Mathematics in Ancient and Medieval India c7.s5.179

c7.s5.179

(i) 61x² + 1 = y² Select an auxiliary equation 61.1² + 3 = 8² (1) Let a₁ = 1, b₁ = 8, k₁ = 3 and N = 61 Then by Bhāskara II's lemma we get $N((a₁m + b₁)/k₁)² + (m² - N)/k₁ = ((b₁m + Na₁)/k₁)$ where m is an arbitrary chosen number. Here m is so chosen that m² - N is least and (a₁m + b₁)/k₁ is an integer Taking m = 7, we get (a₁m + b₁)/k₁ = 5, (m² - N)/k₁ = - 4. $and (b₁m + Na₁)/k₁ = 39$ Let a₂ = 5, b₂ = 39, k₂ = - 4 Hence from (1), we get 61.5² - 4 = 39² (2) Thus in one step, we get an equation Nx² + t = y² when t = - 4 From (2), 61.(5/2)² - 1 = (39/2)²