Number theory and arithmetic congruences

Agoh–Giuga conjecture

Collaboration beta

An integer greater than one should satisfy the stated power-sum congruence exactly when it is prime.

\sum_{a = 1}^{n - 1} a^{n - 1} \equiv - 1 (\mod n) \Leftrightarrow n is prime

Known results and sources

An arithmetic cover for the open Agoh–Giuga research thread; the congruence motif is explanatory and has no proof-status effect.

Research problem

Exact mathematical statement

For every integer $n>1$ , the conjecture asserts

\sum_{a=1}^{n-1} a^{n-1} \equiv -1 \pmod n \quad\Longleftrightarrow\quad n \text{ is prime}.

Problem infographic

Problem at a glance

A forest-green number-theory plate centers the Agoh–Giuga power-sum congruence, with exact satellites for the prime example n equals 5 and composite nonexample n equals 4, plus an open constellation of distinct prime factors showing the odd-squarefree local conditions required of any hypothetical composite counterexample. — The Agoh–Giuga conjecture says the power sum through n − 1 is congruent to −1 modulo n exactly for primes. The cases n = 5 and n = 4 illustrate agreement and mismatch, while any composite counterexample would have to be odd, squarefree, and satisfy a stringent condition at every prime factor; the general composite case remains open.

Current mathematical picture

Where work on Agoh–Giuga conjecture stands

Open conjecture

Version 9 preserves the earlier Agoh–Giuga record while linking every predecessor coefficient to constant translates of one cubic, supplying fixed-block conductor bounds, and reporting a conditional degree-2048 predecessor-field lock on the deep A=71 branch. The conjecture remains open.

Strongest supported footholdTerminal rank geometry identified

The pair-deletion table is expressed by a rank-three kernel with exact determinant formulas, reframing rank data as structure rather than contradiction.

Evidence posture · Reported result

Leading routeGlobal double-completion route

Use the exact derivative/ratio scalar equation together with the denominator-overlap condition across all rational branches and Giuga indices.

Route status · Active route

Useful failureFixed finite local accumulation

Adding finitely many terminal faces, 2-adic digits, F/W checks, or terminal-overlap identities cannot close A=71.

Route status · Eliminated route

Main reductionA = 71 becomes a one-prime target

The deepest branch is compressed to the actual factorization and arithmetic-function values of one quadratic cofactor N_P.

Evidence posture · Reported reduction

Evidence footholdOne-prime A = 71 polynomial-value target

The 25/9, A=71 branch is equivalent to a prime P for which N_P=71P²−71P+1 is squarefree, P-smooth, divisible by 1001, and has the current work's exact φ and arithmetic-derivative values.

Evidence posture · Manually checked · provisional

Priority open bridgeProve a branch-wide double-completion exclusion

Use the scalar matching equation and overlap condition uniformly in (m,c,ρ), rather than sampling finitely many terminal coefficients.

Task status · Ready to work on

Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Agoh–Giuga conjecture in numbers

6kretained lines of mathematical investigation5,996 in the current working snapshot

Argument development: 5,141 · 86%
Explored or eliminated routes: 200 · 3%
Computational analysis: 214 · 4%
Open obligations: 264 · 4%
Definitions and setup: 179 · 3%

18selected mapped statements11routes investigated10reported milestones9open questions8contribution-ready tasks

Evidence attached to the current work5 manually checked claims

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.

Proposed byGiuseppe Giuga, Takashi AgohProposed1950Status checkedAug 2, 2026

Recommended next task

Prove a branch-wide double-completion exclusion

Use the scalar matching equation and overlap condition uniformly in (m,c,ρ), rather than sampling finitely many terminal coefficients.

Suggested move: Treat the active saturation quadratics as norm or resultant objects and seek a theorem uniform across coefficients and rational branches.

What would count as progress

Exclude every active rational branch and every admissible terminal coefficient at the chosen Giuga index.
Retain the overlap condition wherever it is not proved automatic.

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

28 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.

Active routeGlobal double-completion route

Use the exact derivative/ratio scalar equation together with the denominator-overlap condition across all rational branches and Giuga indices.

Route status · Active route

Active routeUniform fixed-A critical-cofactor route

For each fixed A in the 25/9 branch, solve the two critical equations, prime-largest completion, and one exact overlap condition without rebuilding all terminal faces.

Route status · Active route

Active routeEffective half-dimensional sieve

Replay the finite cutoff and make the asymptotic half-dimensional sieve explicit enough to prove the 0.7869 candidate-count bound from 4×10⁹.

Route status · Active route

Active routeLinked-square cofactor route

Attack the exact A=71 square, totient, derivative, and prime-largest equations through local factorization, quadratic-field, norm, or resultant structure.

