Solving 20 Erdős Problems with 20 Parallel Codex Accounts

Answer: A coordinated run of twenty Codex‑based proof assistants solved twenty long‑standing Erdős problems, producing fully formalised Lean 4 proofs for each. The results include exact asymptotics for Erdős #123, #254, #267, #320, #321, #336, #394, #450, #489, #538, #662, #796, #1188, #130, #709, #769, #959, #1186, #521, and #662, many of which were previously open or only partially resolved.

1. Erdős #123 – Sums of Distinct Powers

Result: For any three pairwise‑coprime integers (a,b,c>1), every sufficiently large integer can be written as a sum of distinct terms (a^i b^j c^k) with no term dividing another. The formal theorem in Lean is Erdos123.erdos_123 : Erdos123.IntendedStatement.

Why it matters. This settles a 1970s conjecture about additive bases generated by three multiplicative generators and removes the “finite‑seed” obstruction that blocked earlier induction schemes.

Key ideas.

Work on a homogeneous exponent level (i+j+k=D) so that divisibility becomes a coordinatewise comparison and any subset of a level is automatically primitive.
Construct an exact arithmetic progression of primitive homogeneous subset sums using an edge‑code and van der Waerden’s theorem (derived from Mathlib’s Hales–Jewett).
Translate the progression into a multiplicatively wide interval ([N,RN]) by adding an optional interior shell of unused monomials, which expands the interval without moving its lower endpoint.
Strengthen the residue‑reduction argument to a flexible finite‑seed gate: any interval ([N,CN]) with (N) large forces d‑completeness.

The proof is completely kernel‑checked; the only axioms are the standard Mathlib ones propext, Classical.choice, and Quot.sound.

2. Erdős #254 – Distinct Sums from Sparse Sets

Result: If (A\subset\mathbb N) satisfies (|A\cap[1,2x]|-|A\cap[1,x]|\to\infty) and (\sum_{n\in A}|\theta n|=\infty) for every (0<\theta<1), then every sufficiently large integer is a sum of distinct elements of (A). Formalised as Erdos254.erdos_254 : Erdos254.Statement.

Why it matters. The problem combines additive combinatorics with Diophantine approximation; prior attempts could not reconcile the two hypotheses.

Key ideas.

Show that the set of “bad” phases (those with bounded (\sum_{n\in A}|\theta n|)) is countable, turning an uncountable allocation problem into a diagonalisation over a countable family.
Construct three disjoint syndetic subsets of (A) and a fourth universally phase‑divergent correction set, thereby satisfying the Bergelson–Furstenberg–Weiss (BFW) theorem.
Prove a finite‑cyclic version of BFW by explicit Fourier analysis on (\mathbb Z/N\mathbb Z), avoiding any abstract ergodic‑theory machinery.

All components are verified in Lean with no placeholders.

3. Erdős #267 – Irrationality of Lacunary Fibonacci Reciprocals

Result: For any infinite increasing sequence (n_1<n_2<\dots) with a uniform ratio gap (n_{k+1}/n_k\ge c>1), the series (\sum_k 1/F_{n_k}) is irrational. The Lean statement is Erdos267.erdos_problem_267.

Why it matters. The case (c\ge2) follows from classical lacunary‑series criteria; the breakthrough covers the previously open range (1<c<2).

Key ideas.

Encode the reciprocal Fibonacci series as a locally finite word over (\mathbb Z[\varphi]) and analyse its quadratic‑integer norm.
Reduce to a bounded two‑adic order case by removing dyadic tails, then use a reverse‑window argument to produce a non‑zero quadratic integer with norm between 0 and 1, a contradiction.
The construction yields an explicit bound on the length of the “window” needed, making the argument effective.

The proof uses only the standard axioms listed above.

4. Erdős #320 & #321 – Distinct Unit‑Fraction Subset Sums

Result for #320: The number (S(N)) of distinct values of (\sum_{n\in A}1/n) with (A\subset{1,\dots,N}) satisfies [ c\frac{N}{\log N}P(\log!\log N)\le \log S(N)\le C\frac{N}{\log N}P(\log!\log N), ] where (P) is the fully stopped iterated‑log product. Formalised as ResearchPNT.exists_two_sided_full_product_estimate.

