MathDB

Erdős Problem #315 — Extremal Reciprocal Egyptian Fraction Sequences

About 46 years old · traced to

Define u0=1u_0=1 and ui+1=ui(ui+1)u_{i+1}=u_i(u_i+1), and define the Vardi constant

c0=lim⁡i→∞ui(1/2)i+1=1.264085⋯ .c_0=\lim_{i\to\infty}u_i^{(1/2)^{i+1}}=1.264085\cdots.

For every sequence a:N→Na:\mathbb N\to\mathbb N such that ai>0a_i>0 for all ii, aa is strictly increasing, ai≠ui+1a_i\ne u_i+1 for at least one ii, and

∑i=0∞1ai=1,\sum_{i=0}^{\infty}\frac1{a_i}=1,

must one have

lim inf⁡i→∞ai(1/2)i+1<c0?\liminf_{i\to\infty}a_i^{(1/2)^{i+1}}<c_0?
References

Primary source

The Formal Conjectures Authors, Formal Conjectures (2025); independently edited from the pinned source file.

Additional references

Pinned Formal Conjectures source, Apache-2.0.

number-theoryunit-fractions
InErdős Problems
Posted · edited

Progress summary

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The conjecture is reported proved by two independent mathematicians in 2025, but the retrieved sources do not include a verifiable proof.

The problem asks whether the Sylvester reciprocal decomposition of one uniquely has the largest doubly exponential growth constant among all increasing reciprocal decompositions of one. Erdős Problems records affirmative proofs independently by Kamio and by Li and Tang in 2025.

2025 independent proofs

Kamio, and independently Li and Tang, are reported to have proved that every non-Sylvester decomposition has a strictly smaller limiting growth constant than the Vardi constant. This would settle the problem affirmatively, but no proof artifact was retrieved for verification.

Current status (as of March 2026): The conjecture is reported proved by Kamio and independently by Li and Tang in 2025, but the proof remains unverified from the retrieved sources.

Sources

Solutions 0

No solutions have been posted yet.