Prime-Shell Fertility and Recursive Prime Magic Squares
A Construction Report on Centre-Shell Prime Sourcing, Recursive Block Assembly, and Global Uniqueness Pressure
Authorship and assistance note.
This report is prepared for Roberto Carlo Angelone as a constructional methods paper. AI tools were used for computational assistance, drafting, verification support, failure analysis, and editorial refinement. The mathematical direction, constructional framing, and decision-making belong to the human-directed project. The AI is treated here as an instrument of computation and documentation, not as a mathematical author.
________________________________________
Abstract
This report describes the prime-sourcing and recursive block-assembly methods used in the construction of fully nested unique-prime magic squares of orders 81×81, 243×243, and 729×729. The work does not claim to invent ordinary recursive or composite magic squares; such constructions are known. Instead, the distinctive feature is the use of recursive 3^nmagic-square skeletons as role fields for sourcing and arranging globally distinct primes through local centre-shell grammar and higher-level disjoint block assembly.
The basic local component is the general 3×3magic-square form centred at c:
■(c+a&c-a-b&c+b@c-a+b&c&c+a-b@c-b&c+a+b&c-a)
Each row, column, and diagonal sums to 3c. For a prime-shell construction, the centre cis prime and the eight peripheral values
c±a,c±b,c±(a+b),c±(a-b)
must be distinct primes. The distances a,b,a+b,∣a-b∣need not be prime; they are displacement operators around the centre.
The first completed construction contains 6561globally distinct prime entries in a fully nested 81×81square, with master centre 800011and magic constant 64800891=81×800011. A subsequent 243×243construction contains 59049globally distinct prime entries, centre 10000000033, magic constant 2430000008019, and verifies at all aligned 3×3, 9×9, 27×27, 81×81, and full 243×243levels. A further 729×729construction contains 531441globally distinct prime entries, also centred at 10000000033, with magic constant 7290000024057, and verifies through aligned 243×243blocks and the full 729×729square.
The central operational insight is that the construction is not merely a search for primes, but a search for prime centres and prime blocks with enough coordination capacity to support symmetric prime-pair relations, nested magic constraints, residue-lane restrictions, and global non-reuse. This led to the concepts of prime-shell fertility, usable coordination capacity, residue-lane filtering, global ledger pressure, grammar-versus-template recursion, and disjoint recursive block assembly.
________________________________________
1. Prior-Art Boundary and Scope
Ordinary recursive magic squares, composite magic squares, and inlaid magic-square constructions have long histories. The order-3 Lo Shu square is ancient, and larger magic squares built from smaller magic subsquares are known. The present work does not claim novelty in the ordinary sequential 3^nmagic-square skeleton.
Likewise, prime magic squares themselves are not new. Examples of small prime magic squares appear in early recreational mathematics literature, including Dudeney-style prime-square problems and later refinements by Ondrejka, Johnson, and others.
Recent work is also directly relevant. Skelt, Perkins, and Roach introduced and studied prime strictly concentric magic squares of odd order in Mathematics in 2025. Their work focuses on prime strictly concentric magic squares, including extensive order-5 enumeration and border-pair/complement-style constructions. The present construction is different in scale and grammar: it uses recursive 3^nblock structure, centre-shell displacement sourcing, and disjoint block assembly rather than strictly concentric border enumeration.
The present contribution is narrower:
to use the recursive 3^nordinary skeleton as an operational role field;
to source primes by centre-shell fertility rather than by arbitrary placement;
to enforce global uniqueness across nested scales;
to document verified fully nested unique-prime constructions at orders 81×81, 243×243, and 729×729;
to analyse why naïve local template reuse failed;
to identify disjoint block assembly as the successful scaling correction.
Thus the claim is not:
recursive magic squares were discovered here.
The claim is:
a centre-shell prime-sourcing grammar and disjoint recursive block-assembly strategy were used to construct and verify large fully nested unique-prime magic squares, and the search produced useful concepts for future recursive prime-square construction.
This distinction is essential. Without it, the work would overclaim against well-known prior art. With it, the work is positioned as a computational construction and methodology note.