Result for #321: The maximal size (R(N)) of a set (A\subset{1,\dots,N}) with all reciprocal subset sums distinct satisfies [ R(N)=\Theta!\left(\frac{N}{\log N}\prod_{j=3}^{k(N)}\log_j N\right), ] with (k(N)) the last iterated logarithm above a fixed threshold. Formalised as Erdos321.erdos321_asymptotic.

Why they matter. Both problems ask for precise asymptotics of a highly non‑trivial combinatorial counting function; earlier works only obtained a few iterated‑log factors.

Key ideas.

Reformulate equality of subset sums as a signed relation (\sum \epsilon_n/n=0) with (\epsilon_n\in{-1,0,1}).
Use a “good denominator” sieve based on the largest prime factor to obtain an exact renewal recurrence with coefficient 1, avoiding loss of constants at each level.
Prove a sharp additive comparison between the renewal kernel and the exact (\log\log) difference, ensuring that losses are summable over all iterated‑log depths.
Transfer the upper entropy recurrence and the lower combinatorial recurrence into a common positive Neumann model, then evaluate the model explicitly.

Both the upper and lower bounds are verified in Lean; the only axioms are the three standard ones.

5. Erdős #336 – Exact Order of Asymptotic Bases

Result: For the extremal function (h(r)) (maximal exact order of a basis of variable order (\le r)), the limit [ \lim_{r\to\infty}\frac{h(r)}{r^2}=\frac13 ] holds. Formalised as Erdos336.problem336 : HasProblem336Value (1/3).

Why it matters. The constant (1/3) was conjectured from periodic constructions; the difficulty lay in proving a uniform upper bound for all bases.

Key ideas.

Reduce the infinite problem to a finite cyclic setting via a dyadic high‑power argument.
Classify endpoint configurations of a rectified graph (T\subset\mathbb Z\times\mathbb Z/N) and analyse the Kneser stabiliser of (B+B).
Prove an exact two‑generator lattice inequality (3|G|\le(H+2)^2), which yields the (1/3) coefficient.
Exhaustively treat all possible endpoint defect cases, including a critical three‑point classification that eliminates the only remaining obstruction.

The proof is completely formalised; the axiom audit reports only the standard three.

6. Erdős #394 – Growth of the Least Start of Consecutive Products

Result: There exists (c>0) (explicitly (c=1/2048)) such that [ \sum_{n\le x} t_2(n) \ll \frac{x^2}{(\log x)^c}, ] and for every fixed (k\ge2) [ \sum_{n\le x} t_{k+1}(n)=o!\left(\sum_{n\le x} t_k(n)\right). ] Formalised as erdos394_first_target : FirstQuestion and erdos394_second_target : SecondQuestion.

Why it matters. The functions (t_k(n)) measure the least start of a length‑(k) consecutive product divisible by (n); prior pointwise bounds were far too weak to obtain any average‑order savings.

Key ideas.

Build a finite‑prime “score” (U(m)) counting selected prime divisors of (m) and a two‑level score (W(m)) that also records squared prime factors.
Choose a finite set (S) of large primes with (\sum_{p\in S}1/p>152/\epsilon); periodicity modulo (Q=\prod_{p\in S}p^2) yields uniform control of the three classes of bad integers.
Show that for any interval of length (y\ge C(\epsilon)n) the number of integers having a divisor in ((n,2n]) is at most (\epsilon y).
Use a dense hierarchy of cut‑offs (X_N=16^N) and a uniform inequality (\lfloor\log_2 N\rfloor,S_{k+1}(X)\le3S_k(X)) to obtain the little‑o relation for all (x).

All steps are verified in Lean with only the three core axioms.

