Prime numbers with symmetric decimal digits

Infinitely many decimal palindromic primes

Collaboration beta

Can a decimal digit string read the same in both directions and be prime at arbitrarily large lengths? The packet gives exact prime detectors and a character-sum reduction, but the lower bound that would produce infinitely many such primes remains unproved.

# {p : p is prime and p = rev (p)} = \infty

Known results and sources

Illustration: Mirrored digits describe a palindrome; symmetry alone does not make it prime.

Research problem

Exact mathematical statement

A decimal palindrome is a positive integer whose ordinary base-10 representation has no leading zeros and reads identically in both directions. The target is

\#\{p:p\text{ is an ordinary prime and }p=\operatorname{rev}(p)\}=\infty.

It is enough to obtain a prime at infinitely many digit lengths; positive density is not required. Even-length palindromes are divisible by 11, leaving only the exceptional prime 11 at even length. Apart from finitely many one-digit cases, the relevant lengths are $2m+1$ , $m\ge1$ . The source does not establish this infinitude.

Problem infographic

Problem at a glance

Illustration: Decimal mirror symmetry is separated from primality using 10301, the composite 12321, and the even-length divisibility obstruction. — Illustration: Odd-length prime examples and the even-length obstruction explain the decimal infinitude question without settling it.

Current mathematical picture

Where work on Infinitely many decimal palindromic primes stands

Open problem

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureTurning an exact detector or positive short-divisor term into a lower bound

DPP-A23 identifies that cancellation exactly and leaves the unknown prime sum in the rough remainder. DPP-A18 is a target-equivalent finite-parameter certificate, not a factorization-free algorithm or a theorem that its sum exceeds one half. A signed estimate for the actual rough remainder or appropriately restricted arithmetic coefficients could make the detector positive. Its exactness alone cannot do so.

Route status · Narrowed route

Main reductionThe remaining finite band still needs a signed bound

DPP-H11 reports $\Theta_m=10q-\Re\mathcal Z_m^{\rm deep}+O_\epsilon(q/L^4)$ with the exact stated depth, height and real-part cutoffs and primitive-inducer multiplicities. The finishing inequality $\Re\mathcal Z_m^{\rm deep}\le9q$ infinitely often is sufficient and unproved.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeProve $\Re\mathcal Z_m^{\rm deep}\le9q$ , or another adequate one-sided saving from the $10q$ main term, for infinitely many $m$ .Task status · Ready to work on

Work mapped so far

Infinitely many decimal palindromic primes in numbers

1.3kretained lines of mathematical investigation1,255 in the current working snapshot

Argument development: 972 · 77%
Explored or eliminated routes: 46 · 4%
Computational analysis: 63 · 5%
Open obligations: 84 · 7%
Definitions and setup: 90 · 7%

9selected mapped statements2routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Status checkedSep 7, 2026

Source recordedSep 7, 2026

Recommended next task

Prove $\Re\mathcal Z_m^{\rm deep}\le9q$ , or another adequate one-sided saving from the $10q$ main term, for infinitely many $m$ .

Suggested move: State the exact length weights, character depths, zero heights, real-part ranges, and multiplicities before estimating. If averaging across lengths, preserve the fixed primitive characters and their zeros and show the average forces infinitely many prime-bearing lengths.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

14 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

2 recorded

Narrowed routeTurning an exact detector or positive short-divisor term into a lower bound

Route status · Narrowed route

Narrowed routeForcing the deep sign from magnitude and zero-count upper bounds alone

DPP-O10 places synthetic, conjugate-paired points in the allowed region that align positively with the kernels and produce a contribution much larger than q. They are not zeros of constructed Dirichlet L-functions and do not supply Euler products or cross-length consistency. An argument using genuine arithmetic phase relations remains possible. The synthetic obstruction does not rule out all analytic methods or prove that the target is false.

Route status · Narrowed route

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Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Prove $\Re\mathcal Z_m^{\rm deep}\le9q$ , or another adequate one-sided saving from the $10q$ main term, for infinitely many $m$ .Suggested move: State the exact length weights, character depths, zero heights, real-part ranges, and multiplicities before estimating. If averaging across lengths, preserve the fixed primitive characters and their zeros and show the average forces infinitely many prime-bearing lengths.

Ready to work on

Find genuine arithmetic phase information beyond the retained kernel-magnitude and zero-count upper bounds that can control the signed deep aggregate.Suggested move: Formulate a testable coupling from Euler-product coefficients or proved cross-character or cross-length constraints. Insert it at Q comparable to q and height comparable to Q, and show why the synthetic alignment model no longer satisfies the hypotheses.

Ready to work on

Prove positivity of a source-defined exact prime detector on infinitely many genuine palindrome lengths, rather than only improving its coefficient-extraction accuracy.Suggested move: Use A18 with its even-order factorial condition, identify the rough signed remainder exposed by A23, and state an estimate for the actual arithmetic coefficients. Retain carries and the palindrome constraint; a positive short-divisor main term alone is canceled on certified composites.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Sep 7, 2026

Current statusOpen problem

Infinitely many prime palindromes in base ten remains an open special case of the fixed-base prime-palindrome conjecture. An almost-prime infinitude theorem does not settle it.

