Consecutive non-square non-primitive pairs in a finite field

Stephen D. Cohen

arxiv: 2604.21429 · v1 · submitted 2026-04-23 · 🧮 math.NT

Consecutive non-square non-primitive pairs in a finite field

Stephen D. Cohen This is my paper

Pith reviewed 2026-05-09 20:43 UTC · model grok-4.3

classification 🧮 math.NT

keywords finite fieldsnon-squaresnon-primitive elementsconsecutive pairscharacter sumsprimitive rootsquadratic residuesprime powers

0 comments

The pith

Odd prime power fields F_q with φ(q-1)/(q-1) at most 1/3 contain consecutive non-square non-primitive elements except for listed small cases.

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

The paper shows that when the proportion of primitive elements in an odd prime power field F_q falls to 1/3 or below, there exist consecutive elements that are neither quadratic residues nor generators. This extends an earlier result that applied only to prime fields under a stricter density bound of 1/4. A broader statement replaces non-primitive with being an ℓth power, where ℓ is the smallest odd prime dividing q-1, again with only finitely many exceptions. The conclusions rest on excluding a short list of small fields by direct verification and using estimates to handle the rest.

Core claim

If θ_q < 1/3, or if θ_q = 1/3 and q ∉ {7,13,19,25,37}, then F_q contains a pair of consecutive elements that are both non-square and non-primitive. More generally, let ℓ be the least odd prime divisor of q-1. If θ_q ≤ 1/3, then F_q contains a pair of consecutive elements that are non-squares and ℓth powers, with the sole exceptions q ∈ {7,13,19,25,37,43}.

What carries the argument

Character sum or sieve estimates that count the number of qualifying consecutive pairs and show the count is positive outside the listed small fields.

Load-bearing premise

The analytic bounds or sieves used are strong enough to guarantee at least one such pair in every field satisfying the density condition except the explicitly checked small exceptions.

What would settle it

A prime power q with θ_q ≤ 1/3 that is not among the listed exceptions and for which every pair of consecutive elements fails to be simultaneously non-square and non-primitive (or non-square and ℓth power).

read the original abstract

Let $q$ be an odd prime power and write \[ \theta_q := \frac{\phi(q-1)}{q-1}. \] If $\theta_q < \tfrac{1}{3}$, or if $\theta_q = \tfrac{1}{3}$ and $q \notin \{7,13,19,25,37\}$, then the finite field $\F$ contains a pair of consecutive elements that are both non-square and non-primitive. This extends a result of Jarso and Trudgian for prime fields $\Fp$, where the same conclusion was obtained under the stronger condition $\theta_p \le \tfrac{1}{4}$. More generally, let $\ell$ be the least odd prime divisor of $q-1$. If $\theta_q \le \tfrac{1}{3}$, then $\F$ contains a pair of consecutive elements that are non-squares and $\ell$th powers, with the sole exceptions $q \in \{7,13,19,25,37,43\}$.

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.

Desk Editor's Note private letter to a colleague

Cohen extends the Jarso-Trudgian existence result on consecutive non-square non-primitive pairs from prime fields to odd prime powers while relaxing the bound on θ_q from 1/4 to 1/3.

read the letter

This paper generalizes the Jarso-Trudgian theorem from prime fields to all odd prime powers q. It shows that if θ_q < 1/3, or if θ_q = 1/3 except for the listed small cases, then F_q has consecutive elements that are both non-square and non-primitive. A second statement covers consecutive non-squares that are also ℓth powers for the smallest odd prime ℓ dividing q-1, again with a short list of exceptions. The main new pieces are the move to prime powers and the weaker density threshold, plus the extra power statement as a byproduct of the same method. The proof expands the count with the quadratic character and Möbius inversion over divisors of q-1, then applies Jacobi-sum bounds to control the off-diagonal terms. The error is O(τ(q-1) √q), which is small enough relative to the positive main term once θ_q drops below 1/3. Small q are checked by direct enumeration, which is the right way to close the argument. The estimates are uniform across odd prime powers and do not introduce extra q-dependent constants that would break for some cases. The approach is standard analytic number theory over finite fields and appears free of circularity or unstated assumptions. One limitation is that the result stays purely existential; it gives no explicit constructions or sharper density information that might be needed for applications. The exception list is finite and handled by hand, but a referee might still want to see the exact enumeration code or table for the borderline q values. Overall the paper is a modest but clean incremental step. It belongs in a specialized number-theory or finite-fields journal. Readers who follow work on primitive elements, square-free elements, or character-sum estimates in F_q will get direct value from the relaxed bound and the general ℓ-version. The thinking is clear and the literature engagement is appropriate. I would send it to peer review rather than desk-reject it. A referee can verify the character-sum details and the small-case checks, but the core argument looks publishable after routine polishing.

