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.)
