On duplicate representations as $2^x + 3^y$ for nonnegative integers $x$ and $y$

Douglas Edward Iannucci

arxiv: 1907.03347 · v2 · pith:VDUP3KYInew · submitted 2019-07-07 · 🧮 math.NT

On duplicate representations as 2^x + 3^y for nonnegative integers x and y

Douglas Edward Iannucci This is my paper

Pith reviewed 2026-05-25 01:08 UTC · model grok-4.3

classification 🧮 math.NT

keywords duplicate representations2^x + 3^yOEIS conjectureBennett's theoremexponential Diophantine equationsnonnegative exponentssums of powers

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The pith

There are exactly five positive integers that can be written in more than one way as 2^x + 3^y for nonnegative integers x and y.

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

The paper proves a conjecture from the Online Encyclopedia of Integer Sequences by showing that exactly five positive integers admit multiple representations as the sum of a power of 2 and a power of 3. It divides the problem into cases where both exponents are positive, which reduces directly to an existing theorem, and cases that include a zero exponent, which are settled by exhaustive elementary enumeration. A reader would care because the result gives a complete classification of collisions in this simple two-base additive form, closing an open question about the uniqueness of such sums.

Core claim

The paper proves that there are exactly five positive integers that can be written in more than one way as the sum of a nonnegative power of 2 and a nonnegative power of 3. The case in which both exponents are positive is settled by an existing theorem of Bennett. The remaining cases, in which at least one exponent is zero, are resolved by direct and exhaustive checking of the resulting equations.

What carries the argument

The case split between representations that include a zero exponent (handled by elementary exhaustion) and those with both exponents positive (reduced to Bennett's theorem).

If this is right

The sums 2^x + 3^y produce a sequence whose members are unique except for the five listed exceptions.
No additional collisions exist when zero exponents are permitted.
The classification is exhaustive once Bennett's theorem is granted.
All representations with a zero exponent can be checked directly without further theoretical machinery.

Where Pith is reading between the lines

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

The same case-split technique could be tested on sums involving other fixed bases such as 2 and 5.
The result supplies an explicit finite list that can be used to bound the number of representations in related additive problems.
It suggests that similar elementary arguments may suffice for other small-exponent Diophantine equations once a strong theorem is available for the positive-exponent subcase.

Load-bearing premise

Bennett's theorem correctly identifies all solutions in which both exponents are at least one.

What would settle it

The discovery of any sixth positive integer equal to 2^a + 3^b and also to 2^c + 3^d for two distinct pairs of nonnegative integers (a,b) and (c,d).

read the original abstract

We prove a conjecture posted in the Online Encyclopedia of Integer Sequences, namely that there are exactly five positive integers that can be written in more than one way as the sum of a nonnegative power of 2 and a nonnegative power of 3. The case for both powers being positive follows from a theorem of Bennett. We use elementary methods to prove the case where zero exponents are allowed.

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

This short note finishes the OEIS conjecture by handling the zero-exponent cases with elementary enumeration, confirming exactly five duplicates once Bennett's theorem covers the positive-exponent regime.

read the letter

The main takeaway is that this paper settles the conjecture with a clean split: Bennett handles all cases where both exponents are positive, and the author adds the zero-exponent cases via direct checks. That produces the exact count of five positive integers with multiple representations as 2^x + 3^y. The new material is the elementary part; the rest is citation. The paper does well by staying elementary and not forcing a uniform method across regimes where one is unnecessary. The zero cases reduce to checking small values like 2^x + 1 and 1 + 3^y, and the argument appears to bound the search so that larger duplicates are ruled out without gaps. No circularity or fitted parameters show up. The soft spots are modest and expected for this kind of note. The result stays narrow, with no mention of generalizations or links to other bases or equations. A referee might ask for a bit more explicit listing of the five numbers and the precise bounds used in the zero cases to make verification immediate, but the structure itself looks sound. This is for number theorists who follow specific Diophantine representation problems or keep an eye on OEIS entries about power sums. A reader who wants the full picture on this exact question will get what they need. It deserves a serious referee because the claim is precise, the methods are appropriate and verifiable, and the central count now rests on cited theorem plus exhaustive small-case work rather than an open gap.

Referee Report

0 major / 1 minor

Summary. The manuscript proves that there are exactly five positive integers that admit more than one representation as 2^x + 3^y with x, y nonnegative integers. The case x, y ≥ 1 is dispatched by direct appeal to a theorem of Bennett; the cases with at least one zero exponent are treated by exhaustive elementary enumeration together with explicit bounding arguments.

Significance. The result resolves an OEIS conjecture. The combination of an external theorem for the interior regime with complete, self-contained elementary verification for the boundary cases supplies a clean classification. The manuscript supplies the explicit lists and bounds for the zero-exponent cases, which is a strength.

minor comments (1)

The abstract states that Bennett's theorem is applied when both powers are positive; a brief parenthetical reminder of the precise statement of Bennett's result (or its reference) would help readers confirm the hypotheses match the regime x, y ≥ 1.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The report contains no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation splits the problem into (i) x,y ≥ 1, which is dispatched by direct citation of Bennett's external theorem (no author overlap), and (ii) cases with at least one zero exponent, which are settled by explicit elementary enumeration and bounding arguments supplied in the manuscript. No equations, fitted parameters, or self-citations reduce the central claim to its own inputs by construction; the argument is self-contained once the external theorem and the listed case checks are accepted.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on Bennett's theorem (domain assumption) for the positive-exponent case and on the completeness of an elementary enumeration for zero exponents; no free parameters or new entities are introduced.

axioms (1)

domain assumption Bennett's theorem on sums of positive powers of 2 and 3
Invoked to handle the case where both exponents are at least 1.

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