Expanding Pythagorean Triples to Exponents n > 2
Pythagorean triples are integer solutions to the equation:
a2+b2=c2
As Fermat's Last Theorem posits, there are no integer triples that satisfy
the equation when the exponent is an integer greater than 2.
an+bn=cn (n greater than 2).
However, consider rewriting
a2+b2=c2
to
a2=b2+c2
Then
a2-b2=c2
and
(a+b)(a-b)=c2
Now, let m, n, and p be positiver integers such that m+n=p
Then the permutations of the equation:
(a+b)m(a-b)n=cp
appear to have many relatively prime integer triples for solutions.
For example:
If p=2, then
(a+b)(a-b)=c2 (the usual Pythagorean triples)
If p=3, then
(a+b)2(a-b)=c3
(a+b)(a-b)2=c3
Examples:
|
a
|
b
|
c
|
(a+b)2
|
(a-b)
|
(a+b)2(a-b)
|
c3
|
|
9
|
7
|
8
|
256
|
2
|
512
|
512
|
|
42
|
39
|
27
|
6561
|
3
|
19683
|
19683
|
|
130
|
126
|
64
|
65536
|
4
|
262144
|
262144
|
|
etc.
|
|
|
|
|
|
|
|
a
|
b
|
c
|
(a+b)
|
(a-b)2
|
(a+b)(a-b)2
|
c3
|
|
14
|
13
|
3
|
27
|
1
|
27
|
27
|
|
63
|
62
|
5
|
125
|
1
|
125
|
125
|
|
172
|
171
|
7
|
343
|
1
|
343
|
343
|
|
etc.
|
|
|
|
|
|
|
If p=4, then
(a+b)3(a-b)=c4
(a+b)2(a-b)2=c4
(a+b)(a-b)3=c4
Examples:
|
a
|
b
|
c
|
(a+b)3
|
(a-b)
|
(a+b)3(a-b)
|
c4
|
|
17
|
15
|
16
|
32768
|
2
|
65536
|
65536
|
|
123
|
120
|
81
|
14348907
|
3
|
43046721
|
43046721
|
|
514
|
510
|
256
|
1073741824
|
4
|
4294967296
|
4294967296
|
|
etc.
|
|
|
|
|
|
|
|
a
|
b
|
c
|
(a+b)2
|
(a-b)2
|
(a+b)2(a-b)2
|
c4
|
|
5
|
4
|
3
|
81
|
1
|
81
|
81
|
|
13
|
12
|
5
|
625
|
1
|
625
|
625
|
|
25
|
24
|
7
|
2401
|
1
|
2401
|
2401
|
|
etc.
|
|
|
|
|
|
|
|
a
|
b
|
c
|
(a+b)
|
(a-b)3
|
(a+b)(a-b)3
|
c4
|
|
41
|
40
|
3
|
81
|
1
|
81
|
81
|
|
313
|
312
|
5
|
625
|
1
|
625
|
625
|
|
1201
|
1200
|
7
|
2401
|
1
|
2401
|
2401
|
|
etc.
|
|
|
|
|
|
|
If p=5, then
(a+b)4(a-b)=c5
(a+b)3(a-b)2=c5
(a+b)2(a-b)3=c5
(a+b)(a-b)4=c5
Examples:
|
a
|
b
|
c
|
(a+b)4
|
(a-b)
|
(a+b)4(a-b)
|
c5
|
|
33
|
31
|
32
|
16777216
|
2
|
33554432
|
33554432
|
|
366
|
363
|
243
|
282429536481
|
3
|
847288609443
|
847288609443
|
|
2050
|
2046
|
1024
|
281474976710656
|
4
|
1.13E+015
|
1.13E+015
|
|
etc.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
a
|
b
|
c
|
(a+b)3
|
(a-b)2
|
(a+b)3(a-b)2
|
c5
|
|
66
|
62
|
32
|
2097152
|
16
|
33554432
|
33554432
|
|
1098
|
1089
|
243
|
10460353203
|
81
|
847288609443
|
847288609443
|
|
8200
|
8184
|
1024
|
4398046511104
|
256
|
1.13E+015
|
1.13E+015
|
|
etc.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
a
|
b
|
c
|
(a+b)2
|
(a-b)3
|
(a+b)2(a-b)3
|
c5
|
|
132
|
124
|
32
|
65536
|
512
|
33554432
|
33554432
|
|
3294
|
3267
|
243
|
43046721
|
19683
|
847288609443
|
847288609443
|
|
32800
|
32736
|
1024
|
4294967296
|
262144
|
1.13E+015
|
1.13E+015
|
|
etc.
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
a
|
b
|
c
|
(a+b)
|
(a-b)4
|
(a+b)(a-b)4
|
c5
|
|
122
|
121
|
3
|
243
|
1
|
243
|
243
|
|
1563
|
1562
|
5
|
3125
|
1
|
3125
|
3125
|
|
8404
|
8403
|
7
|
16807
|
1
|
16807
|
16807
|
|
etc.
|
|
|
|
|
|
|
etc.
