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We describe the algorithms which allowed us to show that all the integers congruent to 50 modulo 80 that lie in the interval (0.3651 x 1012, 1.0400x 1012) are sums of five biquadrates, and that all the integers congruent to 67 modulo 80 that lie in the interval (0.3651 x 1012, 9.5956 x 1018) are sums of seven biquadrates. We finally describe some ascent lemmas that we use to deduce from the previous results that every integer not exceeding 10448 is a sum of 19 biquadrates.

Numerical results for sums of five and seven biquadrates and consequences for sums of 19 biquadrates