Irrationality of rapidly converging series: a problem of Erdős and Graham

Abstract: Answering a question of Erdős and Graham, we show that the double exponential growth condition $\limsup_{n\to\infty}a_n^{1/φ^n}=\infty$ for a monotonically increasing sequence of positive integers ${a_n}{n=1}^\infty$, together with the bound $a_n a{n+1}\geq n^{1+τ}$, is sufficient for the series $\sum_{n=1}^\infty 1/(a_n a_{n+1})$ to have an irrational sum; here $φ$ denotes the golden ratio $(1+\sqrt{5})/2$ and $τ>0$. We also provide a positive generalization to $\sum_{n=1}^\infty 1/(a_n^{w_0}\cdots a_{n+d-1}^{w_{d-1}})$, and a negative result showing that some of its instances are essentially optimal. The original problem was autonomously solved by the AI agent \emph{Aletheia}, powered by Gemini Deep Think, while the remaining material is largely a product of human-AI interactions.

• The paper establishes a sharp growth criterion proving the irrationality of series based on double exponential growth conditions in the denominators.
• It extends prior results by analyzing weighted series and providing both positive and negative results that delineate precise thresholds for irrationality.
• The study highlights an innovative human-AI collaboration that adapts classical analytical techniques to resolve a longstanding open problem in number theory.

Irrationality Criteria for Rapidly Converging Series: Resolution of an Erdős-Graham Problem

Problem Background and Prior Work

The paper "Irrationality of rapidly converging series: a problem of Erdős and Graham" (2601.21442) addresses a long-standing question originally posed by Erdős and Graham regarding the irrationality of a class of series with rapidly growing denominators. Specifically, for a strictly increasing sequence of positive integers $\{a_n\}_{n=1}^\infty$ satisfying a double exponential growth condition,

$\liminf_{n \to \infty} a_n^{1/2^n} > 1,$

is the sum

$\sum_{n=1}^\infty \frac{1}{a_n a_{n+1}}$

necessarily irrational?

Erdős and Graham conjectured that the growth hypothesis should suffice to guarantee irrationality, though explicit criteria and sharp thresholds for the irrationality of such series remained unresolved. Prior results including those on Ahmes and Cantor series, and notably variants due to Hančl and Tijdeman, provided only partial answers often requiring supplementary divisibility or relative growth constraints.

Main Theorems and Generalizations

The paper establishes definitive conditions for irrationality of series of the form

$$\sum_{n=1}^\infty \frac{1}{a_n a_{n+1} \cdots a_{n+d-1}}$$

for arbitrary $d \geq 1$, and more general weighted products in the denominators. The central result is as follows:

Theorem (Sharp Growth Criterion).

Let $d \in \mathbb{N}$ and let $\psi>1$ be the unique positive solution to $\psi^d = \psi^{d-1} + 1$. If $\{a_n\}$ is monotonic and satisfies

$\lim_{n \to \infty} a_n^{1/\psi^n} = \infty,$

then the series above is irrational. Conversely, for every $C>1$, there exists a strictly increasing sequence $\{a_n\}$ with $\lim_{n \to \infty} a_n^{1/\psi^n} = C$ such that the series is rational.

For $d=2$, the critical exponent $\psi$ is the golden ratio $\phi = (1+\sqrt{5})/2$, thus the growth condition connects directly with the original Erdős-Graham problem.

The authors further generalize to sums indexed by tuples $\mathbf{w} = (w_0, w_1, ..., w_{d-1})$ with weights in the denominators and numerators, showing:

• A positive result: Under assumptions that $$a_n^{w_0} \cdots a_{n+d-1}^{w_{d-1}} \geq n^{1+\tau}$$ for some $\tau>0$, and a limsup-type double exponential growth in $a_n$, the corresponding series is irrational.
• A negative result: If the growth parameter is below the derived threshold (largest root of an associated polynomial), sequences can be constructed such that the sum is rational.

These results are shown to be sharp.

Proof Techniques

The proof synthesizes classical irrationality criteria, notably Mahler’s approach, adapted for these rapidly convergent series. The method involves the construction of integer denominators $D_N$ capturing partial sums, application of tail bounds, and a detailed analysis of local maxima in the growth sequence using an adaptation of Borel’s "local peak" method.

A key analytic device is the comparison between the growth rate of $a_n$ and the critical threshold arising from the characteristic polynomial associated to the weights in the denominator. The limiting behavior is leveraged to demonstrate irrationality (or construct rational examples) depending on precise asymptotics.

Nontrivial examples and counterexamples are furnished via careful construction of sequences exploiting the sharpness of the threshold.

Collaborative Methodology and AI Involvement

Of particular note is the methodology: the problem's solution originated from an autonomous AI agent ("Aletheia," built using Gemini Deep Think), which identified and articulated plausible generalizations and proofs. The human authors extended and verified these results, establishing both the mathematical foundation and the limits of prior techniques. This mixed-initiative research contributed substantial advances on an open problem and is among the first documented cases of an AI system autonomously solving a significant problem in transcendental number theory.

Implications

Theoretical Implications:

The paper provides a complete solution and precise boundaries for the irrationality of a broad class of rapidly-converging series. The results extend the taxonomy of series for which irrationality can be inferred purely from denominator growth, notably subsuming prior results for Ahmes and Cantor series as special cases.

Practical Implications:

The characterization of rational/irrational infinite sums with weighted rapidly growing denominators may impact computations and theoretical analyses in analytic number theory, ergodic theory, and related fields.

Implications for AI in Mathematics:

Demonstrating rigorous AI-led discovery augurs a new paradigm wherein autonomous agents contribute nontrivial advances to mathematical research. The human-AI collaboration in this work exemplifies the potential for AI, especially in the exploration of open problems, generalization of classical methods, and hypothesis generation. This could presage new workflows and division of labor in mathematical research, particularly for combinatorial and number-theoretic problems.

Future Directions

Further developments may investigate:

• The extension of these criteria to broader families of series with less restrictive growth conditions;
• Automatic theorem proving and refinement of irrationality criteria by AI agents in even more advanced settings, such as transcendence;
• Systematic exploration of Cantor and Ahmes series via AI-driven classification and construction of sequences with prescribed analytic and arithmetic properties.

The interplay between combinatorial number theory and AI problem-solving is likely to intensify, with AI systems expanding the landscape of solvable mathematical problems and facilitating deeper understanding of foundational questions.

Conclusion

This paper delivers a definitive resolution to the Erdős-Graham problem on the irrationality of series with double exponential growth in denominators, establishing sharp criteria and constructing rational counterexamples at the boundaries. Its technical contributions clarify longstanding open questions in number theory, while the methodology underscores the growing capabilities of autonomous AI systems in mathematical research. The results and method together suggest meaningful advances in both analytic number theory and the future of AI-integrated mathematical discovery.