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Bounds for sets of remainders

T0 review · 3 major / 4 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read The paper proves that distinct remainders have an exact linear density c ≈ 0.2296, with an explicit error bound.

desk verdict Rigorous asymptotics for a natural OEIS sequence, with two fixable proof gaps; the main results look correct and the paper deserves refereeing. read the letter →

arxiv 2508.20853 v1 pith:TRTDRYJE submitted 2025-08-28 math.NT

Omkar Baraskar , Ingrid Vukusic This is my paper · ORCID
classification math.NT MSC 11N3711A0711B83
keywords numberofremaindersdistinctresiduesasymptoticdensitysieveargumentconsecutivedifferencesiteratedremaindersetsPierceexpansionsintegersequence
topics The Collatz Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Counting how many different remainders appear when a fixed n is divided by each k up to n/2 looks simple but had no rigorous asymptotic. This paper proves the count s(n) is exactly c·n plus an error of size O(n/(log n log log n)), where c ≈ 0.2296 is a concrete product over primes, confirming a density that had only been conjectured from data. It also shows that moving from n to n+1 can raise the count by at most one, can lower it by arbitrarily large amounts, and never lowers it by more than O(log log n). The same machinery applies to iterated remainder sets, tying their growth to a long-standing open problem about the length of Pierce expansions.

What carries the argument

The mechanism is a least-prime-factor sieve. Lemma 1 recasts membership of a remainder r in S(n) as a divisibility condition on n−r, namely that n−r has a proper divisor at least r+1; this is equivalent to n−r having a prime factor p ≤ (n−r)/(r+1). Counting remainders then reduces to counting integers m = n−r in short intervals that are divisible by p but no smaller prime, via inclusion–exclusion sets D_p with error O(2^{π(p−1)}). The error term of Theorem 1 comes from choosing the cutoff X = log n and bounding the tail by Σ_{p≥X} 1/p². For differences, the 'not transferred' set T(n, n+1) is characterized by r+1 being the largest proper divisor of n−r with prime cofactor, and for odd n the p

What would settle it

Verify Theorem 1 by exact enumeration: if the normalized error (s(n) − c·n)/(n/(log n log log n)) is unbounded over n up to, say, 10^8, the claimed O-term fails; the paper's own data to 10^7 already fits a much smaller error. For the iterated lower bound, the literal base case of Lemma 13 fails at n=4: the proof requires {0,1} ⊆ S0(4), but S0(4) = {1,2}.

Watch

Extended reading notes

Core claim

The paper's central claim is that the number s(n) of distinct remainders n mod k for 1 ≤ k ≤ floor(n/2) has an exact linear asymptotic density: s(n) = c·n + O(n/(log n log log n)), where c = Σ_p 1/(p(p+1)) · Π_{q<p}(1 − 1/q) ≈ 0.2296. This confirms, with a rigorous error term, a density that had previously only been conjectured from numerical data. The paper also proves that s(n+1) is never more than one larger than s(n), that arbitrarily large downward jumps occur, and that every downward jump is at most O(log log n) in size. For the iterated remainder sets S_j(n), it establishes density bounds 1/(j+2)! ≤ liminf s_j(n)/n and limsup s_j(n)/n ≤ 1/(j+2), tying the problem to the length of Pier

Load-bearing premise

The lower-bound proof for iterated remainder sets assumes, in its base case, that S0(n) contains 0, although S0(n) is defined as {1, …, floor(n/2)}; starting the induction at r=1 repairs the argument, but the written proof depends on the uncorrected statement.

Editorial extensions

If this is right

  • The exact density of the remainder-count sequence is no longer a conjecture: any large-n computation of s(n) can be compared against c·n with a rigorous error bound.
  • Because increases are bounded by one while decreases are unbounded, the sequence s(n) has a one-sided Lipschitz character; consecutive-value drops of any fixed size occur infinitely often, yet never faster than log log n at scale n.
  • The upper bound max S_j(n) ≤ n/(j+2) gives a simple universal shrinkage rate for iterated remainder sets, independent of arithmetic properties of n.
  • The lower bound in Theorem 3 supplies a linear-in-n growth rate for iterated remainder sets, which is the same scale as the known upper bounds on Pierce-expansion length.
  • The paper leaves the error term open; if the numerical O(n^{1/3}) suggestion is correct, it would imply s(n) is far more regular than the sieve proof alone guarantees.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same interval-counting sieve should give an asymptotic for remainders in other ranges, such as 1 ≤ k ≤ α n for fixed α, with a density that varies continuously in α.
  • The colour-coded plots for S2(n) and S3(n) suggest the limiting density depends on n modulo small primes; proving Problem 2 would amount to exhibiting two subsequences of n with different limiting densities.
  • Lemma 8 suggests a way to predict large downward jumps: they are produced by multiplying n+1 by a new large prime, as in the recursive construction of Proposition 4, so one can test whether the first drop of size k always occurs for n equal to a product of initial primes minus 1.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies the sequence s(n), the number of distinct remainders n mod k for 1 ≤ k ≤ floor(n/2). The main result (Theorem 1) claims an asymptotic formula s(n) = c n + O(n/(log n log log n)) with c given by an explicit infinite product over primes, approximately 0.2296. The paper also proves that consecutive differences satisfy s(n+1) ≤ s(n)+1 and s(n+1) ≥ s(n) - O(log log n), and that arbitrarily large decreases occur. Finally, it introduces iterated remainder sets S_j(n) and proves upper and lower bounds of sizes n/(j+2) and n/((j+2)!) up to constants, relating them to an open problem on Pierce expansions. The proofs use elementary sieving, the prime number theorem, Dirichlet's theorem, and a recursive construction.

