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Numerical results for sums of five and seven biquadrates and consequences for sums of 19 biquadrates. (English) Zbl 0879.11052

Summary: The authors describe the algorithms which allowed them 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. They also describe some ascent lemmas used to deduce from the previous results that every integer not exceeding \(10^{448}\) is a sum of nineteen biquadrates.

MSC:

11P05 Waring’s problem and variants
11-04 Software, source code, etc. for problems pertaining to number theory
11Y16 Number-theoretic algorithms; complexity
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