)
+
3
(-5)
,
4
(-40)
+
4
(-26)
+
4
3
+
4
6
+
4
(-13)
+
4
(-23)
=
4
(-37)
+
4
(-18)
+
4
(-8)
+
4
(-34)
+
4
(9)
+
4
(-5)
.

External Links

A Five-Power Diophantine Equation
abc Conjecture
Coincidences in Powers of Integers
Diophantine Equation (Wolfram MathWorld)
Seven Points with Integral Distances

Permanent Citation

Minh Trinh Xuan
​
​"A Four-Power Algebraic Identity"​
​http://demonstrations.wolfram.com/AFourPowerAlgebraicIdentity/​
​Wolfram Demonstrations Project​
​Published: May 26, 2022