This report makes four specific contributions. First, it formalizes prime-shell fertility as a centre-based sourcing measure for recursive prime magic-square construction. Second, it distinguishes local shell fertility from usable coordination capacity under global uniqueness and ledger constraints. Third, it documents verified fully nested unique-prime magic squares at orders 81×81, 243×243, and 729×729. Fourth, it records scaling failure modes and the constructional correction that followed, especially the distinction between recursive shell grammar and numerical displacement-template reuse.
________________________________________
1.1 Data Availability
The completed 81×81unique-prime magic square is available as a CSV dataset with companion notes.
Dataset DOI:
https://doi.org/10.5281/zenodo.20005776
The 81×81dataset contains the full grid, consisting of 6561globally distinct prime entries. The later 243×243and 729×729verifier packages should be archived alongside the original 81×81dataset, because they substantially change the construction status from a single verified object to a verified recursive ladder.
A publication-ready archive should include:
CSV grids;
manifests recording order, centre, magic constants, entry counts, and construction metadata;
SHA-256 checksums;
verification scripts;
verification outputs;
residue-lane audits;
shell-displacement audits where applicable;
repetition reports for any non-unique scaffold objects.
For the 81×81, 243×243, and 729×729objects, the datasets are the results. The report is the method and interpretation. Both are needed.
________________________________________
1.2 Terminology Note
Several terms used in this report are constructional terms introduced for this project. Terms such as fertility, coordination capacity, ledger pressure, shell grammar, and block assembly are not standard invariants in analytic number theory. They are operational quantities used to classify candidate centres, shell completions, block compatibility, and global construction pressure.
This distinction matters. The terminology is intended to describe a computational construction method, not to assert new universal laws of prime distribution.
________________________________________
2. The Centre-Shell Primitive
Every nontrivial 3×3magic square can be represented, up to affine transformation and symmetry, by a centre-shell form:
■(c+a&c-a-b&c+b@c-a+b&c&c+a-b@c-b&c+a+b&c-a)
The sum of each row, column, and diagonal is 3c. This form was used as the atomic local mechanism of the prime construction.
For the prime version, the construction begins with a candidate prime centre c, then searches for positive integer displacements a,bsuch that all nine entries are distinct primes.
The required peripheral values are:
c+a, c-a, c+b, c-b, c+a+b, c-a-b, c+a-b, c-a+b.
Equivalently:
c±a,c±b,c±(a+b),c±(a-b).
The displacements themselves are not required to be prime. They are the distances that define four complementary prime-pair tests around c.
This local primitive converts the search problem from:
find primes that happen to fill a magic square,
into:
find prime centres that can support closed symmetric prime shells.
That change is the methodological centre of the work.
A schematic way to read the shell is:
Position role Value
centre c
first symmetric shell pair c±a
second symmetric shell pair c±b
sum-displacement pair c±(a+b)
difference-displacement pair c±(a-b)
Thus the shell is controlled by four displacement tests: a, b, a+b, and ∣a-b∣. Each test must produce prime values after being applied symmetrically around c.
________________________________________
3. Formal Definitions
3.1 Shell-admissible displacement pair
Let cbe an odd prime and let R>0. A pair (a,b)∈Z_(>0)^2is called R-admissible for cif:
a+b≤R
and all eight peripheral values
c±a,c±b,c±(a+b),c±(a-b)
are distinct primes and are also distinct from c.
The positivity condition a,b>0is essential. If a=0or b=0, repeated centre values or degenerate shells can occur.
3.2 Prime-shell fertility
The prime-shell fertility of cat radius Ris:
F_R (c)=#{(a,b)∈Z_(>0)^2:(a,b)" is " R"-admissible for " c}.
This measures local shell availability around a centre before global constraints are imposed.
3.3 Constraint set
Let Cdenote a constructional constraint set. In this project, Cmay include:
primality;
local distinctness;
global non-reuse of primes;
residue-class restrictions;
recursive block role;
compatibility with a centre grid;
reserved-prime ledger constraints;
scale or radius bounds;
block disjointness;
macro-level magic compatibility.
