On Sidon sets with squares, cubes and quartics in short intervals
Pith reviewed 2026-05-16 05:23 UTC · model grok-4.3
The pith
Non-trivial solutions to x^3 + y^3 = z^3 + t^3 are absent from intervals of length roughly the square root of N.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
For any positive integer N the equation x^3 + y^3 = z^3 + t^3 has no solutions in natural numbers with N ≤ x,y,z,t < N + ((38/3)N + 1297/36)^{1/2} + 19/6 unless {x,y} equals {z,t}. The boundary case with ≤ does admit solutions for infinitely many N. Solutions are always guaranteed to exist within an interval of length c N^{2/3} for some absolute c. There are infinitely many N with no solutions even in intervals of length N^{4/7 - ε}. For the equation x^4 + y^4 = z^4 + t^4 the corresponding guaranteed absence holds up to length c N^{3/5} and presence by c N^{12/13}.
What carries the argument
Explicit interval length thresholds for the appearance or absence of non-trivial solutions to x^k + y^k = z^k + t^k derived from estimates on the number of representations in short intervals.
Load-bearing premise
The specific values of the constants and the exponents in the interval lengths depend on the sharpness of the analytic and combinatorial estimates employed in the derivations.
What would settle it
An explicit example of four natural numbers between N and N plus the square root of (38/3 N) whose cubes sum pairwise to the same value in two different ways, for a sufficiently large N, would disprove the main no-solution claim.
read the original abstract
Representative examples of our results are as follows. For any positive integer $N$ the equation $$ x^3+y^3=z^3+t^3, \quad x,y,z,t\in \mathbb{N}, \quad \{x,y\}\not=\{z,t\} $$ has no solutions satisfying $$ N\le x,y,z,t < N+\Bigl(\frac{38}{3}N+\frac{1297}{36}\Bigr)^{1/2}+\frac{19}{6}. $$ The strict inequality ``$<$" can not be substituted by ``$\le$", that is, there exist infinitely many positive integers $N$ such that the equation has a solution with $$ N\le x,y,z,t \le N+\Bigl(\frac{38}{3}N+\frac{1297}{36}\Bigr)^{1/2}+\frac{19}{6}. $$ There is an absolute constant $c>0$ such that for any positive integer $N$ the equation has a solution satisfying $$ N\le x,y,z,t \le N+cN^{2/3}. $$ For any $\varepsilon>0$ there exist infinitely many positive integers $N$ such that the equation has no solutions satisfying $$ N\le x,y,z,t \le N+N^{4/7-\varepsilon}. $$ There is an absolute constant $c>0$ such that for any positive integer $N$ the equation $$ x^4+y^4=z^4+t^4,\quad x,y,z,t\in\mathbb{N}, \quad \{x,y\}\not=\{z,t\}, $$ has no solutions satisfying $$ N\le x,y,z,t \le N+cN^{3/5}. $$ There is an absolute constant $c>0$ such that for any positive integer $N$ this equation has a solution satisfying $$ N\le x,y,z,t \le N+cN^{12/13}. $$
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves explicit bounds showing that the equation x³ + y³ = z³ + t³ has no non-trivial solutions (with {x,y} ≠ {z,t}) for x,y,z,t in [N, N + L) where L = ((38/3)N + 1297/36)^{1/2} + 19/6, and that this length is sharp because solutions exist for infinitely many N when the strict inequality is relaxed to ≤. It further shows existence of solutions in intervals of length c N^{2/3} for some absolute c > 0, non-existence up to length N^{4/7-ε} for infinitely many N, and parallel results for x⁴ + y⁴ = z⁴ + t⁴ with non-existence up to c N^{3/5} and existence up to c N^{12/13}.
Significance. If the derivations hold, the explicit constants (38/3, 1297/36, 19/6 and the exponents 2/3, 4/7, 3/5, 12/13) obtained by direct optimization of quadratic inequalities from difference estimates constitute a concrete advance in the study of Sidon-type properties for power sums in short intervals. The matching lower and upper bounds, together with the parameter-free character of the main length expressions, strengthen the results beyond asymptotic statements and make them directly usable in additive combinatorics.
major comments (1)
- The optimization step that produces the precise coefficients 38/3, 1297/36 and 19/6 from the quadratic inequality in the cube case is load-bearing for the claimed sharpness; the manuscript should display the explicit minimization (or the system of equations solved) in the proof of the main theorem so that the constants can be verified independently.
minor comments (2)
- In the statement of the quartic results, the constant c is left unspecified; a numerical upper bound on c (even if not optimal) would improve readability.
- The notation {x,y} ≠ {z,t} is used without prior definition; a brief sentence clarifying that it means the unordered pairs are distinct would prevent any ambiguity.
Simulated Author's Rebuttal
We thank the referee for the careful reading of our manuscript and the positive assessment. We address the single major comment below.
read point-by-point responses
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Referee: The optimization step that produces the precise coefficients 38/3, 1297/36 and 19/6 from the quadratic inequality in the cube case is load-bearing for the claimed sharpness; the manuscript should display the explicit minimization (or the system of equations solved) in the proof of the main theorem so that the constants can be verified independently.
Authors: We agree that the explicit minimization step is essential for independent verification of the constants. In the revised version we will insert, immediately after the statement of the quadratic inequality in the proof of the main theorem, a self-contained paragraph deriving the coefficients. This will include the system obtained by setting the derivative of the relevant quadratic form to zero (or the equivalent algebraic elimination) together with the verification that the resulting critical point yields the stated values 38/3, 1297/36 and 19/6. revision: yes
Circularity Check
No significant circularity; constants derived explicitly from proof inequalities
full rationale
The paper derives the explicit constants (38/3, 1297/36, 19/6) and exponents (2/3, 4/7, 3/5, 12/13) via direct optimization of quadratic inequalities arising from difference estimates and combinatorial bounds in the proofs of the main theorems. These steps are self-contained analytic derivations with tracked error terms and no hidden parameters. No load-bearing self-citations, self-definitional reductions, fitted inputs renamed as predictions, or ansatzes smuggled via prior work are present. The non-existence and existence statements in short intervals follow from standard estimates without reducing to the target claims by construction.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard axioms and inequalities of the integers together with basic analytic number theory estimates
discussion (0)
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