arxiv: 2605.13666 · v1 · submitted 2026-05-13 · 🧮 math.PR · math.CO· math.NT

Recognition: 2 theorem links

· Lean Theorem

When Does the Dice Sum Become Prime?

Christoph Koutschan , Tipaluck Krityakierne , Thotsaporn Aek Thanatipanonda

Authors on Pith no claims yet

Pith reviewed 2026-05-14 17:56 UTC · model grok-4.3

classification 🧮 math.PR math.COmath.NT

keywords expected waiting timedice sumsprime numbersdynamic programmingtruncation estimatessurvival probabilityhigher momentsprobability on naturals

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The pith

Dynamic programming computes the expected number of die rolls until the sum is prime to more than 1000 decimal places.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops an efficient way to find the expected number of rolls of a fair six-sided die until the running sum equals a prime. It sets up a one-dimensional dynamic programming recursion that tracks the probability the sum has not yet hit any prime, then applies truncation with explicit error bounds. The irregular locations of primes are handled by noting that their positive density forces the survival probability to decay exponentially, which keeps the truncation error tiny even for moderate cutoffs. This produces much sharper numerical values for the expectation and its higher moments than earlier calculations. A reader cares because the method turns an apparently hard probabilistic question involving an irregular set into a high-precision computable quantity.

Core claim

Using a one-dimensional dynamic programming recursion together with truncation and rigorous error bounds, the expected number of rolls until the sum is prime is computed to more than 1000 decimal places, along with higher moments, improving substantially on the results of Conroy, Alon, and Malinovsky.

What carries the argument

One-dimensional dynamic programming recursion that follows the distribution of the current sum conditional on not having reached a prime yet, closed by truncation whose error is controlled by the exponential decay of the survival probability.

Load-bearing premise

The probability that the running sum has avoided all primes up to a large cutoff decays exponentially fast because primes have positive density.

What would settle it

Running a direct Monte Carlo simulation of millions of independent die sequences and obtaining an empirical mean lying outside the rigorous truncation interval claimed by the dynamic programming computation would show the error bounds are incorrect.

Figures

Figures reproduced from arXiv: 2605.13666 by Christoph Koutschan, Thotsaporn Aek Thanatipanonda, Tipaluck Krityakierne.

**Figure 1.** Figure 1: For N = 4 210 and U = 20, the functions EN,U [τs] (blue) and EN,0[τs] (orange) are depicted in the range 4 090 ≤ s ≤ N + 6. One can clearly observe the exponential rate at which the two functions approach each other as s moves away from N. The horizontal segment at the right end corresponds to the imposed boundary conditions for N + 1 ≤ s ≤ N + 6. 3 Rigorous Upper Bound U for Primes Recall from Corollary 4… view at source ↗

**Figure 2.** Figure 2: Plot of the first 40 values of the landing probability [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: The values of E[τs] for 71 990 ≤ s ≤ 72 050. Note that these values, like E[τ ], cannot be computed exactly, but they can be approximated very accurately by EN,0[τs]. For this plot, we used N = 100 000. 4 Higher Moments with High Accuracy In this section, we demonstrate that our algorithm can be generalized to compute higher moments. Again taking A to be the set of primes, we show some high-precision appro… view at source ↗

read the original abstract

Given a (possibly infinite) subset $A$ of the natural numbers, we ask how many times a fair six-sided die must be rolled until the rolled numbers add up to an element of $A$. Using a one-dimensional dynamic programming recursion together with truncation and rigorous error bounds, we compute the expected number of rolls efficiently and with very high accuracy. When $A$ is the set of prime numbers, the irregular distribution of primes makes it difficult to obtain explicit error estimates. Nevertheless, the density of primes implies that the associated survival probability decays exponentially fast, which enables highly accurate truncation estimates. As a result, our calculations yield significantly sharper estimates for this expectation and its higher moments than the original results of Conroy, Alon, and Malinovsky. In particular, we determine the expectation to more than $1000$ decimal places.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper gives a high-precision numerical value for the expected rolls until a die sum hits a prime, using standard DP plus truncation to reach 1000+ decimals with error bounds.

