On sums of two squares and a basis of order 2
Pith reviewed 2026-05-09 23:05 UTC · model grok-4.3
The pith
For large N there exist two intervals of consecutive integers adding to N, each of length roughly log N times (log log N) to a small power, such that no n in them has both n and an + b as a sum of two coprime squares.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
Let R be the set of integers n that can be written n = x² + y² with gcd(x, y) = 1. Fix integers a > 0 and b with a not dividing b. For all sufficiently large N there exist integers n1 and n2 with n1 + n2 = N and an integer m equal to the floor of (log N) (log log N) raised to the power 1/325565 such that the two intervals of 2m + 1 consecutive integers centered at n1 and at n2 lie inside [1, N] and, for every n in their union, at least one of n or an + b lies outside R. In particular, the product n(an + b) lies outside R for every such n.
What carries the argument
The two complementary short intervals I1 and I2 centered at n1 and n2 with n1 + n2 = N that simultaneously avoid having both n and an + b inside R for every n in the union of the intervals.
If this is right
- The product n(an + b) lies outside R throughout both intervals.
- The avoidance property holds for every fixed a > 0 not dividing b.
- The intervals become arbitrarily long as N grows, though the growth is extremely slow.
- R fails to contain both members of the pair (n, an + b) in any of the 2m + 1 consecutive positions inside the chosen blocks.
Where Pith is reading between the lines
- The same method may extend to show that R cannot simultaneously represent three or more numbers in a fixed linear configuration over short intervals.
- Improving the exponent on log log N would require stronger estimates on the distribution of R in short intervals.
- The result supplies a concrete obstruction to representing all large integers as sums of two elements from R when the summands are forced to satisfy a linear relation.
Load-bearing premise
The analytic estimates that control the distribution of sums of two coprime squares continue to hold when the intervals are only logarithmically long in N.
What would settle it
An explicit large N together with a verification that every pair of complementary intervals of the stated length contains at least one n for which both n and an + b belong to R would falsify the claim.
read the original abstract
Let $\mathcal{R}$ denote the set of integers $n$ that can be represented as the sum $n = x^2 + y^2$ with $(x,y) = 1$. Let $a$ and $b$ be integers with $a>0$, $a \nmid b$. We show that for sufficiently large positive integer $N$ there are two strings of consecutive positive integers $I_{1}=\{n_1-m,\ldots, n_1+m\}$ and $I_{2}=\{n_2-m, \ldots, n_2+m\}$ such that $m = [(\log N) (\log \log N)^{1/325565}]$, $I_{1}\cup I_{2} \subset [1, N]$, $N = n_1 + n_2$, and for any $n\in I_{1}\cup I_{2}$ at least one of $n$ or $an+b$ does not lie in $\mathcal{R}$. In particular, we have $n(an+b)\notin \mathcal{R}$ for all $n\in I_{1}\cup I_{2}$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves that for all sufficiently large positive integers N, there exist n1 and n2 with n1 + n2 = N together with intervals I1 and I2 of 2m + 1 consecutive integers centered at n1 and n2 (with m = floor((log N)(log log N)^{1/325565})) lying inside [1, N] such that for every n in I1 ∪ I2 at least one of n or an + b fails to lie in R, the set of sums of two coprime squares. In particular this implies n(an + b) ∉ R for all such n.
Significance. If correct, the result supplies an explicit (though small) lower bound on the gaps in the set S = {n : n ∈ R and an + b ∈ R}. Such quantitative gap statements are useful for additive-basis problems involving the complement of R or linear images of R. The exponent 1/325565 is typical of what arises from iterated applications of zero-density estimates or short-interval sieves for the representation function r(n) = # ways n = x² + y² with gcd(x, y) = 1; the paper therefore demonstrates that current analytic technology is already sufficient to produce super-logarithmic gaps.
major comments (2)
- [Main Theorem (abstract statement)] The main theorem is stated only for 'sufficiently large N' with no explicit N0 and no verification that the implicit constants in the error terms of the short-interval estimates for R keep the derived lower bound on m positive. If the effective constant hidden in the O-term of the density estimate for S exceeds the main term for some arbitrarily large N, the claimed intervals I1, I2 may fail to exist; this is load-bearing for the central existence claim.
- [Proof of the main theorem] The derivation of the specific exponent 1/325565 is not accompanied by a transparent accounting of how many logarithmic factors are lost at each step of the analytic argument (e.g., in the application of the large sieve or zero-density bound). Without an explicit chain of inequalities showing that the final exponent remains positive after all constants are inserted, it is impossible to confirm that the result is not vacuous.
minor comments (2)
- [Abstract] The notation [·] for the floor function is non-standard; replace with ⌊·⌋ throughout.
- [Introduction] The phrase 'a basis of order 2' appears in the title but is never defined or related explicitly to the stated theorem; add a sentence in the introduction clarifying the connection.
Simulated Author's Rebuttal
We thank the referee for the careful reading of the manuscript and for the positive evaluation of its significance. We address each major comment below and indicate the revisions planned for the next version.
read point-by-point responses
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Referee: The main theorem is stated only for 'sufficiently large N' with no explicit N0 and no verification that the implicit constants in the error terms of the short-interval estimates for R keep the derived lower bound on m positive. If the effective constant hidden in the O-term of the density estimate for S exceeds the main term for some arbitrarily large N, the claimed intervals I1, I2 may fail to exist; this is load-bearing for the central existence claim.
Authors: We agree that the statement 'for all sufficiently large N' leaves the dependence on N implicit. Rendering the result fully effective by computing an explicit N0 would require tracking every constant through the zero-density estimates, large-sieve inequalities, and short-interval sieves, many of which are currently known only with ineffective constants. We will add a clarifying sentence in the introduction and in the statement of the main theorem noting that the result is ineffective and that the small exponent 1/325565 is chosen precisely so that the main term eventually dominates the error terms for large N. This addresses the concern without claiming an explicit bound. revision: partial
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Referee: The derivation of the specific exponent 1/325565 is not accompanied by a transparent accounting of how many logarithmic factors are lost at each step of the analytic argument (e.g., in the application of the large sieve or zero-density bound). Without an explicit chain of inequalities showing that the final exponent remains positive after all constants are inserted, it is impossible to confirm that the result is not vacuous.
Authors: The exponent is obtained by accumulating losses from the large sieve (one log factor), zero-density bounds (several log-log factors), and the short-interval estimates in Sections 3–5. While the proof contains all necessary inequalities, we concede that a single consolidated chain is not displayed. In the revised manuscript we will insert a short new subsection (or appendix paragraph) that lists the principal sources of logarithmic loss and verifies that, after these losses, the exponent 1/325565 still leaves a positive power of log log N in the main term. This addition will make the positivity of the final expression transparent. revision: yes
Circularity Check
Existence theorem derived from analytic estimates without circular reduction
full rationale
The paper proves an unconditional existence statement for sufficiently large N using standard analytic number theory tools (likely short-interval estimates or sieves on the set R of coprime sums of two squares). The length m is obtained directly from the error terms in those estimates rather than from any fitted parameter or self-referential definition. No equation or claim reduces the stated conclusion to its own inputs by construction, and the abstract supplies no load-bearing self-citation. The result therefore stands as an independent theorem whose validity rests on the correctness of the underlying estimates, not on circularity.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math The asymptotic density of primitive sums of two squares up to x is asymptotic to c x / sqrt(log x) for an explicit constant c
- standard math Effective versions of sieve or exponential-sum estimates that detect membership in R
discussion (0)
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