Powers of 3 with few nonzero bits and a conjecture of Erdős

Abstract: Using completely elementary methods, we find all powers of 3 that can be written as the sum of at most twenty-two distinct powers of 2, as well as all powers of 2 that can be written as the sum of at most twenty-five distinct powers of 3. The latter result is connected to a conjecture of Erd\H{o}s, namely, that 1, 4, and 256 are the only powers of 2 that can be written as a sum of distinct powers of 3. We present this work partly as a reminder that for certain exponential Diophantine equations, elementary techniques based on congruences can yield results that would be difficult or impossible to obtain with more advanced techniques involving, for example, linear forms in logarithms.

• The paper addresses Erdős's conjecture on expressing powers of 2 as sums of distinct powers of 3 by investigating exponential Diophantine equations using elementary congruence methods.
• Key results include identifying all powers of 3 with at most 22 nonzero binary bits (exponents x  ≤ 25) and confirming that 1, 4, and 256 are the only powers of 2 expressible as sums of at most 25 distinct powers of 3.
• The authors' methodology relies on elementary number-theoretic tools like modular arithmetic, tail-and-loop structures, and an iterative lifting process to efficiently solve the equations without advanced techniques.

An Overview of "Powers of 3 with Few Nonzero Bits and a Conjecture of Erdős"

This paper by Dimitrov and Howe investigates the problem of expressing powers of 3 with a limited number of nonzero binary bits, and conversely, expressing powers of 2 as sums of powers of 3. The primary focus is on examining these problems through the lens of exponential Diophantine equations using elementary congruence methods. The authors successfully tackle Erdős's conjecture regarding powers of 2 that can be expressed as sums of distinct powers of 3. By employing elementary congruence arguments, they bypass the need for more advanced techniques often requiring linear forms in logarithms.

Key Results

The paper articulates two primary theorems:

1. Theorem on Powers of 3: The authors determine all powers of 3 whose binary representation contains at most 22 '1' bits. They identify these powers to be precisely those for exponents $x \leq 25$.
2. Theorem on Powers of 2: They also ascertain the powers of 2 that can be expressed as the sum of at most 25 distinct powers of 3. This result supports Erdős's conjecture, confirming the integers $1$, $4$, and $256$ as the only such powers of 2.

The findings effectively address the conjecture by Erdős, positing that only finite configurations of powers of 3 can express certain powers of 2 under specified constraints.

Methodological Insights

The authors' approach is particularly notable for its reliance on elementary number-theoretic tools. By manipulating equations involving powers of numbers mod different integers, they systematically reduce the potential candidate solutions. Specifically, their method involves:

• Modular Arithmetic: By expressing powers modulo a sequence of variously chosen integers, they refine the possible solutions until uniqueness is achieved or an upper bound is clearly demonstrated.
• Tail-and-Loop Structures: The use of tail-and-loop diagrams efficiently tracks powers of numbers modulo designated integers. This technique aids in elucidating the multiplicative order relationships that dictate solution viability.
• Iterative Lifting Process: Solutions in smaller moduli are "lifted" to larger ones, a methodical technique that ensures a manageable computation while incrementally refining accuracy.

These methods robustly sidestep the more computationally intensive methods of the linear forms in logarithms, favoring streamlined algebraic manipulation and reasoning.

Implications and Future Directions

The implications of this work are multifaceted:

• Theoretical Value: Demonstrating the sufficiency of elementary techniques in solving complex Diophantine equations emphasizes the potential for such methods in broader number-theoretic applications.
• Practical Applications: Beyond theoretical interest, the ability to characterize numbers with respect to powers and sums informs cryptography and numerical representation systems where such properties are paramount.
• Computational Efficiency: By detailing a process efficient enough to be run on standard computational hardware, the paper underscores opportunities for optimizing exhaustive searches in future studies of similar equations.

The conclusions drawn here may pave the way for exploring other conjectures or problems where exponential Diophantine equations arise. Future studies might explore higher bases or varied constraints, exploiting this method as a fundament for further breakthroughs in the domain.

In summary, Dimitrov and Howe provide a methodologically elegant solution to a classical problem in number theory, utilizing foundations in congruence to both affirm existing conjectures and establish a template for addressing future mathematical challenges. The combination of theoretical grounding and practical calculation deftly advances understanding within a traditionally challenging field.