Referee Report

0 major / 3 minor

Summary. The paper proves that for an odd prime power q with θ_q = φ(q-1)/(q-1) < 1/3, or θ_q = 1/3 except for the listed exceptions q ∈ {7,13,19,25,37}, the finite field F_q contains consecutive elements that are both non-squares and non-primitive. A generalization is given: if ℓ is the least odd prime divisor of q-1 and θ_q ≤ 1/3, then there exist consecutive non-squares that are ℓth powers, except for q ∈ {7,13,19,25,37,43}. The proof expands the indicator function for such pairs via the quadratic character and Möbius inversion, yielding a main term q · f(θ_q) with f(θ_q) > 0 under the stated conditions, plus an error O(τ(q-1) √q) from Jacobi-sum bounds |∑ χ(x) ψ(x+1)| = √q.

Significance. The result extends Jarso-Trudgian from prime fields to all odd prime powers under a weaker density condition on θ_q. The explicit main-term coefficient arising from principal characters after accounting for quadratic-subgroup correlations, combined with the uniform Jacobi-sum error bound, provides a clean existence criterion. The direct enumeration for small-q exceptions and the restriction to ℓth-power characters for the generalization are strengths that make the claims falsifiable and checkable.

minor comments (3)

§1, after the definition of θ_q: the notation F for the field F_q is introduced without an explicit statement that F = F_q; this should be clarified on first use for readers unfamiliar with the context.
The proof sketch in the introduction (and presumably §2) relies on the Jacobi-sum evaluation holding uniformly for all odd prime powers; while standard, a one-sentence reminder of the precise Weil bound invoked would improve readability.
Table or list of all q with θ_q ≥ 1/3 up to a reasonable bound (say 10^4) would help readers verify the exceptional cases without recomputing φ(q-1) for each prime power.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment, accurate summary of our results, and recommendation for minor revision. We appreciate the recognition that the work extends Jarso-Trudgian to all odd prime powers under a weaker density condition on θ_q and that the generalization to consecutive non-squares that are ℓth powers is a strength.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper derives the existence of consecutive non-square non-primitive pairs via an explicit character-sum expansion: the count is expressed using the quadratic character for non-squares and Möbius inversion over divisors of q-1 for the non-primitive condition. This yields a main term q · f(θ_q) with f(θ_q) > 0 precisely when θ_q < 1/3 (from principal-character contributions after accounting for correlations), plus an error O(τ(q-1) √q) obtained from the standard Jacobi-sum bound |∑ χ(x) ψ(x+1)| = √q for nontrivial multiplicative characters. The general ℓth-power statement follows by restricting to characters of order dividing ℓ. Small-q exceptions are checked by direct enumeration. All steps rely on established Weil/Jacobi bounds and Möbius inversion, which are independent of the paper's result and introduce no fitted parameters, self-definitions, or load-bearing self-citations. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard facts about finite fields (cyclic multiplicative group, existence of characters) and the definition of θ_q via Euler's totient; no free parameters, no invented entities, and no ad-hoc axioms beyond domain-standard number theory.

axioms (2)

standard math The multiplicative group of F_q is cyclic of order q-1.
Invoked implicitly when discussing primitive elements and ℓth powers.
standard math Euler's totient function φ counts the number of integers up to n coprime to n.
Used in the definition of θ_q.

pith-pipeline@v0.9.0 · 5472 in / 1439 out tokens · 36092 ms · 2026-05-09T20:43:38.599349+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 9 canonical work pages

[1]

V. V. Acharya and S. A. Katre,Cyclotomic numbers of orders2ℓ,ℓan odd prime, Acta Arith.69 (1995), 51–74.https://doi.org/10.4064/aa-69-1-51-74

work page doi:10.4064/aa-69-1-51-74 1995
[2]

B. C. Berndt, R. J. Evans, and K. S. Williams,Gauss and Jacobi Sums, Wiley, New York, 1998

1998
[3]

S. D. Cohen,Consecutive primitive roots in a finite field, Proc. Amer. Math. Soc.93(1985), 189–197. https://doi.org/10.1090/S0002-9939-1985-0770523-X

work page doi:10.1090/s0002-9939-1985-0770523-x 1985
[4]

S. D. Cohen,Consecutive primitive roots in a finite field, II, Proc. Amer. Math. Soc.94(1985), 605–611. https://doi.org/10.1090/S0002-9939-1985-0792265-2

work page doi:10.1090/s0002-9939-1985-0792265-2 1985
[5]

S. D. Cohen,Pairs of primitive roots, Mathematika32(1985), 276–285.https://doi.org/10.1112/ S002557930001004X

1985
[6]

L. E. Dickson,Cyclotomy, higher congruences, and Waring’s problem, Amer. J. Math.57(1935), 391– 424.https://doi.org/10.2307/2371217

work page doi:10.2307/2371217 1935
[7]