7. Erdős #450 – Linear‑Scale Short‑Distance Density

Result: For every fixed (\epsilon>0) there is a constant (C(\epsilon)) such that any interval of length (y\ge C(\epsilon)n) contains at most (\epsilon y) integers having a divisor in ((n,2n]). Moreover, any sufficient threshold must eventually exceed (n); thus the optimal order is (\Theta_\epsilon(n)). Formalised as tur anLinearAnswer_isSufficientScale and sufficientScale_eventually_gt_n.

Why it matters. The problem asks for a uniform bound over all translates, which is much stronger than average‑density results. The linear bound matches the obvious lower obstruction from factorial translates.

Key ideas.

Fix a finite set (S) of large primes (all (\ge5)) with (\mu=\sum_{p\in S}1/p>152/\epsilon).
Define (U(m)) and (W(m)) as in #394; use Chebyshev’s inequality on the periodic score distribution modulo (Q=\prod_{p\in S}p^2) to bound low‑score divisors and high‑score numbers.
Show that any integer (m) with a divisor (d\in(n,2n]) falls into one of three classes, each of which contributes at most a fixed multiple of (y) to the count.
The three‑class estimate yields (#{m\in(x,x+y):\exists d\in(n,2n]\mid m}\le \epsilon y) once (y\ge n(Q+2)).

The proof is fully formalised; the axiom set is the standard three.

8. Erdős #489 – Second‑Moment of Gaps in a Sparse Sieve

Result: If (A\subset\mathbb N) satisfies (|A\cap[1,x]|=o(\sqrt x)) and (B={n:\forall a\in A, a\nmid n}) is infinite with enumeration (b_1<b_2<\dots), then the limit [ \lim_{x\to\infty}\frac{1}{x}\sum_{b_i<x}(b_{i+1}-b_i)^2 ] exists and is finite. Formalised as erdos489_statement.

Why it matters. The expression is a second‑moment of gaps in a sieve set; controlling it requires uniform integrability of long gaps, which is not guaranteed by simple density arguments.

Key ideas.

Prove that the sparse sequence (a_r) of forbidden divisors satisfies (\sum 1/a_r<\infty) and a double‑sum kernel (\sum_{r,s}\frac{\min(r+1,s+1)}{a_ra_s}) is convergent.
In a long gap, select many “primitive” positions whose residues avoid low‑score divisors; coprimality of these positions yields quadratically many ordered pairs.
Use a primitive‑ray capacity bound to show that the total contribution of long gaps is dominated by the convergent kernel, establishing uniform integrability.
For bounded gaps, the sieve becomes periodic after a finite prefix, so the normalized second moment converges by standard Cesàro averaging.

All arguments are checked in Lean; only the three standard axioms appear.

9. Erdős #538 – Reciprocal Sums under Prime‑Factor Restrictions

Result: For fixed (r\ge2) and any (A\subset{1,\dots,N}) with at most (r) representations (m=pa) ((p) prime, (a\in A)) for each (m), the sum (\sum_{a\in A}1/a) satisfies [ \Theta_r!\left(\frac{\log N}{\log\log N}\right). ] Both the upper bound (Erdős 1973) and a matching lower construction are formalised in Erdos538.MatchingOrder.

Why it matters. The problem asks for the optimal order of magnitude; the missing (\log\log N) factor in earlier constructions is supplied by a finite‑field “safe isotropic kernel” that yields a cap‑two family of density (\Omega(1/k)) on the (k)-prime‑factor layer.

Key ideas.

Build a cap‑two (k)-uniform family over (\mathbb F_q) with density (\ge1/(64k)) using favourable linear‑algebraic configurations and a safety condition preventing total isotropy.
Transfer the family to integer layers by colour‑coding prime supports; each layer contributes (\Omega(1/k)) of the harmonic mass.
Sum over (k\le\log\log N) to obtain the lower bound; the classical incidence argument gives the matching upper bound.

The Lean development contains no sorrys; the axiom audit lists only the three standard ones.

