Erdős problem 723 — wave 7i

Accessed and computed 2026-07-27 UTC.

Claim labels used throughout:

• (a) elementary-rigorous: proved below without an imported theorem.
• (b) rigorous-modulo-named-theorem: the imported theorem and its hypotheses

are named explicitly.

• (c) plausible/structural-unverified: interpretation or cost extrapolation,

not a theorem.

• (d) computational-only: a finite result certified by the standalone

program, not promoted to a uniform theorem.

0. Mandatory live-page gate

I fetched the rendered live page through the Bright Data browser path, not

datacenter curl. (d; live-page observation) The page

Erdős Problem #723 displayed:

• status badge FALSIFIABLE, together with the site's open-status disclaimer;
• 0 comments on this problem;
• 0 claimed proofs for this problem;
• Currently working on this problem: None;
• Interested in collaborating: None;
• Likes this problem: Amadeus_wu;
• all four other work/difficulty/formalisation markers as None.

Thus none of the required stop conditions was present. (d)

The live headline statement, verbatim, is:

> If there is a finite projective plane of order \(n\) then must \(n\) be a

> prime power?

The accompanying definition is transcribed exactly, without changing its

content, as

This formula is the page's prose definition: all blocks have size \(n+1\), and

every pair of ground-set elements is in exactly one block. **(a; exact formal

transcription)**

The live page's listed known results are: prime-power orders exist; the

conjecture is settled through \(n=11\); order \(12\) is open; Bruck--Ryser

forces an order congruent to \(1\) or \(2\bmod 4\) to be a sum of two squares;

and order \(10\) was excluded computationally. (b; page-grounded) The

page's LaTeX view identifies the two

citations as:

• R. H. Bruck and H. J. Ryser, *The nonexistence of certain finite projective

planes*, Canadian J. Math. 1 (1949), 88--93,

DOI 10.4153/CJM-1949-009-2.

• C. W. H. Lam, The search for a finite projective plane of order 10,

pp. 335--355 (1997); the author's

full expository article

describes the computer search.

The 1949 paper's Theorem 1 says precisely that, for \(N\equiv1,2\bmod4\), a

prime \(3\bmod4\) in the square-free part of \(N\) prevents a plane. By the

two-squares theorem this is equivalent to the form quoted on the live page.

(b)

1. Literature check and scope

The following primary records were located and checked; the identifiers and

claims below are not reconstructed from memory.

1. Marshall Hall, Jr., Cyclic projective planes, Duke Math. J. 14 (1947),

1079--1090,

DOI 10.1215/S0012-7094-47-01482-8.

Hall's planar multiplier result is the input used below. (b)

2. Daniel M. Gordon, *The prime power conjecture is true for

\(n<2{,}000{,}000\)*, Electron. J. Combin. 1 (1994), R6,

DOI 10.37236/1186. Its abstract explicitly

defines this as the prime-power conjecture for **abelian planar difference

sets and reports verification through two million. (d; bibliographic

verification), (b; reported theorem)**

3. Leonard D. Baumert and Daniel M. Gordon, *On the existence of cyclic

difference sets with small parameters*,

arXiv:math/0304502. Its abstract

explicitly reports that no cyclic projective plane of non-prime-power order

exists through two billion. (d), (b)

4. Daniel M. Gordon, On difference sets with small \(\lambda\), J. Algebraic

Combin. 55 (2022), 109--115,

DOI 10.1007/s10801-020-00992-x,

also arXiv:2007.07292. The introduction

states the First Multiplier Theorem and the fixed-translate/orbit reduction;

Section 3 reports the abelian planar-difference-set conjecture checked

through \(2\cdot10^{10}\). (d), (b)

5. Zvonimir Janko and Tran Van Trung, *The full collineation group of any

projective plane of order 12 is a \(\{2,3\}\)-group*, Geom. Dedicata 12

(1982), 101--110,

DOI 10.1007/BF00147334. The title,

authors, journal, and pages were independently checked against the Crossref

record. (d), (b)

Consequently, the cyclic calculation below is not a new literature bound:

Baumert--Gordon is vastly stronger, Gordon's later abelian result is stronger

still, and Janko--Van Trung already excludes a cyclic group of order \(157\)

from an order-12 plane. (b) Its value here is a small, transparent,

from-scratch certificate and a precise demonstration of why cyclic machinery

does not address the live unrestricted question.

I found no primary source resolving the unrestricted order-12 case. This is a

literature-search report, not a claim that no uncatalogued result exists; the

authoritative live page still marked it open with zero proof claims on

2026-07-27. (d)

2. Elementary reductions

2.1 The block definition forces the usual plane parameters

Let \(v=n^2+n+1\) and \(k=n+1\). For a fixed point \(x\), the blocks through

\(x\), after deleting \(x\), partition the other \(v-1\) points into sets of

size \(k-1=n\). Hence every point is on

blocks. Counting point pairs gives

Two distinct blocks meet in at most one point. A fixed block meets

\(k(r-1)=(n+1)n=v-1\) other blocks, with no double count, so it meets every

other block exactly once. Thus the page's set-system formulation is exactly a

symmetric \(2-(n^2+n+1,n+1,1)\) design, i.e. a finite projective plane.