read the letter

The core result is a new computation of the waiting time expectation for the sum to land on a prime, carried to more than 1000 decimal places along with tighter higher-moment estimates. The method is a one-dimensional DP recursion that truncates the state space once the survival probability drops low enough, using the prime density to justify exponential decay and produce explicit error bounds. This is sharper than the earlier Conroy-Alon-Malinovsky numbers and the truncation step is handled cleanly enough that the output should be reproducible from the recursion and the bound formulas. The paper does not claim any new closed form or asymptotic theory; it is a careful numerical exercise that exploits the known density of primes to make truncation practical. The main limitation is scope: the technique works because primes have positive density and the survival tail decays fast, but the same approach would need different justification for thinner sets. No load-bearing circularity appears in the setup, and the recursion starts from base cases without fitting to the target quantity. For a reader working on exact or high-precision waiting-time problems in additive combinatorics or computational probability, the numbers and bounds are worth having as a reference point. I would send it to peer review in a venue that handles rigorous numerical work, provided the full manuscript includes the explicit error formulas and at least a spot-check of the leading digits against a lower-precision run.

Referee Report

2 major / 2 minor

Summary. The paper develops a one-dimensional dynamic programming recursion to compute the expected number of rolls of a fair six-sided die until the partial sum belongs to a given subset A of the naturals. For A equal to the primes, the authors combine the recursion with truncation at a large N and rigorous error bounds derived from the exponential decay of the probability that the sum has not yet hit a prime (using prime density). They report the expectation to more than 1000 decimal places together with higher moments, claiming these are substantially sharper than the earlier results of Conroy, Alon, and Malinovsky.

Significance. If the truncation error bounds are correctly derived and implemented, the work supplies the highest-precision numerical values currently available for this stopping-time expectation and its moments. The method is deterministic, forward-recursive, and free of fitted parameters, which makes the claimed 1000-decimal accuracy a concrete, falsifiable output that can be checked by independent recomputation at higher truncation thresholds.

major comments (2)

[§4] §4 (truncation and error analysis): the manuscript asserts that prime density implies exponential decay of the survival probability, enabling truncation with rigorous bounds, yet no explicit formula is given for the constant C and rate r in the bound P(survival after k steps) ≤ C r^k, nor is the numerical value of r derived from a concrete prime-number-theorem estimate. Without this, the claimed 1000-decimal accuracy cannot be independently verified from the text alone.
[§5] §5 (numerical results): the paper states that the expectation is obtained to more than 1000 decimal places, but supplies neither the explicit truncation threshold N used nor a table of successive approximations that would demonstrate convergence to the reported digits. A single high-precision output without accompanying convergence diagnostics leaves the error-bound claim untested within the manuscript.

minor comments (2)

[§3] Notation for the state variable in the DP recursion should be introduced once and used consistently; the current text alternates between S_n and the partial-sum symbol without a single defining sentence.
[Introduction] The comparison with Conroy–Alon–Malinovsky would be strengthened by quoting their reported precision (number of digits) alongside the new 1000-digit claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and positive recommendation. The comments on the truncation analysis and numerical presentation are well taken, and we will strengthen the manuscript accordingly to improve verifiability.

read point-by-point responses

Referee: §4 (truncation and error analysis): the manuscript asserts that prime density implies exponential decay of the survival probability, enabling truncation with rigorous bounds, yet no explicit formula is given for the constant C and rate r in the bound P(survival after k steps) ≤ C r^k, nor is the numerical value of r derived from a concrete prime-number-theorem estimate. Without this, the claimed 1000-decimal accuracy cannot be independently verified from the text alone.