10. Erdős #662 – Short Distances in One‑Separated Planar Sets

Result: The conjectured extremality of the triangular lattice is false. Explicit rational oblique lattices provide arbitrarily large one‑separated point sets with more short‑distance pairs than the triangular lattice, both for the closed‑shell and strict‑shell readings. Formal statements are Research.triangular_shell_six_global_average_reading_false and Research.strict_shell_readings_false.

Why it matters. The problem asks whether the triangular lattice maximises the number of pairs at distance (\le t) in a large one‑separated set. The counterexamples show that lattice geometry can beat the densest packing for fixed radii.

Key ideas.

Choose a basis (u=(1,0),;v=(136/305,273/305)) (closed shell) and (u=(1,0),;v=(276/565,493/565)) (strict shell) and verify one‑separation via a positive quadratic form identity.
Count non‑zero lattice offsets of radius (6) (closed) and (\sqrt{300}) (strict) to obtain 128 vs. 126 and 1078 vs. 1074, respectively.
Amplify the excess by taking large rectangular patches; the excess persists for arbitrarily large (n), disproving the conjecture.

All calculations are performed in Lean with exact integer arithmetic; only the three core axioms are used.

11. Erdős #796 – Asymptotics for a Subset‑Sum Extremal Function

Result: For (g_3(n)) (the maximal size of (A\subset{1,\dots,n}) such that every (m) has fewer than three representations (m=a_1a_2) with (a_1<a_2\in A)), the normalised second‑order term converges: [ \frac{g_3(n)-\frac{\log\log n}{\log n}n}{n/\log n}\to M, ] where (M) is an explicit constant (=\text{Mertens.M}+\text{variationalLimit}). Formalised as Erdos796.erdos796_statement.

Why it matters. The problem asked for a precise second‑order term; the result gives an exact constant and confirms the conjectured form.

Key ideas.

Reduce the problem to a smooth‑remainder gate and an extracted‑tail gate, each handling a different range of divisor sizes.
Use a refined sieve to control the contribution of large prime factors and a combinatorial decomposition for the small‑factor part.
Prove that the two gates together capture all contributions, yielding the limit.

The proof is fully formalised; the axiom list is the standard three.

12. Erdős #1188 – Counting Minimal Distinct Covering Systems

Result: The number (F(x)) of minimal distinct covering systems with all moduli (\le x) satisfies [ \frac{\log\log F(x)}{\log x}\to1, ] i.e. (F(x)=\exp\bigl(x^{1+o(1)}\bigr)). Formalised as erdos1188_loglog_ratio_tendsto_one.

Why it matters. Erdős expected (F(x)) to grow very slowly; the result shows it grows almost double‑exponentially.

Key ideas.

Construct a sparse “no‑axis” family of covering systems by assigning residues to a carefully chosen set of primes, avoiding the classical primorial axis.
Show that the number of choices grows like (\exp(x^{1+o(1)})) by counting independent residue assignments.
The upper bound follows from the trivial (\prod_{n\le x}(n+1)) estimate; the lower bound uses the explicit construction.

All steps are verified in Lean with only the three standard axioms.

13. Erdős #130 – Infinite‑Chromatic Integer‑Distance Graph in General Position

Result: There exists an infinite set (A\subset\mathbb R^2) with no three collinear points and no four concyclic points such that the graph joining points at integer distance has infinite chromatic number. Formalised as Erdos130.erdos130_infinite_chromatic.

Why it matters. The problem asked whether the chromatic number can be infinite under strong geometric constraints; the construction shows it can.

Key ideas.

Build finite rational circle‑tangency graphs of arbitrarily high chromatic number using a Hales–Jewett‑type boost.
Apply a rational inversion to place circle centres in general position (no three collinear, no four concyclic).
Assemble countably many translated blocks with carefully chosen translation vectors to avoid creating new collinearities or concyclic quadruples.
Scale the whole configuration so that all tangencies become integer distances.

The Lean proof contains no sorrys; the axiom audit lists only propext, Classical.choice, and Quot.sound.

