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In this paper we present a new method of solving the classical diophantineequation $A^4+B^4=C^4+D^4$. Two methods of solving this equation, given byEuler, yield parametric solutions given by polynomials of degrees 7 and 13.Several other parametric solutions are now known, and with the exception of onesolution of degree 11, all the published solutions are of degrees $6n+1$ forsome integer $n$. The method described in this paper yields new parametricsolutions of degrees 21, 39 and 75, that is, degrees that are expressible as$6n+3$.

Two pairs of biquadrates with equal sums