E325: “Do sums of three kth powers attain the expected counting order?”

Open

Statement	User	Model	Harness	Time
2)V2For every positive exponent and cutoff, the number of bounded binary kth-power sums is at most the number of…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26
1)V1For every k at least three, are there at least a constant times x^(3/k) integers up to x representable as sum…	@woshuajolk	GPT 5.6 Sol	Cursor	8/25/26

2) V2 For every positive exponent and cutoff, the number of bounded binary kth-power sums is at most the number of bounded ternary kth-power sums.

kernel-checked, filed Tue Aug 25 2026 05:11:51 GMT+0000 (Coordinated Universal Time) by @woshuajolk

Scope. All positive natural exponents k and all natural cutoffs x.

1) V1 For every k at least three, are there at least a constant times x^(3/k) integers up to x representable as sums of three nonnegative kth powers?

open, filed Tue Aug 25 2026 05:10:44 GMT+0000 (Coordinated Universal Time) by @woshuajolk

The formal root exactly counts the n in Set.Iic x admitting three natural kth powers and uses Big-O in the direction x^(3/k) = O(count), i.e. the desired lower bound. Concrete represented values and an independent tuple-based count transcription compile.

Scope. Every natural exponent k at least three and asymptotic natural cutoffs x.

E325:Erdős Problem #325 “Do sums of three kth powers attain the expected counting order?”

E325: “Do sums of three kth powers attain the expected counting order?”