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Another inequality inspired by Erdős

This paper presents an elementary arithmetical proof of a specific inequality inspired by Paul Erdős's classical proof of the Bertrand postulate, as part of a broader effort to establish an arithmetically pure proof of the postulate itself.

Original authors: Barbora Batíková, Tomáš J. Kepka, Petr C. Němec

Published 2026-07-21

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CC0 1.0

Original authors: Barbora Batíková, Tomáš J. Kepka, Petr C. Němec

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective working in the vast, quiet library of Number Theory, a branch of mathematics dedicated to the hidden secrets of whole numbers. In this world, numbers aren't just tools for counting apples; they are characters with their own personalities, relationships, and strict rules. One of the most famous rules in this library is the "Bertrand Postulate," a centuries-old observation that says if you pick a number bigger than 6, there is always at least one prime number (a number divisible only by 1 and itself) hiding in a specific neighborhood just below it. For a long time, mathematicians like the legendary Paul Erdős used clever tricks involving binomial coefficients (which are like the numbers you get when you count combinations of items) to prove these rules. But sometimes, the most interesting mysteries aren't the big, loud theorems, but the quiet, strange inequalities that pop up when you try to simplify those proofs. These are the "oddities" that make mathematicians scratch their heads and wonder, "Does this always hold true, or is there a sneaky exception hiding in the numbers?"

This paper, titled "Another Inequality Inspired by Erdős," dives deep into one of those sneaky inequalities. The authors, Barora Batíková, Tomáš J. Kepka, and Petr C. Němec, are investigating a specific mathematical relationship involving positive integers. They define a few special "characters" for every number $n$ : a value $z(n)$ related to how many times 3 fits into $2n$ , a value $m(n)$ related to the square root of $2n$ , and a value $r(n)$ related to powers of 2. They then combine these into a complex expression, $y(n)$ , and ask a simple question: Is this expression always negative, always positive, or does it flip-flop? The paper proves that this expression is never exactly zero (it never lands perfectly on the line), and it maps out exactly which numbers make it negative and which make it positive. It's a bit like checking every single house on a very long street to see if the lights are on or off, finding that while most houses follow a pattern, there are a few specific addresses where the lights behave differently.

The Great Number Hunt

The story begins with a look back at the past. In 1845, a mathematician named J. Bertrand made a bold guess: for any number $n$ bigger than 6, there's always a prime number between $n-2$ and $n^2$ . Later, Paul Erdős, a genius known for his elegant and simple proofs, came up with a new way to prove a slightly different version of this idea. In doing so, he stumbled upon some unusual inequalities. One of these, involving a value called $x(n)$ , was already studied in a previous paper by these same authors. They found that $x(n)$ is usually negative, but it hits exactly zero for a tiny handful of numbers: 436, 451, 529, 545, and 546.

Now, the authors turn their attention to a "kindred" inequality, a sibling to the first one, which they call $y(n)$ . This new expression is a bit more complicated, mixing powers of 2 and powers of $n$ . The goal of this paper is to solve a puzzle: For which positive integers $n$ is $y(n)$ less than zero? And, crucially, does $y(n)$ ever equal exactly zero?

The Two-Pronged Attack

To solve this, the authors use two different strategies, like a detective using both a magnifying glass and a high-tech scanner.

Strategy 1: The Pure Arithmetic Approach
First, they use "pure arithmetic," which means they stick strictly to the rules of whole numbers without using the smooth curves of calculus. They break the infinite number line into manageable chunks. They realize that for certain ranges of numbers, the values of $m(n)$ and $r(n)$ stay the same. This allows them to group numbers into intervals.

They create a giant map of these intervals. For example, they look at numbers from 1 to 403 and check the sign of $y(n)$ in each block. They find that for small numbers (like 1, 2, 3, 4), $y(n)$ is positive. But starting at $n=5$ , it flips to negative. It stays negative for a long stretch, but then, just like a rollercoaster, it dips and rises in specific sections.

Using careful calculations and comparisons of powers (like checking if $2^{20}$ is bigger than $3^{15}$ ), they prove that $y(n)$ is negative for a specific set of ranges:

From 5 up to 335.
From 338 up to 350.
From 365 up to 368.

They also prove that $y(n)$ is never zero. It's a strict "either/or" situation; the number is either positive or negative, never sitting right on the fence. For all other numbers (1 through 4, and then 336, 337, 351 through 364, and everything from 369 upwards), the value is positive.

Strategy 2: The Calculus Approach
To double-check their work and show that these results hold up even when viewed through the lens of continuous mathematics, the authors use elementary calculus. They invent a smooth, wavy function $F(x)$ that mimics the behavior of their discrete integer problem.

They analyze the shape of this curve. They show that the curve eventually starts climbing upward forever. By finding where the curve crosses the "zero line" (the x-axis), they can predict where the integer values of $y(n)$ must be positive or negative.

They prove that if $n$ is very large (specifically $n \ge 404$ ), the value is definitely positive.
They use the curve to narrow down the search area, confirming that any "negative" behavior must happen within a specific window (roughly between 5 and 379).
By checking the specific integer points within this window, they confirm the exact boundaries found in the first strategy.

The Final Verdict

The paper concludes with a definitive map of the territory. The inequality $y(n) < 0$ (meaning the expression is negative) is true if and only if $n$ falls into one of these three groups:

$5 \le n \le 335$
$338 \le n \le 350$
$365 \le n \le 368$

Conversely, $y(n) > 0$ (meaning the expression is positive) if $n$ is between 1 and 4, or in the gaps between the negative zones, or if $n$ is 369 or larger.

Most importantly, the authors prove with absolute certainty that $y(n)$ is never equal to zero for any positive integer. There are no "magic numbers" where the expression vanishes perfectly. It's a strict binary world of positive and negative, with no neutral ground.