Significance. If the proofs are repaired, the paper provides the first rigorous derivation of the linear asymptotic density of s(n) with an explicit constant, confirming and quantifying a heuristic StackExchange/OEIS observation. The results on consecutive differences and iterated remainder sets are new and address a natural sequence that appears to have been little studied. A notable strength is that the constant c is defined a priori as an infinite product, with no fitted parameters; the paper also connects the iterated sets to an existing open problem and gives nontrivial bounds. The central claims are plausible and likely correct, but the current manuscript contains a few load-bearing proof gaps that must be fixed before the results can be considered fully established.

major comments (3)
  1. [Section 3, Eq. (5)] The bound |Mp(n)| ≤ n/(p(p+1)) used for E1 is false. For n = p^2 + 1, the integer m = p^2 lies in the interval [p(n+1)/(p+1), n-1] and is divisible by p but by no smaller prime, so |Mp(n)| ≥ 1, while n/(p(p+1)) < 1. Thus the displayed inequality and the resulting E1 estimate are not justified as written. The gap is repairable: for p > sqrt(n), Mp(n) is actually empty (any m = p a with a ≥ 2 has a prime factor < p, and a = 1 gives m = p < p(n+1)/(p+1)); for p ≤ sqrt(n) one may use |Mp(n)| ≤ (n-p)/(p(p+1)) + 1, and the extra sum over p ≤ sqrt(n) is O(sqrt(n)/log n), absorbed by the claimed error. This additional argument needs to be supplied.
  2. [Section 6, proof of Lemma 13] The base case is internally inconsistent. S0(n) was defined in Section 2 as {1,2,...,floor(n/2)}, which does not contain 0, yet the proof states 'For j=0 the inclusion (14) is clearly true with x0=0, since S0(n) = {0,1,...,floor(n/2)}.' The stated inclusion for j=0 requires 0 to be in S0(n). This can be fixed by starting the inclusion at r=1 for j=0 and treating the j=0 lower bound separately, or by redefining S0(n) to include 0 and adjusting the definitions. As written, the induction base for Theorem 3 is false, and this is load-bearing for the lower bound.
  3. [Section 4, proof of Proposition 4] The formula r' = (r-1)Q + 1 appears to be a typo; it should be r' = (r+1)Q - 1. With the printed formula, r'+1 = (r-1)Q + 2 does not in general divide n(j+1)+1 = (n(j)+1)Q, so condition (i) of Lemma 9 fails. The subsequent claim that r' = Q - 1 works is also consistent with the corrected formula. This is a load-bearing error in the construction proving liminf of differences is -infinity, though it is clearly repairable.
minor comments (4)
  1. [Section 3, Lemma 3 application] Lemma 3 is stated for positive integers a and t, but in the proof of Theorem 1 it is applied with a = n - (n-p)/(p+1) - 1 and t = (n-p)/(p+1), which need not be integers. The proof can be adapted by replacing t with floor(t) and absorbing the O(1) discrepancy into E2, but the formal mismatch should be noted.
  2. [Section 4, Proposition 3] In the proof, 'the prime factorization of n' should read 'of n+1'.
  3. [Section 7, captions] The caption of Figure 2 says 'plots of sj(n) for n = 1,2,3' but the text means j = 1,2,3. Please correct.
  4. [Throughout] There are a few small textual inconsistencies, e.g., the definition of T(n) versus T(n,n+1), and the statement of Lemma 9 uses r+1 > floor((n-2)/3) in one place. These do not affect the arguments.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main asymptotic constant is an explicit prime-sum density, derived by a sieve with standard analytic estimates, and no load-bearing self-citations appear.