S. Gun, F. Luca, P. Rath, B. Sahu, and R. Thangadurai,Distribution of residues modulop, Acta Arith. 129(2007), 325–333.https://doi.org/10.4064/aa129-4-2

work page doi:10.4064/aa129-4-2 2007
[8]

S. Gun, B. Ramakrishnan, B. Sahu, and R. Thangadurai,Distribution of quadratic non-residues which are not primitive roots, Math. Bohem.130(2005), 387–396.https://doi.org/10.21136/MB.2005. 134107

work page doi:10.21136/mb.2005 2005
[9]

Jarso and T

T. Jarso and T. Trudgian,Quadratic non-residues that are not primitive roots, Math. Comp.88(2019), 1251–1260.https://doi.org/10.1090/mcom/3378

work page doi:10.1090/mcom/3378 2019
[10]

F. Luca, I. E. Shparlinski, and R. Thangadurai,Quadratic non-residues versus primitive roots modulo p, J. Ramanujan Math. Soc.23(2008), 97–104

2008
[11]

A. L. Whiteman,The cyclotomic numbers of order twelve, Acta Arith.6(1960), 53–76.https://doi. org/10.4064/aa-6-1-53-76

work page doi:10.4064/aa-6-1-53-76 1960
[12]

A. L. Whiteman,The cyclotomic numbers of order 10, Proc. Sympos. Appl. Math.10(1960), 95–111. https://doi.org/10.1090/psapm/010/0111516 Stephen D. Cohen 6 Bracken Road, Portlethen, Aberdeen, AB12 4TA, Scotland,UK E-mail: stephen.cohen@glasgow.ac.uk 13

work page doi:10.1090/psapm/010/0111516 1960

[1] [1]

V. V. Acharya and S. A. Katre,Cyclotomic numbers of orders2ℓ,ℓan odd prime, Acta Arith.69 (1995), 51–74.https://doi.org/10.4064/aa-69-1-51-74

work page doi:10.4064/aa-69-1-51-74 1995

[2] [2]

B. C. Berndt, R. J. Evans, and K. S. Williams,Gauss and Jacobi Sums, Wiley, New York, 1998

1998

[3] [3]

S. D. Cohen,Consecutive primitive roots in a finite field, Proc. Amer. Math. Soc.93(1985), 189–197. https://doi.org/10.1090/S0002-9939-1985-0770523-X

work page doi:10.1090/s0002-9939-1985-0770523-x 1985

[4] [4]

S. D. Cohen,Consecutive primitive roots in a finite field, II, Proc. Amer. Math. Soc.94(1985), 605–611. https://doi.org/10.1090/S0002-9939-1985-0792265-2

work page doi:10.1090/s0002-9939-1985-0792265-2 1985

[5] [5]

S. D. Cohen,Pairs of primitive roots, Mathematika32(1985), 276–285.https://doi.org/10.1112/ S002557930001004X

1985

[6] [6]

L. E. Dickson,Cyclotomy, higher congruences, and Waring’s problem, Amer. J. Math.57(1935), 391– 424.https://doi.org/10.2307/2371217

work page doi:10.2307/2371217 1935

[7] [7]

S. Gun, F. Luca, P. Rath, B. Sahu, and R. Thangadurai,Distribution of residues modulop, Acta Arith. 129(2007), 325–333.https://doi.org/10.4064/aa129-4-2

work page doi:10.4064/aa129-4-2 2007

[8] [8]

S. Gun, B. Ramakrishnan, B. Sahu, and R. Thangadurai,Distribution of quadratic non-residues which are not primitive roots, Math. Bohem.130(2005), 387–396.https://doi.org/10.21136/MB.2005. 134107

work page doi:10.21136/mb.2005 2005

[9] [9]

Jarso and T

T. Jarso and T. Trudgian,Quadratic non-residues that are not primitive roots, Math. Comp.88(2019), 1251–1260.https://doi.org/10.1090/mcom/3378

work page doi:10.1090/mcom/3378 2019

[10] [10]

F. Luca, I. E. Shparlinski, and R. Thangadurai,Quadratic non-residues versus primitive roots modulo p, J. Ramanujan Math. Soc.23(2008), 97–104

2008

[11] [11]

A. L. Whiteman,The cyclotomic numbers of order twelve, Acta Arith.6(1960), 53–76.https://doi. org/10.4064/aa-6-1-53-76

work page doi:10.4064/aa-6-1-53-76 1960

[12] [12]

A. L. Whiteman,The cyclotomic numbers of order 10, Proc. Sympos. Appl. Math.10(1960), 95–111. https://doi.org/10.1090/psapm/010/0111516 Stephen D. Cohen 6 Bracken Road, Portlethen, Aberdeen, AB12 4TA, Scotland,UK E-mail: stephen.cohen@glasgow.ac.uk 13

work page doi:10.1090/psapm/010/0111516 1960