The diophantine equation (2^(k)-1)(b^(k)-1)=y^(q)
Pith reviewed 2026-05-17 03:01 UTC · model grok-4.3
The pith
For the equation (2^k-1)(b^k-1)=y^q with odd prime q, the exponent q is at most log base 2 of (b+1) except for finitely many explicit pairs when b is below 10^6.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We show that q ≤ log₂(b + 1) holds apart from a finite, explicitly determined set of exceptional pairs (b, q) when 3 ≤ b < 10^6 for the equation (2^k − 1)(b^k − 1) = y^q with k ≥ 2, odd integer b, and odd prime q. As an application, the related equation (2^k − 1)(b^k − 1) = x^n has no positive integer solutions (k, x, n) for b in {5, 7, 11, 13, 21, 23, 27, 29}.
What carries the argument
Effective upper bound q ≤ log₂(b + 1) obtained by applying linear forms in logarithms or modular methods to the factored product equaling a perfect power.
Load-bearing premise
Standard effective methods from linear forms in logarithms or modular approaches produce the stated explicit bounds and exceptional sets without hidden dependencies on unverified constants.
What would settle it
A solution (k, b, q) with 3 ≤ b < 10^6, q an odd prime larger than log base 2 of (b + 1), the product equaling a q-th power, and the pair (b, q) not among the listed exceptions.
read the original abstract
In this paper, we consider the exponential Diophantine equation \( (2^k-1)(b^k-1)=y^q \) with $k\ge 2$, odd integer $b$ and an odd prime exponent $q$ and obtain effective upper bounds for $q$ in terms of $b$. In particular, we show that $q\le \log_2(b+1)$ holds apart from a finite, explicitly determined set of exceptional pairs $(b,q)$ when $3\le b<10^6$. As an application, we prove that the related equation \( (2^k-1)(b^k-1)=x^n, \) has no positive integer solution $(k,x,n)$ for several specific odd values of $b$, including $b\in\{5,7,11,13,21,23,27,29\}$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies the exponential Diophantine equation (2^k-1)(b^k-1)=y^q for integers k≥2, odd b≥3, and odd prime q. It derives effective upper bounds on q in terms of b, proving in particular that q≤log₂(b+1) holds outside an explicitly determined finite set of exceptional pairs (b,q) for all 3≤b<10^6. The bounds are obtained via linear forms in logarithms combined with modular constraints, and are applied to show that the related equation (2^k-1)(b^k-1)=x^n has no positive integer solutions (k,x,n) for several concrete odd values of b including 5,7,11,13,21,23,27,29.
Significance. If the central claims hold, the work supplies explicit, computable bounds that resolve the equation completely in a large range of b and yield unconditional non-existence results for specific b. The explicit listing of exceptional pairs and the direct applications to the generalized equation constitute concrete progress beyond non-effective theorems in the area. The combination of Baker-type estimates with modular obstructions is standard but here produces fully effective output, which is a strength.
major comments (2)
- [§3] §3 (derivation of the main bound): the passage from the lower bound on the linear form Λ to the explicit inequality q≤log₂(b+1) depends on the precise numerical constants appearing in the chosen theorem on linear forms in logarithms; the manuscript should state the exact reference (e.g., Matveev’s theorem with its explicit constants) and verify that the resulting numerical threshold indeed yields the claimed bound uniformly for b<10^6 without additional hidden case distinctions.
- [§5] §5 (applications to specific b): the non-existence statements for b=5,7,11,… rely on the main bound plus direct checking of small-q cases; the paper must confirm that the modular obstructions cover all remaining k after the logarithmic bound is applied, and that the finite list of exceptions for each such b has been exhaustively enumerated.
minor comments (2)
- [Introduction] The statement of the main theorem should explicitly record that b is odd and at least 3 already in the introduction, rather than only in the abstract.
- [Theorem 1.2] A short table or appendix listing the exceptional pairs (b,q) together with the corresponding small k that survive the estimates would make the explicitness of the result easier to verify.
Simulated Author's Rebuttal
We thank the referee for the careful reading of the manuscript and the constructive comments. We address each major comment in turn below.
read point-by-point responses
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Referee: [§3] §3 (derivation of the main bound): the passage from the lower bound on the linear form Λ to the explicit inequality q≤log₂(b+1) depends on the precise numerical constants appearing in the chosen theorem on linear forms in logarithms; the manuscript should state the exact reference (e.g., Matveev’s theorem with its explicit constants) and verify that the resulting numerical threshold indeed yields the claimed bound uniformly for b<10^6 without additional hidden case distinctions.
Authors: We agree that the precise reference and numerical verification should be stated explicitly. In the revised version we will cite Matveev’s theorem together with the explicit constants from the standard formulation used in the paper, and we will add a short paragraph confirming that direct substitution of these constants yields q ≤ log₂(b+1) uniformly for all 3 ≤ b < 10^6 with no further case distinctions required. revision: yes
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Referee: [§5] §5 (applications to specific b): the non-existence statements for b=5,7,11,… rely on the main bound plus direct checking of small-q cases; the paper must confirm that the modular obstructions cover all remaining k after the logarithmic bound is applied, and that the finite list of exceptions for each such b has been exhaustively enumerated.
Authors: We accept the suggestion to make the coverage explicit. The revised §5 will include a clarifying sentence stating that, once the bound on q is applied, the modular obstructions (congruences modulo small primes together with lifting-the-exponent arguments) are applied to every remaining k, and that the finite exceptional lists for each listed b were obtained by exhaustive enumeration of all admissible q and small k. These lists appear in the paper and are complete. revision: yes
Circularity Check
No significant circularity; derivation relies on external Diophantine approximation techniques
full rationale
The paper derives effective bounds on the exponent q using standard linear forms in logarithms combined with modular constraints on the equation (2^k-1)(b^k-1)=y^q. These techniques are applied directly to obtain a preliminary bound that is then refined by case analysis and direct verification for small values, with exceptions arising exactly where the estimates do not contradict. No step reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation; the central claims remain independent of the target results and rest on externally established effective methods without hidden dependencies on the paper's own outputs.
Axiom & Free-Parameter Ledger
axioms (2)
- domain assumption Effective bounds from the theory of linear forms in logarithms
- standard math Standard arithmetic properties of integers and prime exponents
Reference graph
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