On permutations derived from integer powers x^n
Pith reviewed 2026-05-25 16:12 UTC · model grok-4.3
The pith
Integer powers x^n induce bijective mappings on each residue class mod p via an explicit coding shift α.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The theorem shows that the integer part of x^p * p^{-α} provides a bijective mapping on the set of x in each residue class mod p with x < p^{l+1}, where α is the coding shift with an explicit formula given depending on n and p. For n coprime to p, α equals 1 from group theory, but the formula extends it generally, including to the zero class, demonstrating that such bijective encodings can be found for any finite l and all x < p^{l+1}.
What carries the argument
The coding shift α, a positive integer depending on n and p with an explicit formula, that makes floor(x^p * p^{-α}) a bijection on the residue classes.
Load-bearing premise
An explicit formula for the coding shift α exists and produces bijections for arbitrary n and p, including non-coprime cases and the zero residue class.
What would settle it
Pick small values such as p=2, n=4, l=2, compute α from the formula, apply floor(x^4 * 2^{-α}) to each x in a residue class with x<2^3, and check whether the results are all distinct and stay inside the same class.
read the original abstract
We present a general theorem characterizing the relationship between the prime base $p$ representations of non-negative integers $x$ and their positive integer powers, $x^n$. For any positive integer $l$, the theorem establishes the existence of bijective mappings (permutations) between all $p^l$ members $x$ of each non-zero residue class mod $p$ satisfying $x < p^{l+1}$. These mappings are obtained as the integer part of ${x^p}{p^{-\alpha}}$ for a particular positive integer $\alpha$, depending on $n$ and $p$, called the "coding shift", for which an explicit formula is given. For relatively prime $n$ and $p$, $\alpha = 1$ and the result follows directly from properties of the multiplicative group of invertible elements modulo $p^{l+1}$. We extend our result for general $n$ also to identify the coding shift required to obtain such bijective mappings for members of the zero residue class mod $p$, demonstrating that such bijective mappings (or encodings) can be found for any finite $l$ and for all positive integers $x < p^{l+1}$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript claims a general theorem establishing, for any positive integer l, the existence of bijective mappings on the p^l elements of each residue class modulo p (with 0 ≤ x < p^{l+1}) realized by the map x ↦ ⌊x^n ⋅ p^{-α}⌋, where α is an explicit positive integer 'coding shift' depending only on n and p. When gcd(n,p)=1 the result reduces to the standard fact that exponentiation by n is bijective on (ℤ/p^{l+1}ℤ)^* with α=1; the text asserts an extension of the same construction to the case gcd(n,p)>1 together with a separate explicit construction for the zero residue class.
Significance. If the explicit formula for α is correctly derived and shown to produce the stated uniform-in-l bijections for arbitrary n (including when p divides n) and for the zero class, the result would supply a concrete, parameter-free encoding of residue classes via power maps. The coprime case is routine from group theory; a verified general formula would be the novel contribution and could be of interest for constructive questions in p-adic number theory or combinatorial number theory.
major comments (2)
- [Abstract and main theorem statement] Abstract / main theorem: the assertion that an explicit formula for the coding shift α produces bijections when gcd(n,p)>1 is load-bearing for the general claim, yet the provided text supplies no derivation showing why this α compensates for the failure of bijectivity of the power map on the units or why the same α works uniformly for every l. The coprime case is justified by standard properties of (ℤ/p^{l+1}ℤ)^*, but the extension step is only asserted.
- [Extension to zero residue class] Extension to zero residue class: the abstract states that a coding shift is identified for the zero class mod p, but no explicit formula, construction, or argument establishing that ⌊x^n ⋅ p^{-α}⌋ permutes the p^l elements inside [0,p^{l+1}) for arbitrary l is visible. This is required for the claim that such encodings exist for all residue classes.
minor comments (1)
- [Abstract] Abstract contains the expression 'x^p' while the title and surrounding text refer to x^n; this appears to be a typographical inconsistency that should be corrected for clarity.
Simulated Author's Rebuttal
We thank the referee for the careful reading and for identifying points where the derivations need to be made more explicit. We address each major comment below and will revise the manuscript accordingly to strengthen the presentation of the general case and the zero-class construction.
read point-by-point responses
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Referee: [Abstract and main theorem statement] Abstract / main theorem: the assertion that an explicit formula for the coding shift α produces bijections when gcd(n,p)>1 is load-bearing for the general claim, yet the provided text supplies no derivation showing why this α compensates for the failure of bijectivity of the power map on the units or why the same α works uniformly for every l. The coprime case is justified by standard properties of (ℤ/p^{l+1}ℤ)^*, but the extension step is only asserted.
Authors: We agree that the manuscript asserts the existence of the explicit formula for α and states that it produces the claimed uniform-in-l bijections, but does not supply a self-contained derivation of why this particular α compensates for the loss of bijectivity when p divides n. The coprime case is handled via the standard group-theoretic argument, while the general case relies on the formula for α together with an inductive argument on the p-adic valuation that is only sketched. We will revise the main theorem section to include a complete, step-by-step derivation showing both the compensation mechanism and the uniformity in l. revision: yes
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Referee: [Extension to zero residue class] Extension to zero residue class: the abstract states that a coding shift is identified for the zero class mod p, but no explicit formula, construction, or argument establishing that ⌊x^n ⋅ p^{-α}⌋ permutes the p^l elements inside [0,p^{l+1}) for arbitrary l is visible. This is required for the claim that such encodings exist for all residue classes.
Authors: The manuscript does supply an explicit formula for the coding shift in the zero class (distinct from the non-zero case) and asserts that the resulting map is bijective for each fixed l. However, the referee is correct that a detailed construction and proof of the permutation property for arbitrary l is not fully expanded. We will add a dedicated subsection that gives the explicit α for the zero class together with the inductive argument establishing the bijection on the p^l elements of [0, p^{l+1}). revision: yes
Circularity Check
No circularity: explicit formula and group-theoretic base case are independent of the claimed bijections
full rationale
The paper states an explicit formula for the coding shift α and derives the coprime case directly from the known bijectivity of the power map on the multiplicative group of units modulo p^{l+1}. No step reduces a prediction to a fitted parameter, renames a known result, or relies on a self-citation chain whose content is unverified outside the paper. The construction for the general-n and zero-residue cases is presented as a direct (if non-obvious) extension whose validity rests on the supplied formula rather than on redefinition or tautology.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Properties of the multiplicative group of invertible elements modulo p^{l+1}
invented entities (1)
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coding shift α
no independent evidence
discussion (0)
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