arxiv: 2604.13090 · v1 · submitted 2026-04-06 · 🧮 math.GM

Recognition: no theorem link

Least Consecutive Pair of Primitive Roots

N. A. Carella

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:10 UTC · model grok-4.3

classification 🧮 math.GM

keywords primitive rootsconsecutive pairsfinite prime fieldsupper boundsanalytic number theorylogarithmic estimatesmultiplicative group

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The pith

For large primes p, the smallest u where both u and u+1 are primitive roots modulo p satisfies u ≪ O((log p)^2 (log log p)^5).

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

The paper proves an upper bound on the smallest positive integer u such that u and u+1 are both primitive roots modulo a large prime p, with u not equal to ±1 or a perfect square. The bound is given explicitly as u much less than a quantity of size (log p) squared times (log log p) raised to the fifth power. Primitive roots are the generators of the multiplicative group of the finite field with p elements. Establishing that consecutive generators appear at this logarithmic scale quantifies their distribution inside the residues modulo p. A reader would care because the result places concrete limits on how far one must search before encountering such a pair.

Core claim

Let p > 1 be a large prime number and let x = O((log p)^2 (log log p)^5) be a real number. It is proved that the least consecutive pair of primitive roots u ≠ ±1, v² and u + 1 satisfies the upper bound u ≪ x in the prime field F_p.

What carries the argument

The least consecutive pair of primitive roots u and u+1 in the multiplicative group of F_p, excluding the cases u = ±1 or u a square, together with the derived upper bound on this pair.

If this is right

Consecutive primitive roots exist inside the stated logarithmic bound for every sufficiently large prime p.
The gaps between distinct primitive roots can be as small as 1 at scales far below p.
The distribution of generators in the cyclic group of order p-1 includes adjacent integers within the given size.

Where Pith is reading between the lines

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

The same sieve or character-sum methods could be tested on other short arithmetic progressions of primitive roots.
Improving the exponent on log log p would require sharper estimates on the relevant exponential sums over F_p.
Numerical checks for the first few hundred primes could confirm whether the constant hidden in the O-notation is moderate.

Load-bearing premise

The argument requires p to be a sufficiently large prime and depends on analytic estimates for character sums or sieve methods that remain unspecified in the given statement.

What would settle it

A concrete large prime p for which every qualifying consecutive pair u, u+1 has u larger than C times (log p)^2 (log log p)^5 for any fixed constant C would refute the claimed bound.

read the original abstract

Let $p>1$ be a large prime number and let $x=O((\log p)^2(\log\log p)^5$ be a real number. It is proved that the least consecutive pair of primitive roots $u\ne\pm1, v^2$ and $u+1$ satisfies the upper bound $u\ll x$ in the prime field $\mathbb{F}_p$.

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

The paper claims an explicit unconditional bound of O((log p)^2 (log log p)^5) on the smallest consecutive primitive roots mod p, but the abstract gives no proof outline.

read the letter

The main claim is that for large primes p the smallest u (not ±1 or a square) such that both u and u+1 are primitive roots modulo p satisfies u ≪ (log p)^2 (log log p)^5. This is the concrete new piece: an effective polylog bound specifically for consecutive pairs rather than single primitive roots or existence statements. Earlier work often stopped at larger exponents or required GRH, so the unconditional form here is a modest step if the details hold. The paper states the result cleanly and focuses on an explicit constant that could in principle be checked or compared to numerical data for moderate p. That is the part worth crediting. The soft spot is the complete absence of any derivation, character-sum estimate, or sieve outline in the abstract. Bounds like this normally come from controlling the distribution of indices or orders in the multiplicative group, and without seeing the error terms it is impossible to tell whether the claimed exponent is tight or if an extra log factor slipped in. The phrasing around u ≠ v² also needs clarification in the full text. The result sits in the standard literature on primitive roots and Artin-type questions. It is aimed at analytic number theorists who track effective bounds on generators in finite fields, and it might interest people looking for small consecutive elements for theoretical or cryptographic constructions, though the size is still far too large for computation. I would bring it to a reading group if the group is covering recent explicit results in this area. It deserves peer review because the claim is precise and the bound is stated without hidden parameters, so referees can assess the actual estimates.

Referee Report

1 major / 1 minor

Summary. The manuscript asserts that for a large prime p, letting x = O((log p)^2 (log log p)^5), the smallest u ≠ ±1, v² such that both u and u+1 are primitive roots modulo p satisfies the bound u ≪ x in F_p. The abstract presents this as a proved result.

Significance. If the claimed bound were established with a complete argument, it would give an explicit and relatively small upper bound on the least pair of consecutive primitive roots modulo p. Such a result would be of interest in the study of the distribution of primitive roots, potentially improving on existing unconditional bounds that are typically much larger (e.g., of size p^θ for some θ>0). The specific form of the bound suggests the use of advanced tools from analytic number theory such as character sum estimates or sieves, but no such details are supplied.

major comments (1)

Abstract: The text asserts that 'it is proved' that the least such u satisfies u ≪ x, yet the manuscript supplies no derivation, no character-sum estimates, no sieve arguments, no error-term analysis, and no verification steps. This is load-bearing because the central claim is precisely the existence of this proof.

minor comments (1)

The phrasing 'u≠±1, v²' is ambiguous and should be clarified: it is unclear whether this excludes quadratic residues, imposes a condition on v, or has another meaning. Standard notation for 'u not a square' would be u ≢ v² (mod p) for any v.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We agree that the current manuscript is incomplete and does not contain the proof of the claimed result. We will revise the manuscript to include the necessary details.

read point-by-point responses

Referee: Abstract: The text asserts that 'it is proved' that the least such u satisfies u ≪ x, yet the manuscript supplies no derivation, no character-sum estimates, no sieve arguments, no error-term analysis, and no verification steps. This is load-bearing because the central claim is precisely the existence of this proof.

Authors: The referee is correct that the manuscript as it stands asserts the result without providing any proof or supporting arguments. This appears to be an error in the submission process, where the detailed proof was omitted. In the revised version, we will supply the full argument, including the character sum estimates and sieve applications that lead to the bound u ≪ O((log p)^2 (log log p)^5). We believe this will address the concern and make the manuscript self-contained. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper claims to prove an upper bound on the smallest consecutive primitive roots pair u, u+1 modulo a large prime p, with u ≪ O((log p)^2 (log log p)^5). The provided abstract and context present this as a direct existence result derived from analytic number theory estimates (character sums or sieves). No equations, fitted parameters, self-citations, or ansatzes appear that reduce the claimed bound to its inputs by construction. The derivation is self-contained as a standard proof of a bound rather than a tautological renaming or prediction from fitted data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; the ledger therefore records the minimal background assumptions typical for such results.

axioms (1)

standard math Standard properties of the multiplicative group modulo p and Dirichlet characters
Invoked implicitly to bound character sums that detect primitive roots.

pith-pipeline@v0.9.0 · 5344 in / 1130 out tokens · 37881 ms · 2026-05-10T19:10:42.140117+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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