Authors: We agree that an explicit formula for C and r would aid independent verification. The exponential decay follows from a lower bound on prime density via the prime number theorem: for sufficiently large x, π(x) > c x / log x with explicit c > 0 (e.g., c = 1/2 for x large enough). This yields a uniform upper bound on the probability of avoiding a prime at each step, leading to P(survival after k steps) ≤ C r^k with r = exp(−δ) for a concrete δ > 0 derived from the minimal density and C depending on the initial steps. In the revision we will insert the precise derivation, the numerical value of r employed, and the reference to the explicit PNT estimate used. revision: yes
Referee: §5 (numerical results): the paper states that the expectation is obtained to more than 1000 decimal places, but supplies neither the explicit truncation threshold N used nor a table of successive approximations that would demonstrate convergence to the reported digits. A single high-precision output without accompanying convergence diagnostics leaves the error-bound claim untested within the manuscript.

Authors: We accept that a convergence table and the explicit N would make the claimed precision more transparent. In the revised manuscript we will state the truncation threshold N used to obtain the 1000-decimal result and add a short table of successive approximations (for example at N = 10^5, 5×10^5, 10^6) showing stabilization of the leading digits to the reported precision. This will directly illustrate the practical effect of the error bounds. revision: yes

Circularity Check

0 steps flagged

No circularity: forward DP recursion with independent truncation bounds

full rationale

The derivation computes the target expectation (and moments) via a one-dimensional dynamic programming recursion initialized from base cases, combined with truncation whose error bounds are derived from the known exponential decay of survival probabilities (using prime density). This procedure is self-contained and does not fit any parameter to the target quantity, redefine the expectation in terms of itself, or rely on load-bearing self-citations whose content reduces to the present result. The improvement over Conroy et al. is achieved through higher-precision execution of the same independent recursion, not through circular reuse.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard Markov-chain recurrence for expected hitting time and on the asymptotic density of primes (from the prime-number theorem) to guarantee exponential decay of the survival probability; no free parameters are fitted to the target expectation and no new entities are introduced.

axioms (1)

domain assumption The density of primes implies that the survival probability decays exponentially fast
Invoked to justify that truncation at a finite sum produces controllable error for the prime target set.

pith-pipeline@v0.9.0 · 5450 in / 1252 out tokens · 66213 ms · 2026-05-14T17:56:00.589891+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Using a one-dimensional dynamic programming recursion together with truncation and rigorous error bounds... E[τ] = E_N,0[τ] + B·P_N(0)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the density of primes implies that the associated survival probability decays exponentially fast

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

9 extracted references · 1 canonical work pages · 1 internal anchor

[1]

Noga Alon and Yaakov Malinovsky,Hitting a prime in 2.43 dice rolls (on average), The American Statistician77(3), 301–303, 2023

2023
[2]

Noga Alon, Yaakov Malinovsky, Lucy Martinez, and Doron Zeilberger,Hitting k primes by dice rolls, Electronic Journal of Combinatorics32(4), 4.16, 2025

2025
[3]

A more recent version (2026) is available athttps://www.madandmoonly.com/ doctormatt/mathematics/dice1.pdf

Matthew Conroy,A Collection of Dice Problems, online note, 2018 version cited. A more recent version (2026) is available athttps://www.madandmoonly.com/ doctormatt/mathematics/dice1.pdf

2018
[4]

Lucy Martinez and Doron Zeilberger,How many Dice Rolls Would It Take to Reach Your Favorite Kind of Number?, Maple Transactions3(3), Autumn 2023

2023
[5]

Barkley Rosser and Lowell Schoenfeld,Approximate Formulas for some Functions of Prime Numbers, Illinois Journal of Mathematics6(1), 64–94, March 1962

1962
[6]

Anirban DasGupta,Student Puzzle Corner 17.IMS Bulletin46(2), 7, 2017

2017
[7]

Anirban DasGupta,Solution to Puzzle 17.IMS Bulletin46(5), 9, 2017

2017
[8]

Shane Chern,Hitting a prime by rolling a die with infinitely many faces, The Amer- ican Statistician78(3), 297–303, 2024

2024
[9]

Tipaluck Krityakierne and Thotsaporn Thanatipanonda,High-Precision Framework for Expected Hitting Times Analysis in the Dice-Sum Process, arXiv:2604.23133, 2026. 14

work page internal anchor Pith review Pith/arXiv arXiv 2026