(a)

At \(n=12\), this is a \(2-(157,13,1)\) design with 157 blocks and 13 blocks

through each point. (a)

2.2 A point-regular cyclic plane is a perfect difference set

Call a plane cyclic here if a cyclic collineation group \(C_v\) acts

regularly on its \(v\) points. Label the points by \(\mathbb Z_v\). A line

stabiliser has order dividing both \(v\) and \(k\), but

\(\gcd(v,k)=\gcd(n^2+n+1,n+1)=1\); hence it is trivial. The translates of one

line \(D\subset\mathbb Z_v\) are therefore all the lines. Unique incidence of

each point pair says exactly

So \(D\) is a cyclic \((v,n+1,1)\) difference set (a perfect difference set),

and conversely its translates form the plane. (a)

3. The multiplier certificate

3.1 The only imported lemma

First Multiplier Theorem, planar case. If \(D\) is an abelian

\((v,k,1)\) difference set of order \(n=k-1\), then every prime \(p\mid n\)

with \(p\nmid v\) is a numerical multiplier: \(pD=D+g_p\) for some \(g_p\).

Here \(p>1=\lambda\), and \(\gcd(n,n^2+n+1)=1\), so all hypotheses hold.

(b; Hall's theorem)

There is no hidden “common translate” assumption. Let

\(S=\sum_{d\in D}d\pmod v\). Since \(\gcd(k,v)=1\), translate by

\(-k^{-1}S\), obtaining \(E\) with element-sum zero. Summing

\(pD=D+g_p\) shows

and direct substitution gives \(pE=E\). Thus this one \(E\) is fixed by the

group \(H\leq\mathbb Z_v^\times\) generated by the distinct prime divisors of

\(n\). (a), conditional only on the preceding (b) theorem

3.2 Order 12

For \(n=12\), \((v,k,\lambda)=(157,13,1)\), and \(2\) is a multiplier.

The checker recomputes

Multiplication by 2 therefore has one orbit \(\{0\}\) and three nonzero

orbits of size 52 on \(\mathbb Z_{157}\). An invariant set can only have size

a subset sum of \(1,52,52,52\), never 13. Therefore:

> **No projective plane of order 12 admits a point-regular cyclic

> collineation group.**

This is (b), with every finite calculation independently checked by the

script. Using both prime multipliers 2 and 3 gives the still simpler orbit

partition \(1+156\); the script verifies

\(\operatorname{ord}_{157}(3)=78\) and generates the subgroup directly.

3.3 Exact bounded computation

For a general scanned \(n\), a residue \(x\in\mathbb Z_v\) of additive order

\(m\mid v\) has an \(H\)-orbit of size \(|H_m|\), where \(H_m\) is the image

of \(H\) in \(\mathbb Z_m^\times\). There are

\(\varphi(m)/|H_m|\) such orbits. Unit scaling shows that all orbits in this

stratum have the same internal-difference behaviour. (a)

An orbit is unusable if it is longer than \(k=n+1\), or if two distinct

ordered pairs inside it give the same nonzero difference: a perfect

difference set permits each nonzero difference only once. The checker:

1. factors \(n\) and \(v\) by literal trial division;

2. enumerates every divisor \(m\mid v\) and computes \(\varphi(m)\);

3. generates \(H_m\) by graph traversal;

4. checks all ordered internal differences;

5. performs an exact bounded subset-sum with the orbit multiplicities.

No probabilistic primality test, SAT solver, database, NumPy, SymPy, or

third-party library is used. (d)

For \(2\le n\le10{,}000\), the exact table is:

| class | count |

|---|---:|

| prime powers (not counterexample targets) | 1280 |

| non-prime-powers excluded by Bruck--Ryser | 3012 |

| remaining non-prime-powers orbit-scanned | 5707 |

| orbit-size survivors | 1, namely \(1322\) |

The counts sum to 9999. The deterministic transcript digest is

80522e383c5e89590d83afde5695c89053331b8f0cb89aab97522e762c3838ae.

The fast additive-order calculation was independently compared with direct

residue-by-residue orbit enumeration for all 94 relevant orders through 200.

(d)

The sole size survivor is closed separately. The program recomputes

and finds exactly these internally collision-free multiplier-orbit types:

| additive order \(m\) | orbit size | number of orbits |

|---:|---:|---:|

| 1 | 1 | 1 |

| 73 | 9 | 8 |

| 1387 | 18 | 72 |

Solving \(z+9a+18b=1323\) within these multiplicities gives only

or exactly

orbit unions. Direct ordered-difference checks show that no two of the 72

size-18 orbits are mutually compatible. Every candidate needs at least 70 of

them, so none is a difference set. (d), with the implication (a)

Combining Bruck--Ryser, Hall's theorem, and the finite certificates proves:

> **For every non-prime-power \(2\le n\le10{,}000\), there is no cyclic

> projective plane of order \(n\).**

This is (b)+(d) and is deliberately not presented as new in view of the

much stronger cited work.

4. Exactly what remains

The cyclic reduction requires a point-regular cyclic collineation group.

