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

Abstract:

We describe the algorithms which allowed us to show that all the integers congruent to 50 modulo 80 that lie in the interval $(0.3651 \times {10^{12}},1.0400 \times {10^{12}})$ are sums of five biquadrates, and that all the integers congruent to 67 modulo 80 that lie in the interval $(0.3651 \times {10^{12}},9.5956 \times {10^{18}})$ 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 ${10^{448}}$ is a sum of 19 biquadrates.