Erdős–Graham #203 — finite prime-fibre obstruction

The problem asks whether there exists an integer m>=1, (m,6)=1, such that

[ 2^k3^\ell m+1 ]

is composite for every k,\ell>=0.

This repository studies a sufficient construction using finitely many prime fibres. For a prime p>3, the exponent pairs eliminated by p form a coset of

[ R_p={(u,v):2^u3^v\equiv1\pmod p}. ]

CRT makes the phase choices for distinct primes independent, turning the construction into a finite geometric covering problem followed by one arithmetic realization step.

Main finite theorem

For the frozen 31-fibre U_5040 base:

Any finite prime-fibre cover extending that base requires at least 12 outside prime fibres.

The corrected replay includes the previously omitted induced-index classes d=8 and d=10 and still eliminates every extension using at most 11 outside fibres.

The exact replay is:

• docs/CANONICAL_R12_LOWER_BOUND_2026-09-14.md — theorem and proof architecture;
• tools/replay_r12_lower_bound_2026_09_14.py — executable exact calculation;
• receipts/replay-r12-lower-bound-2026-09-14.json — matching receipt.

Finite obstruction calculus

Let

n_p = |<2,3> mod p|
d_p = n_p / gcd(n_p,5040)

and define

L = 1/d_p          local density on a 5040 replica
O = d_p/n_p        replica occupancy
G = 1/n_p          global density

so

[ G=LO. ]

The frozen 31-fibre base has raw density

[ 143/140. ]

A phase-independent overlap forced by p=5 costs 257/1680, so the base union has density at most

[ 1459/1680. ]

Any repair family must therefore supply at least

[ 221/1680 ]

of additional global raw fibre mass.

For the low 3-coprime induced-index classes d=2,4,5,7,8,10, the combined replica occupancy of every available fibre is only

[ 53/140. ]

The complete d=11 census contains 24 fibres with direction multiplicities

4,3,3,3,2,2,2,2,2,1

which rules out the equality case requiring eleven parallel 1/11 lines.

Independent finite results

tools/replay_eg203_gold.py recomputes four earlier exact calculations:

result	value
primes p>3 with `n_p	5040`
all phases for the N=5040 fibre family	union density at most 823/840 < 1
{5,7,11,13} core on the 60×60 torus	2,880 assignments; maximum union density 353/720
primes p<=10^6 with n_p<=1000	238 fibres, raw density about 1.83048759933

Reproduce

python tools/check_gold_arithmetic.py
python tools/replay_eg203_gold.py
python tools/replay_d8_branch_2026_09_13.py
python tools/replay_no_d3_branch_2026_09_13.py
python -m pip install sympy
python tools/replay_r12_lower_bound_2026_09_14.py

Scope

The finite prime-fibre cover is a sufficient route to a YES solution of Erdős–Graham #203. A successful finite cover followed by CRT realization would solve the problem positively.

The converse is not proved: failure of this finite-cover construction would not by itself prove that no suitable m exists.

The headline finite obstruction theorems are presently computational/written results; they have not yet been formalized in Lean in this repository.

Author: Jared Wilder. License: Apache-2.0.