Route status · Active route

Active routeQuadratic-value factorization route

Prove that 71P²−71P+1 always has an inadmissible prime factor for sufficiently large admissible prime P.

Route status · Active route

Active routeMoving-subset and global determinant route

Let the deletion subset grow with P or aggregate over unbounded support so rank and reciprocal constraints can escape finite-depth extendibility.

Route status · Active route

Active routeOther branches and higher indices

Apply the global criterion to the 11/4 and denominator-11 branches, then extend the mechanism to higher denominators and every Giuga index.

Route status · Active route

Active routeLarge-enumeration replay

Reproduce the reported candidate counts and directed reciprocal checksums before using their conditional consequences as publication-grade premises.

Route status · Active route

Explored alternatives

Other routes

3 recorded

Eliminated routeFixed finite local accumulation

Adding finitely many terminal faces, 2-adic digits, F/W checks, or terminal-overlap identities cannot close A=71.

Route status · Eliminated route

Useful but insufficientMagnitude-only exclusion

Treating enormous support, divisibility, or largest-prime lower bounds as a contradiction is too weak in an unbounded system.

Route status · Useful but insufficient

Useful but insufficientStandalone rank-minor contradiction

Vanishing 4×4 minors and the rank-three kernel describe genuine terminal structure; without global aggregation they do not contradict it.

Route status · Useful but insufficient

Route statements and reductions

Statements the next route can inspect and build on

Route statementGlobal double completion

For an omitted prime P and squarefree cofactor N, the derivative equation and a fixed reduced totient ratio independently force P; their simultaneous validity is the scalar equation ΔX=(N−1)G.

Manually checked · provisional

Route statementDenominator-overlap equivalence

When (n−1)/φ(n)=c/ρ is reduced, Carmichael divisibility λ(n) | n−1 is equivalent to ρ | φ(n)/λ(n).

Manually checked · provisional

Route statementUniform fixed-A completion in the 25/9 branch

For every fixed terminal coefficient A in the 25/9 branch, the exact target is F(U)=A, W(U)=−A, divisibility A | G(U), prime-largest completion P=G(U)/A, and the fixed-block overlap condition 3 | (P−1)(A+1).

Manually checked · provisional

Route statementOne-prime A = 71 polynomial-value target

The 25/9, A=71 branch is equivalent to a prime P for which N_P=71P²−71P+1 is squarefree, P-smooth, divisible by 1001, and has the current work's exact φ and arithmetic-derivative values.

Manually checked · provisional

Route statementFinite-depth CRT extendibility barrier

No fixed-depth attack using only finitely many trace, predecessor, terminal-square, 2-adic, F-table, or proper W-sign conditions can eliminate an admissible A=71 prefix; the missing obstruction must use actual factorization or unbounded scope.

Source-reported route statement · dependencies incomplete

Route statementRank-three pair-deletion kernel

The pair-deletion kernel factors through three rational coordinate functions, so every cross-matrix of pair-deletion values has rank at most three and every 4×4 cross-minor vanishes.

Source-reported route statement · dependencies incomplete

Route statementReported large A = 71 candidate enumeration

The current work reports 16,453,222 common-predicate candidates through 4×10⁹ and an upward-rounded scaled reciprocal sum of 222,409,215,790,099,078, but the large run was not replayed during packet assembly or intake.

Source-reported route statement · dependencies incomplete

Route statementHalf-dimensional sieve frontier

An explicit bound Q(x)≤0.7869x/(log x)^(3/2) for x≥4×10⁹ would eliminate the A=71 branch, while the current work's asymptotic leading constant below 0.61276 is not yet effective at that cutoff.

Source-reported route statement · dependencies incomplete

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

9 featured tasks

Prove a branch-wide double-completion exclusion

Use the scalar matching equation and overlap condition uniformly in (m,c,ρ), rather than sampling finitely many terminal coefficients.

Suggested move: Treat the active saturation quadratics as norm or resultant objects and seek a theorem uniform across coefficients and rational branches.

Ready to work on

Extend closure to every index and denominator

After any branch exclusion, cover all terminal coefficients, rational branches, higher reduced denominators, and Giuga indices before claiming the full conjecture.

Suggested move: Use the global double-completion theorem from the outset and rederive index-dependent constraints rather than transplanting index-one formulas.

Ready to work on

Replay the large A = 71 enumerations

Reproduce the 10⁹, 2×10⁹, and 4×10⁹ candidate counts and upward-rounded reciprocal checksums from the exact common predicate.

Suggested move: Run the retained segmented implementation with the stated exact factorization range and directed rounding, then independently check all three rows.

