A Three-Term Algebraic Identity with Squares or Quartics

​
k =
2
4
x
7
y
4
A
B
C
3069
1289
1619
D
E
F
649
2099
2979
4
A
+
4
B
+
4
C
4
D
+
4
E
+
4
F
98344195796483
98344195796483
Let
x
,
y
be two arbitrary numbers.
Set
A=(x+y)
2
x
-10xy-3
2
y

,
B=3
3
x
+5
2
x
y-15x
2
y
+15
3
y
,
C=
3
x
+11
2
x
y-13x
2
y
+9
3
y
,
D=(x+y)3
2
x
+2xy-9
2
y

,
E=
3
x
+3
2
x
y+19x
2
y
-15
3
y
,
F=
3
x
-
2
x
y+27x
2
y
-3
3
y
.
Then for
k=2,4
,
k
A
+
k
B
+
k
C
=
k
D
+
k
E
+
k
F
.
In this Demonstration,
x
and
y
are integers.
For example,
2
19
+
2
103
+
2
133
=
2
7
+
2
107
+
2
131
,
4
19
+
4
103
+
4
133
=
4
7
+
4
107
+
4
131
.

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 Three-Term Algebraic Identity with Squares or Quartics"​
​http://demonstrations.wolfram.com/AThreeTermAlgebraicIdentityWithSquaresOrQuartics/​
​Wolfram Demonstrations Project​
​Published: January 17, 2023