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On the Digits of Partition Functions

This paper establishes significantly improved upper bounds for the smallest integer $n$ such that the partition function $p(n)$ begins with a specified string of digits in base $b$ , resolving a problem posed by Douglass and Ono through an elementary discrepancy framework.

Original authors: Siddharth Iyer

Published 2026-03-27

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Siddharth Iyer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ 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 have a giant, magical vending machine. Every time you put in a number $n$ , it doesn't just give you a snack; it spits out a massive number called a partition.

In math, a "partition" is just a way of breaking a number down into smaller pieces. For example, if you put in the number 4, the machine might tell you there are 5 ways to break it down:

4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1

The number of ways to do this is called $p(n)$ . As $n$ gets bigger, $p(n)$ explodes in size. The number for $n=100$ is already huge, and for $n=1,000$ , it's a number with so many digits that it would take a library to write them all down.

The Big Question: "Can I find a specific number?"

The paper tackles a fun puzzle proposed by mathematicians Douglass and Ono.

Imagine you are looking for a specific sequence of digits, like "739", hiding at the very beginning of one of these massive partition numbers.

Does there exist a number $n$ such that $p(n)$ starts with "739..."?
If yes, what is the smallest $n$ you have to feed into the machine to get that result?

This is the core problem: How far do we have to search before we find a partition number that starts with our chosen digits?

The Old Way vs. The New Way

The Old Way (The "Sledgehammer"):
Previous mathematicians (like Luca) tried to solve this using very heavy, complex tools from "transcendence theory" (a branch of math dealing with numbers that can't be roots of simple equations). Their answer was technically correct, but the number they gave for the search limit was astronomically huge.

Analogy: It's like trying to find a specific grain of sand on a beach by checking every single grain on every beach on Earth, even though you know the sand is right there in the bucket next to you. Their estimate was so big it was practically useless for real-world calculation.

The New Way (The "Smart Compass"):
The author, Siddharth Iyer, uses a much simpler, more "everyday" approach. He treats the problem like a game of hunting for a specific spot on a clock.

The Clock Analogy:
Imagine the leading digits of a number are determined by where a hand points on a clock. If the hand points between 7:00 and 7:39, the number starts with "739...".
The author realizes that as you feed larger numbers ( $n$ ) into the partition machine, the "hand" spins around the clock in a very predictable, smooth pattern. It doesn't jump randomly; it glides.
The "Gap" Strategy:
The author uses a simple math trick (the Mean Value Theorem) to prove that the hand must sweep over every single spot on the clock face if you spin it far enough. He calculates exactly how far you have to spin it to guarantee you hit your target "739" zone.
The Result:
Because his method is so much more direct, his answer is drastically smaller.
- Analogy: Instead of checking every grain of sand on Earth, he figured out that if you just check the first 100 grains in the bucket, you are guaranteed to find your target.

What Did He Actually Prove?

The paper gives two main results, one for standard partitions ( $p(n)$ ) and one for "plane partitions" (which are like 3D stacks of blocks, a slightly more complex version of the game).

For Standard Partitions: He proved that if you are looking for a string of $t$ digits (like "739" where $t=3$ ), you will definitely find it within a number roughly proportional to $b^{2t}$ (where $b$ is your base, usually 10).
- In plain English: If you want a 3-digit code, you won't have to search past a number with about 6 digits. If you want a 10-digit code, you won't have to search past a number with about 20 digits. It's a very manageable search.
For Plane Partitions: The math is slightly different (the numbers grow even faster), but the logic holds. He found a new, much tighter limit for these 3D stacks as well.

Why Does This Matter?

It's a "Benford's Law" Mystery: The paper mentions that these numbers follow "Benford's Law," which says that in many real-world datasets, the digit "1" appears as the first digit about 30% of the time, while "9" appears less often. This paper proves that partition numbers play by these rules and, more importantly, tells us exactly how fast they cycle through all possible starting digits.
Simplicity Wins: The biggest takeaway is that you don't always need the most complex, "heavy-duty" math to solve a problem. Sometimes, looking at the problem with a fresh, elementary perspective (like looking at fractional parts of logs) yields results that are not only correct but millions of times better than the old methods.

Summary

Siddharth Iyer took a question about finding specific numbers in a giant, chaotic list of mathematical partitions. Instead of using a sledgehammer to smash through the problem, he used a compass. He showed that the list of partition numbers is actually very orderly, and you can find any starting digit sequence you want much, much sooner than anyone previously thought possible.

1. Problem Statement

The paper addresses a specific number-theoretic problem concerning the leading digits of partition functions. Specifically, it seeks to find the smallest integer $n$ such that the partition function $p(n)$ (or the plane partition function $PL(n)$) begins with a specified string of digits $f$ in a given base $b$ .

