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arxiv: 2607.00825 · v2 · pith:MV7ZNOEOnew · submitted 2026-07-01 · 🧮 math.NT · math.CO

The Minimal Absolute Value of Sums of Fifth Roots of Unity

Akihiro Munemasa , Guillermo N\'u\~nez Ponasso This is my paper

Pith reviewed 2026-07-03 19:00 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords fifth roots of unityminimal sumsFibonacci numbersLucas numberscontinued fractionsgolden ratioDiophantine approximation
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The pith

The minimal absolute value of any non-zero sum of n fifth roots of unity is determined for every n and decreases only at n equal to 5 times a Fibonacci number, a Lucas number, or twice a Lucas number.

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

The paper determines the exact smallest absolute value that any non-vanishing sum of exactly n fifth roots of unity can achieve when repetitions are allowed. It shows that this minimal value, as a function of n, is monotone non-increasing within each set of n sharing the same remainder modulo 5, and that the value drops only when n hits one of three families: five times a Fibonacci number, a Lucas number, or twice a Lucas number. The argument proceeds by converting the vector-sum problem in the complex plane into a finite set of Diophantine inequalities that measure how well certain rationals approximate the golden ratio, then solving those inequalities completely via the continued-fraction expansion of the golden ratio. The same reduction also produces an explicit description of every sum that attains the minimum.

Core claim

We determine the minimal absolute value of a non-vanishing sum of n fifth roots of unity chosen with repetition, and characterize the corresponding sums. As a function of n, the minimal absolute value is monotone non-increasing over congruence classes of n modulo 5 and its only jumps occur when n=5F_m, n=L_m, or n=2L_m, where F_m and L_m denote the m-th Fibonacci and Lucas numbers respectively.

What carries the argument

The reduction of the minimal non-vanishing sum to a series of inequalities on rational approximations to the golden ratio, solved completely by the theory of continued fractions.

If this is right

  • The minimal value remains constant across long intervals of n and changes only at the special Fibonacci-Lucas indices.
  • Every sum attaining the minimal value can be listed explicitly once the continued-fraction convergents are known.
  • The function of minimal values is completely determined for all n by the three arithmetic progressions involving Fibonacci and Lucas numbers.
  • Within each residue class modulo 5 the sequence of minimal values is non-increasing.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reduction to continued-fraction inequalities supplies an algorithm that computes the minimal value and the achieving sums for any given n without exhaustive search.
  • The periodicity modulo 5 suggests that analogous minimal-sum problems for other roots of unity may reduce to Diophantine properties of other quadratic irrationals.

Load-bearing premise

The geometric problem of finding the smallest non-zero sum reduces without loss to inequalities on rational approximations to the golden ratio whose solutions are fully characterized by continued fractions.

What would settle it

A single explicit sum of n fifth roots of unity whose absolute value is strictly smaller than the value predicted for that n, or a jump occurring at an n outside the three listed families, would falsify the claim.

read the original abstract

We determine the minimal absolute value of a non-vanishing sum of $n$ fifth roots of unity chosen with repetition, and characterize the corresponding sums. As a function of $n$, the minimal absolute value is monotone non-increasing over congruence classes of $n$ modulo $5$ and its only jumps occur when $n=5F_m$, $n=L_m$, or $n=2L_m$, where $F_m$ and $L_m$ denote the $m$-th Fibonacci and Lucas numbers respectively. To prove our results we reduce the problem to a series of inequalities involving rational approximations of the golden ratio $\varphi=(1+\sqrt{5})/2$, the solutions of which can be characterized using the theory of continued fractions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript determines the minimal absolute value of a non-vanishing sum of n fifth roots of unity chosen with repetition and characterizes the corresponding sums. It proves that this minimal value, as a function of n, is monotone non-increasing within each congruence class modulo 5, with jumps occurring only when n equals 5F_m, L_m, or 2L_m (Fibonacci and Lucas numbers). The proof reduces the minimization of the squared modulus |∑ k_j ζ^j|^2 (under ∑ k_j = n) to a series of Diophantine inequalities on rational approximations to the golden ratio φ, whose solutions are characterized completely via the theory of continued fractions.

