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Mathematical lemma
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
{\displaystyle \,Nx^{2}+k=y^{2}\implies \,N\left({\frac {mx+y}{k}}\right)^{2}+{\frac {m^{2}-N}{k}}=\left({\frac {my+Nx}{k}}\right)^{2}}
for integers
m
,
x
,
y
,
N
,
{\displaystyle m,\,x,\,y,\,N,}
and non-zero integer
k
{\displaystyle k}
.
The proof follows from simple algebraic manipulations as follows: multiply both sides of the equation by
m
2
−
N
{\displaystyle m^{2}-N}
, add
N
2
x
2
+
2
N
m
x
y
+
N
y
2
{\displaystyle N^{2}x^{2}+2Nmxy+Ny^{2}}
, factor, and divide by
k
2
{\displaystyle 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
{\displaystyle \,Nx^{2}+k=y^{2}\implies Nm^{2}x^{2}-N^{2}x^{2}+k(m^{2}-N)=m^{2}y^{2}-Ny^{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
{\displaystyle \implies Nm^{2}x^{2}+2Nmxy+Ny^{2}+k(m^{2}-N)=m^{2}y^{2}+2Nmxy+N^{2}x^{2}}
⟹
N
(
m
x
+
y
)
2
+
k
(
m
2
−
N
)
=
(
m
y
+
N
x
)
2
{\displaystyle \implies N(mx+y)^{2}+k(m^{2}-N)=(my+Nx)^{2}}
⟹
N
(
m
x
+
y
k
)
2
+
m
2
−
N
k
=
(
m
y
+
N
x
k
)
2
.
{\displaystyle \implies \,N\left({\frac {mx+y}{k}}\right)^{2}+{\frac {m^{2}-N}{k}}=\left({\frac {my+Nx}{k}}\right)^{2}.}
So long as neither
k
{\displaystyle k}
nor
m
2
−
N
{\displaystyle m^{2}-N}
are zero, the implication goes in both directions. (The lemma holds for real or complex numbers as well as integers.)
References [ edit ]
C. O. Selenius, "Rationale of the chakravala process of Jayadeva and Bhaskara II", Historia Mathematica , 2 (1975), 167-184.
C. O. Selenius, Kettenbruch theoretische Erklarung der zyklischen Methode zur Losung der Bhaskara-Pell-Gleichung , Acta Acad. Abo. Math. Phys. 23 (10) (1963).
George Gheverghese Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (1975).
External links [ edit ]