Nothing in the definition supplies any nonidentity collineation. In fact, the

Janko--Van Trung theorem says the full collineation group of a hypothetical

order-12 plane is a \(\{2,3\}\)-group, so a \(C_{157}\) action is already

impossible. Deleting the cyclic hypothesis therefore deletes the difference

set \(D\), the multiplier action, and every orbit used above. (b)

The exact symmetry-free target is any one of the following equivalent

objects:

For the middle reduction, remove a line and its 13 points. The 156 remaining

lines split into 13 parallel classes of 12; recording which line of every

class contains each of the 144 affine points produces the orthogonal array.

Conversely, adjoining one point per class and a line at infinity reverses the

construction. Fixing two OA columns as row and column coordinates leaves 11

MOLS. (a)

This is the missing uniform/computational step: construct such an OA, or

certify that none exists. The standard determinant obstruction is silent.

For an incidence matrix \(A\),

So even the required square determinant is already present. (a)

A raw block exact-cover model has

possible blocks before incidence filtering. At the intentionally generous

rate \(10^9\) blocks per core-second, merely streaming that list once would

take about \(942{,}721\) core-hours (107.54 core-years), without selecting or

proving anything. (c; arithmetic itself is (d)) This is why I did not run

a symmetry-free block search.

The OA formulation is syntactically smaller but not known to be easy. After

fixing two coordinate columns, a direct one-hot model has 19,008 primary

variables. One straightforward conjunction encoding for pairwise

orthogonality adds 1,140,480 row/pair/symbol-pair indicators, for 1,159,488

variables before clauses. (a; encoding count) No responsible core-hour

estimate for solving this unrestricted instance is available: producing such

a certificate would resolve the open order-12 case itself. (c) The precise

wall is therefore not “more CPU” in the abstract; it is the absence of a

search decomposition or structural lemma that reduces

\(\mathrm{OA}(144,13,12,2)\) without assuming forbidden symmetry.

5. Reproduction

The standalone checker is

erdos723_wave7i_verify.py. Run:

python3 runs/erdos723_wave7i_verify.py

On the report VM it used Python's standard library only and completed in

34.5 seconds. Its final run printed the order-12 orbit certificate, agreement

of two independent orbit enumerators on 94 cases, all scan counts and digest,

the \(n=1322\) certificate, and the unrestricted wall arithmetic. (d)

The complete source is included in the standalone file; the mathematically

operative core is reproduced here so the report itself records the algorithm:

from collections import Counter
from dataclasses import dataclass
from itertools import combinations
from math import comb, gcd, isqrt

@dataclass(frozen=True, order=True)
class OrbitType:
    modulus: int
    size: int
    count: int

def factor(n):
    ans, p = [], 2
    while p*p <= n:
        if n % p == 0:
            e = 0
            while n % p == 0:
                n //= p
                e += 1
            ans.append((p, e))
        p += 1 if p == 2 else 2
    if n > 1:
        ans.append((n, 1))
    return tuple(ans)

def divisors_with_phi(fac):
    out = [(1, 1)]
    for p, emax in fac:
        old = tuple(out)
        pp = 1
        for e in range(1, emax+1):
            pp *= p
            ph = (p-1)*p**(e-1)
            out.extend((d*pp, q*ph) for d, q in old)
    return tuple(sorted(out))

def subgroup_mod(m, generators, cap=None):
    seen, stack = {1}, [1]
    while stack:
        x = stack.pop()
        for g in generators:
            y = x*g % m
            if y not in seen:
                seen.add(y)
                if cap is not None and len(seen) > cap:
                    return None
                stack.append(y)
    return tuple(sorted(seen))

def sidon(points, modulus):
    differences = set()
    for i, x in enumerate(points):
        for j, y in enumerate(points):
            if i == j:
                continue
            d = (x-y) % modulus
            if d in differences:
                return False
            differences.add(d)
    return True

def valid_orbit_types(n):
    k, v = n+1, n*n+n+1
    generators = tuple(p for p, _ in factor(n))
    answer = [OrbitType(1, 1, 1)]
    for m, phi_m in divisors_with_phi(factor(v)):
        if m == 1:
            continue
        H = subgroup_mod(m, generators, cap=k)
        if H is not None and sidon(H, m):
            answer.append(OrbitType(m, len(H), phi_m//len(H)))
    return tuple(sorted(answer))

def reachable(target, types):
    bits, mask = 1, (1 << (target+1))-1
    for typ in types:
        left, power = typ.count, 1
        while left:
            take = min(power, left)
            bits = (bits | (bits << (take*typ.size))) & mask
            left -= take
            power <<= 1
    return bool((bits >> target) & 1)

The standalone file additionally contains the independent direct-residue

enumerator, the full scan loop, exact candidate counting, explicit coset

construction for \(n=1322\), all assertions, digesting, and wall arithmetic;

those verification layers are not omitted from the delivered code.

PARTIAL: The unrestricted problem remains open; verified here are an explicit Hall-multiplier proof excluding cyclic order 12 and a from-scratch exact exclusion of every non-prime-power cyclic order through 10,000, with the surviving symmetry-free OA(144,13,12,2) step isolated.