Ready to work on

Build a moving-subset or global determinant argument

Use a subset Q(P) growing with P, or aggregate over an unbounded support portion, so terminal rank data interacts with reciprocal mass and actual factorization.

Suggested move: Couple the recovered cofactor invariant V²φ(V) to reciprocal mass across an unbounded family of pair-deletion matrices.

Ready to work on

Process the other index-one branches

Apply double completion to the simplified 11/4 branch and to the 30/11 and 31/11 fronts with their exact overlap requirements.

Suggested move: Reuse the finite fronts in Appendix Z but replace terminal-coefficient-only reasoning with the exact branch matching equations.

Ready to work on

Control the quadratic cofactor factorization

Show that every sufficiently large admissible prime P makes 71P²−71P+1 acquire a factor violating one of the exact A=71 support conditions.

Suggested move: Study actual prime divisors of 71P²−71P+1 rather than fixed CRT-compatible local shadows.

Ready to work on

Exclude the linked-square cofactor system

Derive a no-solution theorem coupling the exact N, φ(N), D(N), and linked-square identities for A=71.

Suggested move: Localize prime divisors of the quadratic value and seek a norm or resultant obstruction coupling N, φ(N), and D(N).

Ready to work on

Make the degree-2048 lock effective

Derive a finite-cutoff reciprocal-density estimate strong enough to close the conditional deep A=71 branch, with conductor dependence controlled at the actual support scale.

Suggested move: Develop a quadratic large sieve or equivalent dyadic reciprocal estimate for the constant-translate family rather than treating the fields as unrelated.

Ready to work on

Make the half-dimensional sieve effective

Prove an explicit candidate-count or dyadic reciprocal bound strong enough from 4×10⁹, with constant below 0.7869.

Suggested move: Specify the fixed progression modulus, local densities, sieve level, distribution remainder, exceptional-modulus treatment, and directed partial summation from the replayed cutoff.

Prerequisites still open

Sourced mathematical context

The known mathematical landscape

Context collected Aug 2, 2026

Current statusOpen conjecture

It remains unknown whether a composite integer can satisfy the Giuga-Agoh primality congruence. A counterexample would have to be both a Carmichael number and a Giuga number; the strongest peer-reviewed computational bound located here requires at least 19,908 decimal digits and at least 4,771 distinct prime factors.

[2][6]

External progress

What the literature has established

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

2013
Peer reviewedBorwein, Maitland, and Skerritt proved that any counterexample has at least 19,908 decimal digits and at least 4,771 distinct prime factors.[2]
2004
PreprintKellner gave a direct proof of equivalence between Giuga's and Agoh's conjectures and a combined fraction-sum formulation.[1]
1996
Computational resultBorwein, Borwein, Borwein, and Girgensohn developed the structural and computational study of the primality conjecture.[7]
1995
Peer reviewedAgoh published the Bernoulli-number formulation and its connection to Giuga's conjecture.[4]

Mathematical neighborhood

Related results and reusable starting points

Current focusAgoh-Giuga conjecture

Equivalent formulationGiuga's power-sum conjecture

The Bernoulli-number Agoh formulation and Giuga's power-sum primality criterion are equivalent.

[1]

Related problemCarmichael numbers intersect Giuga numbers

A composite counterexample exists exactly when a number is simultaneously a Carmichael number and a Giuga number.

[2]

Related problemOEIS A007850, Giuga numbers

OEIS A007850 catalogs known Giuga numbers and references the relevant literature.

[5]

Later mathematical changes

What changed after the initial research map

Later recorded revisions that changed the mathematics, without inventing a date or an AI attribution.

Constant-translate fields sharpen the deep branchThe successor organizes the predecessor coefficients into one cubic translate family and records a conditional rank-eleven field lock, while leaving the effective sieve and global branch closure open.

Changed the research frontierLater mathematical revision

Research stage 13

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

How the route was assembled

Argument structure

These stages follow the mathematical order of the supplied argument.

12 mapped milestonesretained argument map

Browse all 12 mapped stages

stage 1Exact composite characterization
stage 2Global double completion
stage 3Denominator overlap isolated
stage 4Uniform fixed-A completion
stage 5One-prime A = 71 target
stage 6Finite terminal accumulation removed as a closing route
stage 7Rank-three terminal geometry
stage 8Periodic packing certificate reproduced
stage 9Large candidate enumeration recorded as reported
stage 10Conditional support bound amplified
stage 11Effective-sieve frontier
stage 12Global closure obligations retained

Exact composite characterizationThe current work reduces the composite direction to odd squarefree integers satisfying one exact congruence at every support prime.