Context: It is known that the leading digits of $p(n)$ and $PL(n)$ follow Benford's Law.
The Question: For a digit string $f$ of length $t$ in base $b$ (where $f$ does not start with 0), let $N_p(f, b)$ be the smallest positive integer such that $p(N_p(f, b))$ starts with $f$ . The goal is to establish non-trivial upper bounds for $N_p(f, b)$ .
Previous State of the Art:
- For $p(n)$ , Luca [5] used exponential sums and transcendence theory to prove $N_p(f, b) \leq \exp(2 \cdot 10^{25}(t+12)(\ln b)^2)$ .
- For $PL(n)$, Luca [4] used properties of prime numbers to prove $N_{PL}(f, b) \leq b^{51t+688}$ .
- These bounds are extremely large (exponential or super-exponential in $t$ ).

2. Methodology

The author employs an elementary discrepancy framework that avoids the heavy machinery of transcendence theory or deep analytic number theory used in previous works. The methodology relies on three main pillars:

A. Logarithmic Formulation

The problem of finding the leading digits is transformed into a problem about the fractional parts of logarithms.

Let $\{x\}$ denote the fractional part of $x$ .
A number $n$ starts with digits $f$ in base $b$ if and only if:
$\{ \log_b n \} \in [\log_b f - t + 1, \log_b(f+1) - t + 1)$
where $t$ is the length of the string $f$ .

B. Asymptotic Approximations

The paper utilizes precise asymptotic expansions for the logarithms of the partition functions to approximate $\log_b p(n)$ and $\log_b PL(n)$ :

For $p(n)$ : Using the Hardy-Ramanujan asymptotic formula, the author derives an approximation of the form:
$\log_b p(n) \approx \frac{\pi\sqrt{24}}{6\ln b}\sqrt{n} - \frac{\ln n}{\ln b} + C$
with a rigorous error bound of $O(n^{-1/2})$ .
For $PL(n)$: Using the asymptotic formula for plane partitions, the approximation is:
$\log_b PL(n) \approx \frac{3(A/4)^{1/3}}{\ln b} n^{2/3} - \frac{25}{36 \ln b} \ln n + C'$
with a rigorous error bound of $O(n^{-2/3})$ .

C. General Discrepancy Bound (Proposition 3.1)

The core technical contribution is a general proposition that guarantees the existence of an integer $m$ within a specific range such that the fractional part of a function $g(m)$ falls into a target interval $[a, a+\delta)$ .

The function $g(x)$ is modeled as $c_1 x^\theta + c_2 \ln x + c_3 + E(x)$ .
The proof utilizes the Mean Value Theorem to analyze the derivative of the main term.
It establishes that if the interval length is sufficient relative to the derivative's growth, the function's values will "sweep" across the target fractional interval, ensuring an integer solution exists.
The bound for the smallest such integer $m$ is derived explicitly in terms of the constants $c_1, c_2, c_3, \theta$ and the error term bound.

3. Key Contributions and Results

The paper derives significantly sharper upper bounds for $N_p(f, b)$ and $N_{PL}(f, b)$ compared to previous results.

Main Theorem (Theorem 1.2)

Let $b \geq 2$ be the base and $t$ be the length of the digit string $f$ .

For the Partition Function $p(n)$ :
$N_p(f, b) \leq 288 \cdot b^{2t} + 2$
- Improvement: This is a polynomial bound in $b^t$ , replacing the previous double-exponential bound.
For the Plane Partition Function $PL(n)$:
$N_{PL}(f, b) \leq 130 \cdot b^{3t} + 29400 \cdot b^{3t/2}$
- Improvement: This is a polynomial bound (dominated by $b^{3t}$ ), replacing the previous bound of $b^{51t+688}$ .

Technical Derivation Details

For $p(n)$ : The dominant term in the discrepancy bound comes from $L_2$ , which scales as $(1/\delta)^{1/\theta}$ . Since $\theta = 1/2$ and $\delta \approx b^{-t}$ , the bound scales as $(b^t)^2 = b^{2t}$ .
For $PL(n)$: Here $\theta = 2/3$ . The bound involves two competing terms: one scaling as $(b^t)^{3/2}$ and another as $(b^t)^3$ . The final bound reflects the sum of these dominant terms.

4. Significance

Elementary Approach: The paper demonstrates that deep results in partition theory (specifically regarding leading digits) can be obtained using elementary tools (Mean Value Theorem, fractional parts, and asymptotic estimates) rather than complex transcendence theory.
Drastic Improvement: The reduction from bounds involving $10^{25}$ or exponents like $51t$ to simple polynomial coefficients ($288$ and $130$) represents a massive improvement in the quantitative understanding of these sequences.
Generalizability: The framework established in Proposition 3.1 is general and can likely be applied to other sequences where the logarithm has a similar asymptotic expansion involving a power term and a logarithmic correction.
Practicality: The new bounds suggest that the "first occurrence" of any digit string in partition functions happens much earlier than previously thought, making computational verification and further statistical analysis more feasible.

In summary, Iyer's work provides a rigorous, elementary, and highly efficient solution to the problem of finding the first occurrence of specific digit patterns in partition functions, fundamentally improving the known upper bounds by several orders of magnitude.