Quartic diophantine chains

On certain Diophantine systems with infinitely many parametric solutions and applications

“…h is an elliptic curve with nonzero discriminant) has infinitely many solutions in rationals. In this connection, we mention the work of Choudhry and Wróblewski [1], who showed that if h(x, y) = x 4 − y 4…”

“…Similar results were proved in [8,9] and [3], where instead of f (x, y), we considered g(x, y) = (y 2 − x 3 )/x and g(x, y) = (y 2 − x 5 )/x respectively. In the light of these results, it is natural to ask what can be said about a general system of the form (1) h(x 1 , y 1 )…”

A diophantine problem on biquadrates revisited

“…On substituting the valus of p, q, r given by (10) and the value of t given by (11) in (4) and choosing k suitably, we obtain the following solution of Eqs. ( 1) and ( 2):…”

“…As a numerical example, when a = 1, b = −1, we get the two triads of biquadrates {7 4 , 133 4 , 153 4 } and {17 4 , 49 4 , 171 4 } with equal sums and equal products.…”