Symmetric ideal degree-3 PTE solutions are asymptotically counted as (4 log 2 / 3 π²) H³ log H + O(H³) due to the second moment of the sum-of-two-squares representation function.
Proof of the main conjecture in Vinogradov's mean value theorem for degrees higher than three
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abstract
We prove the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three. This will be a consequence of a sharp decoupling inequality for curves
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Arithmetic Symmetry in Ideal Prouhet-Tarry-Escott Solutions
Symmetric ideal degree-3 PTE solutions are asymptotically counted as (4 log 2 / 3 π²) H³ log H + O(H³) due to the second moment of the sum-of-two-squares representation function.