3.4 Coordination capacity
The coordination capacity of cunder constraints Cand radius Ris:
K_R (c;C)=#{(a,b):(a,b)" produces a valid centre-shell satisfying " C}.
Thus F_R (c)measures local fertility, while K_R (c;C)measures usable fertility after constructional constraints are imposed.
A centre may have high F_R (c)but low K_R (c;C)if its locally valid shells collide with primes already used elsewhere.
3.5 Block-disjoint assembly
A collection of magic blocks B_1,…,B_9is block-disjoint if no prime entry appearing in one block appears in any other block. A 3×3block assembly is valid when:
each block is internally a nested prime magic square;
the nine block centres form a valid 3×3centre-shell magic square;
the block entry sets are pairwise disjoint;
the resulting larger square verifies at all required aligned scales.
This definition describes the successful scaling strategy used for the 243×243and 729×729constructions.
________________________________________
4. Prime Sourcing Method
The prime-sourcing process used five nested filters.
4.1 Prime centre selection
A candidate centre cfirst had to be prime. But primality alone was insufficient. The centre had to support one or more closed shell completions.
Centres were therefore treated as possible expansion nodes. Around each centre, the search tested displacement pairs (aⓜ,b)and accepted them only if the eight shifted values were prime.
4.2 Shell generation
For each candidate centre, displacement pairs were generated subject to radius and residue constraints. A pair (aⓜ,b)was locally valid only when it produced a complete prime shell.
The shell condition is stricter than ordinary symmetric prime-pair search. It requires simultaneous prime-pair closure at four distances:
a,b,a+b,∣a-b∣.
Each distance defines a symmetric pair around the same centre. The construction therefore filters centres by their ability to support several compatible complementary prime pairs.
4.3 Local distinctness
Each 3×3shell had to contain nine distinct primes. This removed degenerate displacement choices and prevented local repetition.
4.4 Global uniqueness
A shell that was locally valid could still be rejected if any of its primes had already been used elsewhere in the square. This global ledger constraint became the dominant practical difficulty.
The problem was therefore not merely primality testing. It was a global packing problem: many locally valid shells had to be selected without overlap.
4.5 Recursive nesting
The centres themselves had to lie in a recursively magic centre grid. In the verified 81×81construction, a nested 27×27prime magic square supplied the 729shell centres. Each centre was then expanded into a disjoint 3×3prime shell.
The sourcing process therefore had two levels:
source prime centres that form a valid recursive centre grid;
source disjoint prime shells around those centres.
The successful construction required both.
________________________________________
4.6 Actual Construction Algorithm Used
The completed 81×81object was produced by a human-directed computational search with AI assistance. The construction should be described as centre-first and shell-expansion-based rather than as a blind fill of 6561cells.
At a high level, the algorithm was:
Fix the recursive target structure. Use the aligned 3^nblock hierarchy, with the final 81×81object decomposed into 729aligned 3×3shells.
Construct or select a nested 27×27prime magic centre grid. These 729centre values become the centres of the local 3×3shells.
For each centre c, generate candidate shell pairs (aⓜ,b). Candidate pairs were tested through the centre-shell formula.
Apply primality tests to all eight peripheral values. A shell candidate was rejected unless c±a, c±b, c±(a+b), and c±(a-b)were all prime.
Apply local distinctness. Reject shells with repeated local entries.
Apply the global forbidden-prime ledger. Reject shells containing primes already used elsewhere.
Lock accepted shells into their aligned 3×3positions. The accepted shell preserves the local magic sum 3c.
Run full recursive verification. Verify primality, uniqueness, full-square magic sums, and all aligned nested block sums.
The later 243×243and 729×729constructions used the block-disjoint assembly strategy:
construct or select nine mutually disjoint nested prime magic blocks of the previous order;
arrange their centres as a 3×3centre-shell macro-square;
assemble the blocks as macro-entries;
verify global uniqueness across the union of all block entries;
verify all aligned nested levels in the larger square.