14. Erdős #709 – Improved Upper Bound for the Divisibility Packing Function

Result: For the minimal function (f(n)) guaranteeing that any (n)-element set (A\subset[2,\infty)) appears as divisors in a block of (f(n)\cdot\max A) consecutive integers, we have [ f(n)\le 14,n^{3/7} ] improving the classic Erdős–Surányi (\sqrt n) bound. Formalised as ScaleWorks n (7*(Nat.nthRoot 7 (n^3)+1)) and the inequality f(n) ≤ 14 * n^(3/7).

Why it matters. The exponent (3/7) is the first improvement over the long‑standing (1/2) exponent.

Key ideas.

Use a Katz–Tao four‑projection argument to partition (A) into blocks with controlled residue classes.
Apply an exact Hall‑type matching to select representatives, then bound the total length of the required interval.
The analysis yields the explicit constant 14 and the exponent (3/7).

The proof is fully checked in Lean; only the three standard axioms appear.

15. Erdős #769 – Asymptotics of Cubical Decompositions

Result: The conjecture (c(n)\gg n^n) is false. For every odd (n\ge201), any (k\ge n,2^n,\lceil49n/100\rceil^n+2) allows a decomposition of the (n)-cube into (k) homothetic cubes. Hence (c(n)=o(n^n)) along odd dimensions. Formalised as Erdos769.erdos769_lower_bound_false.

Why it matters. The problem asked whether the minimal number of cubes needed grows super‑exponentially; the result shows a sub‑exponential construction.

Key ideas.

Construct explicit homothetic tilings using a recursive subdivision scheme that respects odd dimensions.
Analyse the edge‑length growth to obtain the bound (n,2^n,\lceil49n/100\rceil^n+2), which is asymptotically smaller than (n^n).

The Lean development contains no placeholders; the axiom audit reports only the three core axioms.

16. Erdős #959 – Superlinear Gap Between Top Two Distance Multiplicities

Result: There exists a constant (c=1/50000) such that for all sufficiently large (n), [ M(n)\ge n^{1+\frac{c}{\log\log n}}, ] where (M(n)) is the maximum possible difference (f(d_1)-f(d_2)) between the two most common distances among (n) planar points. Formalised as Erdos959.erdos959_superlinear_lower_bound.

Why it matters. The previous best lower bound was (\Omega(n\log n)); this result shows a genuinely superlinear growth.

Key ideas.

Build a configuration of replicated lattice disks indexed by subsets of primes (\equiv1\pmod4) to create many points sharing a common distance.
Use a refined prime‑distribution estimate to control the number of distinct distances and amplify the multiplicity gap.
Apply a careful counting argument to obtain the (n^{1+\frac{c}{\log\log n}}) bound.

All arguments are verified in Lean; the only axioms are the standard three.

17. Erdős #1186 – Exact Minimum Density of Monochromatic 3‑APs

Result: The exact constant is [ \delta_3=\frac{117}{2192}, ] so every two‑colouring of ({1,\dots,n}) contains at least ((117/2192+o(1))n^2) monochromatic 3‑term arithmetic progressions. Formalised as erdos1186_explicit_bounds together with a verified sum‑of‑squares certificate.

Why it matters. Graham offered a prize for this exact value; the result settles the conjecture of Parrilo–Robertson–Saracino (2008).

Key ideas.

Reduce the discrete problem to a continuum quadratic form (Q(x)) over a fixed 548‑cell partition.
Show that the extremal colouring corresponds to a specific 12‑block pattern whose quadratic form value is (-10/137).
Produce an exact sum‑of‑squares (SOS) certificate proving that (Q(x)\ge-10/137) for all colourings; the certificate is checked by an independent exact integer‑arithmetic verifier and also formalised in Lean.
Translate the SOS bound back to the discrete setting, obtaining the exact density.

The Lean proof contains no sorrys; the axiom audit lists only the three standard axioms.

18. Erdős #521 – Almost‑Sure Law for Real Roots of Random ±1 Polynomials

Result: The conjectured almost‑sure convergence [ \frac{R_n}{\log n}\to\frac{2}{\pi}\quad\text{a.s.} ] fails. In fact, (R_n/\log n) does not converge almost surely; the fluctuations are of order (\log n) along a sparse subsequence. Formalised as erdos_521_negative : ¬ Claim.

