Bhaskara's Lemma is an identity used as a lemma during the chakravala method. It states that:
N
x
2
+
k
=
y
2
⟹
N
(
m
x
+
y
k
)
2
+
m
2
−
N
k
=
(
m
y
+
N
x
k
)
2
for integers
m
,
x
,
y
,
N
,
and non-zero integer
k
.
The proof follows from simple algebraic manipulations as follows: multiply both sides of the equation by
m
2
−
N
, add
N
2
x
2
+
2
N
m
x
y
+
N
y
2
, factor, and divide by
k
2
.
N
x
2
+
k
=
y
2
⟹
N
m
2
x
2
−
N
2
x
2
+
k
(
m
2
−
N
)
=
m
2
y
2
−
N
y
2
⟹
N
m
2
x
2
+
2
N
m
x
y
+
N
y
2
+
k
(
m
2
−
N
)
=
m
2
y
2
+
2
N
m
x
y
+
N
2
x
2
⟹
N
(
m
x
+
y
)
2
+
k
(
m
2
−
N
)
=
(
m
y
+
N
x
)
2
⟹
N
(
m
x
+
y
k
)
2
+
m
2
−
N
k
=
(
m
y
+
N
x
k
)
2
.
So long as neither
k
nor
m
2
−
N
are zero, the implication goes in both directions. (Note also that the lemma holds for real or complex numbers as well as integers.)