The search was primarily constructive and ledger-filtered. It should not be presented as a proof-producing algorithm for arbitrary scale. It produced verified finite objects, not a theorem that all such objects exist.
Implementation details should be recorded in any publication package. The search and verification workflow used Python-style computational tooling, with primality testing applied to each candidate shell entry and a global forbidden-prime ledger implemented as a set-like membership structure. Candidate shells were accepted only after passing primality, local distinctness, and global non-reuse checks. Larger-scale attempts used constructive iteration with rejection and repair rather than a closed-form proof-generating algorithm.
Runtime depended strongly on search radius, centre fertility, block availability, and collision pressure. The original exploratory search did not preserve a clean machine-readable runtime log, so exact construction runtime is not claimed here. Verifier runtimes, however, are recorded in the verifier packages.
For reproducibility, a publication version should include the exact scripts used or a reconstructed equivalent verifier/generator, including programming language, primality-test method, ledger rules, search-radius settings, and runtime details. The current report records the algorithmic logic but should still be paired with executable verification code.
________________________________________
5. Complete Modulo-30 Residue Analysis
For primes greater than 5, the possible residues modulo 30are:
1,7,11,13,17,19,23,29(mod30).
For a symmetric pair c-d, c+dto remain coprime to 30, the displacement dmust avoid sending either value into a residue divisible by 2, 3, or 5.
Because c-dand c+dmust both be odd and not divisible by 3or 5, admissible displacement classes are restricted.
5.1 Primary lane
The primary lane is:
d≡0(mod30).
This preserves the centre residue:
c-d≡c+d≡c(mod30).
This lane is always residue-compatible for prime-admissible centre residues. It is the safest lane because it keeps the pair in the same residue class as the centre.
5.2 Secondary lanes
The nonzero compatible displacement classes split according to centre residue.
For:
c≡1,11,19,29(mod30),
the compatible nonzero displacement classes are:
d≡12,18(mod30).
For:
c≡7,13,17,23(mod30),
the compatible nonzero displacement classes are:
d≡6,24(mod30).
Thus the full residue table is:
Centre residue c(mod30) Primary lane Secondary lanes
1 0 12, 18
7 0 6, 24
11 0 12, 18
13 0 6, 24
17 0 6, 24
19 0 12, 18
23 0 6, 24
29 0 12, 18
5.3 Primary versus secondary selected patterns
In the observed 81×81construction, many selected shells used displacements in the primary lane:
a≡b≡0(mod30).
Then automatically:
a+b≡0(mod30),
and:
a-b≡0(mod30).
This preserves the centre residue for all shell entries.
Secondary-lane shells are also possible, but require more careful compatibility because the four shell distances a,b,a+b,a-bmust each produce prime-compatible residues around c. It is not enough for aalone to be admissible; the whole displacement closure set must be admissible.
The 243×243and 729×729constructions show a stronger selected pattern: all entries are congruent to 13(mod30). This implies strict lane purity at those scales and indicates that disjoint block assembly can operate inside a single residue lane when the block library is chosen appropriately.
This residue analysis does not prove primality, but it removes impossible candidates before primality testing and explains why the 13(mod30)lane becomes structurally attractive in the larger constructions.
________________________________________
6. Verified Unique-Prime Constructions
The project now contains three verified fully nested unique-prime constructions.
6.1 The 81×81construction
The first completed object was a fully nested 81×81prime magic square with:
81^2=6561
entries.
The verified properties were:
all entries are prime;
all entries are globally distinct;
master centre is 800011;
global magic constant is:
64800891=81×800011;
729aligned 3×3blocks verified with 0failures;
81aligned 9×9blocks verified with 0failures;
9aligned 27×27blocks verified with 0failures;
the full 81×81square verified across all rows, all columns, and both main diagonals.
The construction used a nested 27×27prime magic square as the centre grid, then expanded each of its 729entries into a disjoint 3×3prime shell.
6.2 The 243×243construction
The later 243×243construction solved the global uniqueness problem at the next scale.