Why it matters. Erdős–Offord proved the expectation and convergence in probability; the almost‑sure statement was open.

Key ideas.

Analyse the crossing count of the random polynomial’s sign changes and show that, conditioned on earlier coefficients, the probability of a large deviation of (R_n) remains bounded away from zero infinitely often.
Use a fourth‑moment gate to convert the almost‑sure claim into an inequality that is violated by the conditional crossing mean.
Construct explicit record degrees where the deviation exceeds any prescribed fraction of (\log n), yielding a contradiction.

The proof is fully formalised in Lean; only the three core axioms appear.

19. Erdős #522 – Almost‑Sure Root Count Inside the Unit Disk

Result: For i.i.d. fair (\pm1) coefficients, the number (R_n) of roots of (\sum_{k=0}^n\epsilon_k z^k) inside the closed unit disk satisfies [ \frac{R_n}{n/2}\to1\quad\text{a.s.} ] formalised as erdos_522 : Erdos522Claim.

Why it matters. Yakir proved convergence in probability; the almost‑sure result required controlling deviations uniformly over all (n).

Key ideas.

Establish a high‑moment bound for an angular‑cosine statistic (M_{n,s,t}) of the polynomial, with explicit exponents (2048) and (8192).
Apply a van der Corput‑type estimate to the oscillatory sums arising from the cosine moments.
Choose parameters (q=1024) and (H=8192) so that the resulting tail bound is summable over (n).
Invoke Borel–Cantelli to obtain almost‑sure convergence of (R_n/(n/2)) to 1.

All components are verified in Lean; the axiom list is the standard three.

How the Parallel‑Codex System Worked

Problem selection. Twenty Erdős problems were chosen from the database, each with a clearly stated formal target.
Automated prompting. For each problem a dedicated Codex‑5.6 instance received the problem statement, any available comments, and a template asking for a Lean 4 proof sketch.
Iterative refinement. The generated sketches were fed back to the model with targeted prompts ("expand the inductive step", "explain the use of Hales–Jewett", etc.) until a complete proof script was produced.
Kernel verification. Each script was compiled with lake build. Any occurrence of sorry, admit, or custom axioms caused the job to be discarded and re‑prompted.
Cross‑checking. Independent Rust and Python checkers verified all large‑scale combinatorial or SDP certificates (e.g., the SOS proof for #1186).
Aggregation. The twenty verified theorems were collected into a single repository, with a unified axiom audit confirming that every proof depends only on propext, Classical.choice, and Quot.sound.

The entire pipeline ran on a cluster of 20 × high‑CPU VMs, completing all twenty proofs in under 48 hours of wall‑clock time.

Significance of the Batch Solution

Breadth. The solved problems span number theory, combinatorics, geometry, and probability, demonstrating that the approach is not limited to a single subfield.
Depth. Many of the results settle long‑standing conjectures (e.g., #1186, #521) or improve decades‑old bounds (e.g., #709, #538).
Reliability. Every proof is machine‑checked; the axiom audit guarantees that no hidden assumptions were introduced.
Reproducibility. All source files, build scripts, and verification logs are publicly available, enabling anyone to re‑run the verification from scratch.

These achievements illustrate that large‑scale, AI‑assisted formal mathematics can make genuine progress on deep open problems, turning speculative conjectures into rigorously verified theorems.

References & Further Reading

The full Lean source code for each problem is hosted on the public verified_math GitHub organization.
Detailed verification logs and axiom audits are included in each project’s README.md.
For a high‑level overview of the Codex‑based proof‑generation pipeline, see the accompanying blog post on Star Fleet Math.

This article summarises the outcomes of the twenty‑problem batch. Each section is self‑contained and can be quoted independently, providing answer‑first statements, the mathematical significance, and a concise sketch of the proof technique.

Solving 20 Erdős Problems with 20 Parallel Codex Accounts

Solving 20 Erdős Problems with 20 Parallel Codex Accounts