LOL Wardaddy...maybe that's the solution for Saffa's rugby...game plan...pity we don't have Fermat for a coach....it's very simple and we all understand it...Bahahahahahahaha
Reciprocal integers (inverse Fermat equation)
The equation
a
1
/
m
+
b
1
/
m
=
c
1
/
m
{\displaystyle a^{1/m}+b^{1/m}=c^{1/m}}
can be considered the "inverse" Fermat equation. All solutions of this equation were computed by Lenstra in 1992.[141] In the case in which the mth roots are required to be real and positive, all solutions are given by[142]
a
=
r
s
m
{\displaystyle a=rs^{m}}
b
=
r
t
m
{\displaystyle b=rt^{m}}
c
=
r
(
s
+
t
)
m
{\displaystyle c=r(s+t)^{m}}
for positive integers r, s, t with s and t coprime.
Rational exponents
For the Diophantine equation
a
n
/
m
+
b
n
/
m
=
c
n
/
m
{\displaystyle a^{n/m}+b^{n/m}=c^{n/m}}
with n not equal to 1, Bennett, Glass, and Székely proved in 2004 for n > 2, that if n and m are coprime, then there are integer solutions if and only if 6 divides m, and
a
1
/
m
{\displaystyle a^{1/m}}
,
b
1
/
m
,
{\displaystyle b^{1/m},}
and
c
1
/
m
{\displaystyle c^{1/m}}
are different complex 6th roots of the same real number.[143]
Negative exponents
n = –1
All primitive integer solutions (i.e., those with no prime factor common to all of a, b, and c) to the optic equation
a
?
1
+
b
?
1
=
c
?
1
{\displaystyle a^{-1}+b^{-1}=c^{-1}}
can be written as[144]
a
=
m
k
+
m
2
,
{\displaystyle a=mk+m^{2},}
b
=
m
k
+
k
2
,
{\displaystyle b=mk+k^{2},}
c
=
m
k
{\displaystyle c=mk}
for positive, coprime integers m, k.
n = –2
The case n = –2 also has an infinitude of solutions, and these have a geometric interpretation in terms of right triangles with integer sides and an integer altitude to the hypotenuse.[145][146] All primitive solutions to
a
?
2
+
b
?
2
=
d
?
2
{\displaystyle a^{-2}+b^{-2}=d^{-2}}
are given by
a
=
(
v
2
?
u
2
)
(
v
2
+
u
2
)
,
{\displaystyle a=(v^{2}-u^{2})(v^{2}+u^{2}),}
b
=
2
u
v
(
v
2
+
u
2
)
,
{\displaystyle b=2uv(v^{2}+u^{2}),}
d
=
2
u
v
(
v
2
?
u
2
)
,
{\displaystyle d=2uv(v^{2}-u^{2}),}
for coprime integers u, v with v > u. The geometric interpretation is that a and b are the integer legs of a right triangle and d is the integer altitude to the hypotenuse. Then the hypotenuse itself is the integer
c
=
(
v
2
+
u
2
)
2
,
{\displaystyle c=(v^{2}+u^{2})^{2},}
so (a, b, c) is a Pythagorean triple.
Integer
n < –2There are no solutions in integers for
a
n
+
b
n
=
c
n
{\displaystyle a^{n}+b^{n}=c^{n}}
for integers n < –2. If there were, the equation could be multiplied through by
a
|
n
|
b
|
n
|
c
|
n
|
{\displaystyle a^{|n|}b^{|n|}c^{|n|}}
to obtain
(
b
c
)
|
n
|
+
(
a
c
)
|
n
|
=
(
a
b
)
|
n
|
{\displaystyle (bc)^{|n|}+(ac)^{|n|}=(ab)^{|n|}}
, which is impossible by Fermat's Last Theorem.