[1]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2024-08-08
Peer reviewedTuxanidy and Panario prove infinitely many palindromes with at most six prime factors, counted with multiplicity, in every fixed integer base at least two.[1]
2004-05-04
PreprintBanks, Hart and Sakata prove that the prime fraction among palindromes tends to zero in each fixed base; this is compatible with infinitely many prime palindromes.[2]

Mathematical neighborhood

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions2 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction3 of 93
lemma2 of 92
equivalence1 of 91
negative result2 of 92

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.25 displayed rows · 2 routes included

retained route statementDo palindromic primes occur at arbitrarily large decimal lengths?
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementInfinitely many genuine odd lengths sufficeintermediate
retained route statementIndividual cancellation requires exact decimal depthintermediate
retained route statementThe remaining finite band still needs a signed boundintermediate
retained route statementAn exact-count detector is not a lower boundintermediate
retained route statementLeading moments survive prime deletion in a bounded growing rangeintermediate
retained route statementSynthetic alignment isolates missing arithmetic informationintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureTurning an exact detector or positive short-divisor term into a lower boundreported failure
Useful failureForcing the deep sign from magnitude and zero-count upper bounds alonereported failure
Research targetProve $\Re\mathcal Z_m^{\rm deep}\le9q$ , or another adequate one-sided saving from the $10q$ main term, for infinitely many $m$ .open
Research targetFind genuine arithmetic phase information beyond the retained kernel-magnitude and zero-count upper bounds that can control the signed deep aggregate.open
Research targetProve positivity of a source-defined exact prime detector on infinitely many genuine palindrome lengths, rather than only improving its coefficient-extraction accuracy.open
ComputationThe source says its inherited 269,351-row dataset and earlier interval certificates were authenticated rather than re-enumerated in the v4 pass. New exact checks cover strip margins, sampling multipliers, a sieve capacity margin, and 48 rational-dilation cases with 26,029 changed matches.These are source-reported finite checks. The new suite computes no genuine Dirichlet zeros; NumPy results remain non-interval diagnostics. No finite count or certificate establishes the missing infinite-subsequence sign bound. · reported unreproduced
Narrowed routeTurning an exact detector or positive short-divisor term into a lower boundDPP-A23 identifies that cancellation exactly and leaves the unknown prime sum in the rough remainder. DPP-A18 is a target-equivalent finite-parameter certificate, not a factorization-free algorithm or a theorem that its sum exceeds one half. A signed estimate for the actual rough remainder or appropriately restricted arithmetic coefficients could make the detector positive. Its exactness alone cannot do so.
Narrowed routeForcing the deep sign from magnitude and zero-count upper bounds aloneDPP-O10 places synthetic, conjugate-paired points in the allowed region that align positively with the kernels and produce a contribution much larger than q. They are not zeros of constructed Dirichlet L-functions and do not supply Euler products or cross-length consistency. An argument using genuine arithmetic phase relations remains possible. The synthetic obstruction does not rule out all analytic methods or prove that the target is false.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeProve $\Re\mathcal Z_m^{\rm deep}\le9q$ , or another adequate one-sided saving from the $10q$ main term, for infinitely many $m$ .

2 approaches have already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

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Prepared starting pointProve $\Re\mathcal Z_m^{\rm deep}\le9q$ , or another adequate one-sided saving from the $10q$ main term, for infinitely many $m$ .

Infinitely many decimal palindromic primes · ready to start

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Research contextPrepared context for any AI agent

Can a decimal digit string read the same in both directions and be prime at arbitrarily large lengths? The current work gives exact prime detectors and a character-sum reduction, but the lower bound that would produce infinitely many such primes remains unproved.

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Sources and references3 cited works · next context review by Dec 7, 2026

The mathematical context was checked on Sep 7, 2026. Status can be refreshed sooner after a material result or claim.

1
Infinitude of Palindromic Almost-Prime Numberspeer reviewed result · Aleksandr Tuxanidy, Daniel Panario · International Mathematics Research Notices · 2024-08-08 · DOI 10.1093/imrn/rnae174 · accessed Sep 7, 2026
2
Almost all palindromes are compositepreprint · William D. Banks, Derrick N. Hart, Mayumi Sakata · 2004-05-04 · ARXIV math/0405056 · accessed Sep 7, 2026
3
A002385 — Decimal palindromic primessoftware or dataset · The OEIS Foundation Inc. · OEIS A002385 · accessed Sep 7, 2026

Important qualifications

An original proposer and proposal year were not established.
The 2004 primary manuscript is classified as a preprint source here; its journal publication is not separately bound in this draft.
No linked dataset, computation, or formal proof was run. Empty formalization fields do not establish that no formalization exists.
This private draft grants no mathematical acceptance, rights, authorship, contributor credit, or publication authority.

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Infinitely many decimal palindromic primes

Exact mathematical statement

Problem at a glance

Where work on Infinitely many decimal palindromic primes stands

Infinitely many decimal palindromic primes in numbers

Prove $\Re\mathcal Z_m^{\rm deep}\le9q$ , or another adequate one-sided saving from the $10q$ main term, for infinitely many $m$ .

How the current approaches connect

Research route map

Other routes

Open questions

The known mathematical landscape

What the literature has established

Related results and reusable starting points

Formal and computational footholds

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Contribute

Important qualifications

Compare another research frontier

Exact mathematical statement

Problem at a glance

Where work on Infinitely many decimal palindromic primes stands

Infinitely many decimal palindromic primes in numbers

Prove ℜZmdeep≤9q\Re\mathcal Z_m^{\rm deep}\le9q, or another adequate one-sided saving from the 10q10q main term, for infinitely many mm.

How the current approaches connect

Research route map

Other routes

Open questions

Claims, milestones, and routes in the current map

Conditions that would advance the current route

Important qualifications

Compare another research frontier

Prove $\Re\mathcal Z_m^{\rm deep}\le9q$ , or another adequate one-sided saving from the $10q$ main term, for infinitely many $m$ .