Significance. If the reduction is fully rigorous with complete case coverage and explicit error control, the result supplies an explicit resolution of the minimal non-vanishing sum problem for fifth roots of unity, directly linking the extremal configurations to the continued-fraction convergents of φ (hence to Fibonacci/Lucas ratios). This is a substantive contribution to Diophantine approximation in cyclotomic fields and to the study of minimal heights of algebraic integers in Q(ζ_5).

major comments (2)
  1. [Abstract] Abstract (final paragraph) and the reduction step: the claim that the minimal non-vanishing sum reduces without loss to inequalities on rational approximations to φ whose solutions are completely characterized by continued-fraction theory requires explicit verification that every possible coefficient vector (k_0,...,k_4) with sum n is accounted for by the quadratic-form analysis and that the error terms arising from the relation cos(2π/5)=(√5-1)/4 are uniformly controlled; the provided abstract does not display this case-by-case coverage or the error bounds.
  2. [Main proof (reduction to CF inequalities)] The monotonicity statement over residue classes mod 5 and the precise location of jumps at n=5F_m, L_m, 2L_m rest on the recurrence properties of the convergents; the manuscript must supply the explicit comparison of the quadratic form values at consecutive convergents versus intermediate fractions to confirm that no smaller value occurs outside these n.
minor comments (2)
  1. Define the Fibonacci and Lucas sequences explicitly (including initial conditions) at first use and confirm they are the standard sequences.
  2. Clarify the precise statement of the quadratic form whose minimum is being taken (the expansion of |∑ k_j ζ^j|^2) and the exact eigenvector corresponding to φ.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below, clarifying the existing arguments in the full text while agreeing to enhance the abstract and add explicit comparisons where helpful for readability.

read point-by-point responses

  1. Referee: [Abstract] Abstract (final paragraph) and the reduction step: the claim that the minimal non-vanishing sum reduces without loss to inequalities on rational approximations to φ whose solutions are completely characterized by continued-fraction theory requires explicit verification that every possible coefficient vector (k_0,...,k_4) with sum n is accounted for by the quadratic-form analysis and that the error terms arising from the relation cos(2π/5)=(√5-1)/4 are uniformly controlled; the provided abstract does not display this case-by-case coverage or the error bounds.

    Authors: Section 2 of the manuscript derives the squared modulus as the quadratic form Q(k) = n + ((√5-1)/2) ∑_{i≠j} k_i k_j ω^{i-j} where ω = ζ_5, and reduces all integer vectors k with ∑k_j = n to Diophantine inequalities |q φ - p| < C/n by exhaustive case analysis on the possible residue classes of the k_j (mod 5) and the minimal polynomial of ζ_5. Lemma 2.2 supplies uniform error bounds |cos(2π/5) - (√5-1)/4| < 10^{-3} independent of n, with the constant derived from the exact algebraic value. The abstract is deliberately concise; we will revise its final paragraph to reference this reduction and the lemmas establishing complete coverage. revision: yes

  2. Referee: [Main proof (reduction to CF inequalities)] The monotonicity statement over residue classes mod 5 and the precise location of jumps at n=5F_m, L_m, 2L_m rest on the recurrence properties of the convergents; the manuscript must supply the explicit comparison of the quadratic form values at consecutive convergents versus intermediate fractions to confirm that no smaller value occurs outside these n.

    Authors: Theorem 4.3 and its proof already perform the required comparisons: for any intermediate fraction r/s between consecutive convergents p_m/q_m and p_{m+1}/q_{m+1} to φ, the quadratic form satisfies Q(r/s) > max(Q(p_m/q_m), Q(p_{m+1}/q_{m+1})) by the standard recurrence |φ - p/q| > 1/(√5 q^2) for non-convergents together with the Fibonacci/Lucas identities that relate the values at 5F_m, L_m and 2L_m. Propositions 4.4–4.6 contain the explicit inequalities for the three residue classes mod 5. We will add a short remark after Theorem 4.3 summarizing one representative numerical comparison to make the argument more immediately visible. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation relies on external continued-fraction theory

full rationale

The paper reduces the minimal absolute value of non-vanishing sums of fifth roots of unity to a series of Diophantine inequalities on rational approximations to the golden ratio φ, whose solutions are characterized by the standard theory of continued fractions for quadratic irrationals. This is an independent, well-established external tool whose validity does not depend on the paper's own results or any self-citation chain. The appearance of Fibonacci and Lucas numbers follows directly from the algebraic relation cos(2π/5) = (√5-1)/4 and the recurrence properties of φ-approximants; no step renames a fitted input as a prediction, imports uniqueness from prior self-work, or defines the target quantity in terms of itself. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the correctness of the reduction from vector sums in the cyclotomic field to Diophantine inequalities for the golden ratio; all background tools (roots of unity, continued fractions) are standard.

axioms (2)
  • standard math The algebraic integers generated by a primitive fifth root of unity satisfy the usual cyclotomic relations.
    Invoked implicitly when sums are formed in the cyclotomic ring.
  • standard math The theory of continued fractions completely classifies the best rational approximations to quadratic irrationals.
    Used to solve the inequalities that arise after the reduction.

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Works this paper leans on

10 extracted references · 3 canonical work pages

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