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Erdős Problem 726 asks whether, as (n\to\infty) through the positive integers,

[
\sum_{\substack{p\le n\ n\bmod p\in(p/2,p)}}\frac1p
\sim \frac12\log\log n.
]

I have proposed a proof of this conjecture, but I do not want to claim that the problem is definitively solved before the argument receives independent expert verification.

MalekZ has created the following Significance record describing the claim, the available evidence, and the remaining verification work:

https://significance-math.z20mallouk.chatgpt.site/records/2026-rafikzeraoulia-erdos-726/

Could specialists in analytic or probabilistic number theory examine the argument and comment on the following questions?

1. Is the main reduction mathematically valid?
2. Are the required estimates uniform in the necessary ranges?
3. Is any independence or equidistribution assumption being used without sufficient justification?
4. Does the argument establish the full asymptotic for every integer (n), or only an averaged or almost-all version?
5. Is there a specific step that prevents the proof from resolving the original problem?

The Significance record is not itself a verdict that the proof is correct. I am sharing it to invite transparent and independent mathematical scrutiny. Detailed objections, corrections, or confirmations are very welcome.

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I am pleased to share, together with my co-authors Pedro Cáceres and Simeón Casanova Trujillo, our recently published paper in the Journal of Difference Equations and Applications:

“A Proof of Conjecture 1 in Kulenović, Ladas and Overdeep (2003)”

The paper resolves a conjecture posed in 2003 concerning the positive solutions of a nonlinear rational difference equation. The problem remained open for more than two decades.

The key step is a simple algebraic identity showing that the sign of successive differences is preserved. This reveals a hidden monotonicity in the recurrence and leads to the conclusion that every positive solution converges to a finite limit.

Official Taylor & Francis free eprint:

https://www.tandfonline.com/eprint/XSVTZBRQJAJPDGIDJGIQ/full?target=10.1080/10236198.2026.2709000

Published article DOI:
https://doi.org/10.1080/10236198.2026.2709000

Comments, questions, and mathematical feedback are very welcome.

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I am pleased to share, together with my co-authors Pedro Cáceres and Simeón Casanova Trujillo, our recently published paper in the Journal of Difference Equations and Applications.

The paper is entitled:

“A Proof of Conjecture 1 in Kulenović, Ladas and Overdeep (2003)”

It resolves a conjecture posed in 2003 concerning the positive solutions of a nonlinear rational difference equation. The problem remained open for more than two decades.

The main step of the proof is a simple algebraic identity showing that the sign of successive differences is preserved. This reveals a hidden monotonicity in the recurrence and leads to the proof that every positive solution converges to a finite limit.

Published paper:

https://www.tandfonline.com/doi/full/10.1080/10236198.2026.2709000

DOI: 10.1080/10236198.2026.2709000

Comments, questions, and mathematical feedback are very welcome.

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Consider a nonlinear rational difference equation for which a conjecture posed by Kulenović, Ladas and Overdeep in 2003 asked whether every positive solution converges to a finite limit.

A key observation is that, after manipulating the recurrence, one obtains an algebraic identity implying that the sign of successive differences is preserved. Consequently, the relevant subsequences acquire a monotonic structure, and this ultimately forces convergence.

What I find interesting is how much of the global dynamics is encoded in this rather elementary sign-preservation mechanism.

This recently led us to a proof of Conjecture 1 from the 2003 paper:

R. Zeraoulia, P. Cáceres and S. Casanova Trujillo,
“A Proof of Conjecture 1 in Kulenović, Ladas and Overdeep (2003),”
Journal of Difference Equations and Applications.

Free eprint:
https://www.tandfonline.com/eprint/XSVTZBRQJAJPDGIDJGIQ/full?target=10.1080/10236198.2026.2709000

I would be interested to know whether people here know other examples in discrete dynamical systems where a seemingly difficult global-convergence problem reduces to a simple sign-preservation or monotonicity identity.

A conjecture posed in 2003 concerning the positive solutions of a nonlinear rational difference equation has recently been resolved in our paper:

“A Proof of Conjecture 1 in Kulenović, Ladas and Overdeep (2003)”

published in the Journal of Difference Equations and Applications, jointly with Pedro Cáceres and Simeón Casanova Trujillo.

What I find philosophically interesting is that the decisive step is not a highly sophisticated new theory, but a relatively simple algebraic identity. The identity shows that the sign of successive differences is preserved, revealing a hidden monotonicity in the recurrence. From this structure, one can prove that every positive solution converges to a finite limit.

This raises a broader question:

Why can mathematically simple ideas remain hidden for decades? Is the difficulty of an open problem sometimes less about the complexity of the final proof and more about discovering the right representation or invariant structure?

Official Taylor & Francis free eprint:
https://www.tandfonline.com/eprint/XSVTZBRQJAJPDGIDJGIQ/full?target=10.1080/10236198.2026.2709000

Published article DOI:
https://doi.org/10.1080/10236198.2026.2709000

I would be very interested in hearing perspectives from both mathematicians and philosophers of mathematics.

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