It has:
243^2=59049
entries.
Verified properties:
all 59049entries are prime;
all 59049entries are globally distinct;
centre is 10000000033;
magic constant is:
2430000008019=243×10000000033;
all entries are congruent to 13(mod30);
all aligned 3×3, 9×9, 27×27, and 81×81blocks verify as magic;
the full 243×243square verifies across rows, columns, and both main diagonals.
The construction was obtained by assembling nine disjoint 81×81prime magic blocks as a 3×3macro-square. Each 81×81block was internally nested and unique, and the nine block centres were arranged so that the macro-level 3×3structure also satisfied the magic condition. This solved the earlier repeated-core problem by replacing reused block cores with disjoint block libraries.
6.3 The 729×729construction
A further 729×729unique-prime construction has also been verified.
It has:
729^2=531441
entries.
Verified properties:
all 531441entries are prime;
all 531441entries are globally distinct;
centre is 10000000033;
magic constant is:
7290000024057=729×10000000033;
all entries are congruent to 13(mod30);
all aligned 3×3, 9×9, 27×27, 81×81, and 243×243blocks verify as magic;
the full 729×729square verifies across rows, columns, and both main diagonals.
The 243×243and 729×729constructions show that the centre-shell sourcing method, when combined with disjoint block assembly and strict residue-lane discipline, scales beyond the original 81×81object. This changes the status of the work from a single construction to a verified recursive construction ladder.
________________________________________
7. Verification Methodology
Verification was treated as an exhaustive computational audit, not as visual inspection or sampling.
For each verified object, the same type of verification was applied at the relevant order.
The verifier checks:
the grid has the claimed shape;
every entry is an integer;
every entry is prime;
all entries are globally distinct;
the centre value is recorded;
the magic constant equals order times centre;
all full-square row sums equal the magic constant;
all full-square column sums equal the magic constant;
both full-square main diagonals equal the magic constant;
every aligned lower-level 3^k×3^kblock verifies as magic;
each aligned block has local magic constant equal to block order times its centre value;
optional residue-lane checks verify whether entries lie in a specified congruence class.
For the 81×81object, this means checking aligned 3×3, 9×9, and 27×27blocks. For the 243×243object, this extends to aligned 81×81blocks. For the 729×729object, this extends to aligned 243×243blocks.
For an aligned block of order mand centre c_B, the expected local magic constant is:
M_B=m" " c_B.
The verification checked rows, columns, and both main diagonals against this expected value at each aligned scale.
The verified 243×243package records overall pass status with all 59049entries prime, unique, and 13(mod30). The verified 729×729package records overall pass status with all 531441entries prime, unique, and 13(mod30).
This methodology is important because nested magic-square claims are easy to state but easy to under-verify. A single full-square magic constant is not enough; the recursive claim requires aligned block verification at every declared scale.
________________________________________
8. Scale Separation and Displacement Behaviour
During the 81×81construction, many successful shells appeared to use two displacements of different scales: one much smaller than the other.
Operationally, this suggested that the two displacements were often playing different roles:
one displacement acted as a local correction or lane-adjustment displacement;
the other acted as a broader band-separation displacement.
A preliminary audit of the 729aligned 3×3shells in the 81×81dataset gives the ratio:
ρ=(min(∣a∣,∣b∣))/(max(∣a∣,∣b∣)).
Observed summary:
median ρ≈0.188;
first quartile ρ≈0.091;
third quartile ρ≈0.381;
minimum ρ≈0.000656;
maximum ρ≈0.994.
This supports the claim that strong scale separation occurs often, but not universally.
Example shells from the first aligned row of 3×3blocks include:
Block Centre c a b ρ c" " mod" " 30 a" " mod" " 30 b" " mod" " 30
(1,1) 802603 2574 26040 0.099 13 24 0
(1,2) 797389 4182 31020 0.135 19 12 0
(1,3) 803911 11760 26538 0.443 1 0 18
(1,4) 799921 11508 28890 0.398 1 18 0
(1,5) 790651 9972 26040 0.383 1 12 0
(1,6) 800281 10260 24918 0.412 1 0 18
(1,7) 806671 4512 18630 0.242 1 12 0
(1,8) 789883 5460 19704 0.277 13 0 24
(1,9) 809869 3828 41700 0.092 19 18 0
(1,10) 479317 10626 49920 0.213 7 6 0
Across all 729shells in this audit, the most common selected displacement lane for both aand bwas 0(mod30), though secondary lanes also occurred.