full rationale

Theorem 1 does not reduce to its inputs. The constant c is defined explicitly as sum_p (1/(p(p+1))) * prod_{q<p}(1-1/q), and the proof derives s(n) = c n + O(n/(log n log log n)) by a disjoint decomposition S(n)\setminus{0} = union_p M_p(n), by counting M_p(n) with inclusion-exclusion (Lemma 3), and by bounding the error terms E1, E2, E3 using standard estimates (PNT, Rosser-Schoenfeld, Lemma 5). No parameter is fitted to values of s(n), and the error bound is not used to define c. The paper contains no self-citation that carries a load-bearing argument: the cited Pierce-expansion work of Shallit and Erdos-Shallit is used only for context and for the separate iterated-remainder discussion, not to force Theorem 1 or Theorem 2. The base-case notation in Lemma 13 is internally inconsistent (S0(n) is defined without 0 but the proof writes S0(n) = {0,1,...,floor(n/2)}), but this is a repairable correctness issue rather than circularity, since the positive elements claimed are present by definition and the induction uses only positive k. Likewise, the reviewer-flagged inequality |Mp(n)| <= n/(p(p+1)) in Eq. (5) is a proof gap in the E1 estimate, not a circular step: even if that displayed bound is false for p near sqrt(n), neither c nor the claimed asymptotic is defined in terms of that bound. The derivation chain is therefore self-contained with respect to external benchmarks, and no step is equivalent to its input by construction.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters are fitted; X(n)=log n in Theorem 1 is a proof device, not a parameter of the result. Standard analytic number theory theorems are used. The iterated sets S_j(n) are new definitions, not physical/hypothetical entities.

assumptions (3)
  • standard math Prime number theorem (or Chebyshev-type prime density estimates) used to bound sums over primes in Lemma 5.
    Used to prove sum_{p>n} 1/p^2 = O(1/(n log n)) in Lemma 5.
  • standard math Dirichlet's theorem on primes in arithmetic progressions.
    Used in Proposition 4 to construct n(j+1) as a prime in the progression a_x = (n(j)+1)(xP+y)-1.
  • standard math Rosser-Schoenfeld bound π(n) ≤ 1.3 n/log n [7].
    Used in Theorem 1 proof to bound 2^{π(log n)}.

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Cite this review

Pith. "Pith review of Bounds for sets of remainders." pith.science (2026). https://pith.science/paper/TRTDRYJE

@misc{pith2026250820853,
  author       = {Pith},
  title        = {Pith review of: Bounds for sets of remainders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TRTDRYJE}},
  note         = {Machine review of arXiv:2508.20853}
}
abstract

Let $s(n)$ be the number of different remainders $n \bmod k$, where $1 \leq k \leq \lfloor n/2 \rfloor$. This rather natural sequence is sequence A283190 in the OEIS and while some basic facts are known, it seems that surprisingly it has barely been studied. First, we prove that $s(n) = c \cdot n + O(n/(\log n \log \log n))$, where $c$ is an explicit constant. Then we focus on differences between consecutive terms $s(n)$ and $s(n+1)$. It turns out that the value can always increase by at most one, but there exist arbitrarily large decreases. We show that the differences are bounded by $O(\log \log n)$. Finally, we consider ''iterated remainder sets''. These are related to a problem arising from Pierce expansions, and we prove bounds for the size of these sets as well.

Figures

Figures reproduced from arXiv: 2508.20853 by the authors.

Figure 1
Figure 1. The function 2n 1/3 and the largest deviations of s(n) from c · n. For the iterated remainder sets, recall that Theorem 3 says that 1 (j + 2)! ≤ lim inf n→∞ sj (n) n ≤ lim sup n→∞ sj (n) n ≤ 1 j + 2 . In particular, there is a gap between our lower and our upper bound. Indeed, numerical experiments strongly suggest that limn→∞ sj (n)/n does not exist for j ≥ 2. The three plots in [PITH_FULL_IMAGE:figures/full_fig_p… view at source ↗
Figure 2
Figure 2. Plots of sj (n) for n = 1, 2, 3; colours according to divisibility by 2 and 3. Problem 2. Prove that for j ≥ 2 the limit limn→∞ sj (n)/n does not exist. Acknowledgements We thank Jeffrey Shallit for his encouragement, for sharing his numbers up to s(107 ) with us, and for suggesting to consider iterated sets [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. s2(n) for n ≡ 1 mod 6; the points where n is divisible by 5 are colored red. References [1] Z. Chase and M. Pandey. On the length of pierce expansions, 2022, 2211.08374. [2] P. Erd˝os and J. O. Shallit. New bounds on the length of finite Pierce and Engel series. S´em. Th´eor. Nombres Bordeaux (2), 3(1):43–53, 1991. URL: http://jtnb.cedram.org/item?id=JTNB_1991__3_1_43_ 0. [3] J. Hoseana and R. Aziz. The generalized … view at source ↗

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Works this paper leans on

10 extracted references · 8 canonical work pages

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