This work is a testament to the power of combining old-school number crunching with modern analytical tools. While the result might seem like a niche puzzle to the uninitiated, it represents the kind of meticulous, step-by-step verification that keeps the foundations of mathematics solid. It shows that even in the world of abstract numbers, every single case matters, and sometimes, the most interesting discoveries are knowing exactly where the exceptions don't exist.

Technical Summary of "Another Inequality Inspired by Erdős"

Problem Statement
The paper investigates a specific inequality involving positive integers, defined through a set of auxiliary functions derived from the properties of powers of 2 and square roots. The authors aim to determine the sign of the function $y(n)$ for all positive integers $n$ .

The function $y(n)$ is defined as:
$y(n) = 2^{2n - 2z(n) - m(n) + 2} - n^{m(n) - 1}$
where:

$z(n)$ is the largest integer such that $3z(n) < 2n$ .
$m(n)$ is the largest integer such that $m(n)^2 \le 2n$ .
$r(n)$ is the least non-negative integer such that $n \le 2^{r(n)}$ .

The core problem is to solve the inequality $y(n) < 0$ and identify all $n$ for which $y(n) = 0$ . This work follows a previous examination by the same authors of a related inequality $x(n) < 0$ , where $x(n) = z(n) - (r(n) + 1)m(n)$ . The authors frame this investigation within the context of elementary arithmetic, motivated by the desire to find proofs for number-theoretic statements (like the Bertrand Postulate) that rely on elementary methods, potentially within the fragment of "Elementary Function Arithmetic."

Methodology
The paper employs two distinct methodological approaches to solve the inequality:

Purely Arithmetical Approach (Section 2):
- Interval Partitioning: The authors define intervals where the values of $m(n)$ and $r(n)$ remain constant. They construct a sequence of intervals $[g(i), g(i+1)-1]$ where these parameters are fixed, allowing for the analysis of $x(n)$ and $y(n)$ as non-decreasing or monotonic sequences within these bounds.
- Comparative Bounds: Using lemmas, the authors establish bounds comparing powers of 2 ( $2^{c(n)-m(n)}$ ) against powers of $n$ ( $n^{m(n)-1}$ ). They utilize specific inequalities to show that if certain conditions on $c(n)$ , $m(n)$ , and $r(n)$ are met, $y(n)$ must be strictly negative or strictly positive.
- Case-by-Case Verification: For the range where general bounds are insufficient (specifically $17 \le n \le 403$ ), the authors perform a detailed, interval-by-interval analysis. This involves calculating specific values of $c(n)$ and comparing them against calculated powers of $n$ to determine the sign of $y(n)$ .
Elementary Calculus Approach (Section 3):
- Continuous Approximation: The authors define continuous functions $F(x)$ that approximate the discrete behavior of $x(n)$ and $y(n)$ . These functions involve terms like $x^{2/3}$ , $\sqrt{x}$ , and $\log_2 x$ .
- Convexity Analysis: By analyzing the second derivatives ( $F''(x)$ ), the authors prove that these functions are convex for sufficiently large $x$ . This convexity implies that the functions have at most one local minimum and cross zero at most once in specific domains.
- Bounding the Roots: The authors calculate specific values to locate the roots of these continuous functions (e.g., finding $k$ such that $f(k) \approx 0$ ). They then use the relationship between the continuous function and the discrete integer sequence to bound the range of integers where the inequality holds.

Key Contributions and Results

Complete Solution to Inequality (2): The paper provides a definitive classification of the sign of $y(n)$ for all positive integers $n$ .
- $y(n) < 0$ : This holds if and only if $5 \le n \le 335$ , $338 \le n \le 350$ , or $365 \le n \le 368$ .
- $y(n) > 0$ : This holds if and only if $1 \le n \le 4$ , $336 \le n \le 337$ , $351 \le n \le 364$ , or $n \ge 369$ .
- $y(n) = 0$ : The authors prove that $y(n)$ is never zero for any positive integer $n$ .
Refinement of Previous Work (Inequality 1): While the primary focus is on $y(n)$ , the paper also revisits the inequality $x(n) < 0$ using the calculus method. It re-confirms the result from their previous work [1] that $x(n) = 0$ only for $n \in \{436, 451, 529, 545, 546\}$ and $x(n) < 0$ for $n \le 435$ , $n=450$ , or $513 \le n \le 528$ .
Methodological Demonstration: The paper demonstrates that complex inequalities in positive integers can be tackled using "elementary calculus" (involving logarithms and roots) to narrow the search space, followed by finite arithmetic verification.

Significance and Claims
The authors position this work as a methodological and didactical exercise. They do not claim to solve a major open conjecture but rather to explore "unusual inequalities in positive integers" that arise from attempts to simplify proofs of the Bertrand Postulate.

The significance claimed is twofold:

Arithmetic Purity: The authors argue that the methods used can be "transmogriﬁed" to yield demonstrations within or close to "Elementary Function Arithmetic," a weak fragment of arithmetic. This aligns with the goal of finding proofs for number-theoretic assertions that do not rely on heavy analytic machinery (calculus), even though the paper itself utilizes calculus as a tool to guide the arithmetic proof.
Didactical Value: The inclusion of both a purely arithmetical solution and a calculus-based solution serves to illustrate different ways of approaching such problems. The authors note that while the arithmetic proof is "pure," the calculus approach offers a way to "facilitate" the calculation and restrict the range of integers requiring manual verification.

The paper concludes modestly, stating that the results provide a complete solution to the specific inequalities presented, without extrapolating to broader applications or future research directions beyond the immediate context of these specific integer inequalities.

The Great Number Hunt

The Two-Pronged Attack

The Final Verdict

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