The later 243×243and 729×729constructions introduce a different phenomenon: global lane purity. At those scales, all entries lie in the 13(mod30)lane. This suggests that scale separation in the initial local shells and residue purity in later block assembly are related but distinct pressures.
________________________________________
9. Failure of Direct 243×243Template Reuse and the Corrective Strategy
The attempted direct lift from 81×81to 243×243exposed an important failure mode.
A natural but flawed strategy was to reuse displacement templates from the successful 81×81construction. Since 81×81contains 729local 3×3shells, the idea was to reuse the corresponding displacement vocabulary while expanding a larger grid of centres.
This failed badly. In one diagnostic attempt, exact reuse of the 729displacement templates from the solved 81×81construction starved 6550of the 6561centres in the proposed larger expansion.
The conclusion was not that higher-order construction is impossible. The conclusion was that recursion does not preserve exact displacement values.
The likely failure mechanisms include scale mismatch, global uniqueness saturation, residue crowding, and local-to-global mismatch.
The resulting methodological correction was:
recursion repeats shell syntax, not displacement numbers.
The reusable structure is:
"centre-shell grammar"+"residue admissibility"+"fertility filtering"+"global ledger control",
not a fixed list of (aⓜ,b)pairs.
The successful 243×243construction came from changing strategy. Instead of reusing local displacement templates across all 6561centres, the construction used disjoint 81×81prime magic blocks as macro-entries in a 3×3structure. This preserved recursive grammar while avoiding the collision collapse caused by literal displacement-template reuse.
The successful 729×729construction continued this correction by assembling nine disjoint 243×243blocks.
Thus the failed 243×243attempt was not a dead end. It identified the wrong scaling mechanism. The correction was to scale by disjoint block assembly rather than by literal displacement-template reuse.
________________________________________
10. Repeated-Core 243×243Scaffold
A separate higher-scale object was first constructed as a repeated-core scaffold.
A repeated-core scaffold is a recursively magic prime-valued square assembled from smaller verified prime magic blocks, but with some lower-level prime entries reused across the full object. Such an object may verify all row, column, diagonal, and aligned nested magic conditions while failing global uniqueness.
The 243×243repeated-core scaffold was produced by block-as-entry assembly: the full square was treated as a 3×3macro-square of 81×81prime magic blocks. Each 81×81block was internally valid, and the nine block centres were arranged so that the macro-level 3×3structure also satisfied the magic condition. Because the block library reused same-core or related 81×81structures, many entries repeated globally even though local and nested magic verification passed.
In the repeated-core scaffold:
total entries were 243^2=59049;
all entries were prime;
rows, columns, diagonals, and aligned nested levels verified;
aligned levels included 3×3, 9×9, 27×27, 81×81, and full 243×243;
distinct primes numbered 16570;
repeat occurrences beyond first numbered 42479.
This object demonstrated that recursive prime-valued structure could be lifted by block assembly. However, it did not solve the globally unique-prime problem. It is now best understood as a transitional scaffold superseded by the later globally unique 243×243construction.
The historical importance of the scaffold is that it separated two questions:
Can recursive prime-valued block assembly preserve magic structure?
Can the same process preserve global uniqueness?
The repeated-core scaffold answered the first question. The later 243×243construction answered the second.
________________________________________
11. Main Insights from the Search
11.1 Prime sourcing is not prime collecting
The construction does not treat primes as isolated entries. A prime is useful only when it can cooperate with other primes under a centre-shell relation or block-assembly role.
The hierarchy is:
"prime"→"symmetric prime-pair centre"→"shell-admissible centre"→"recursively useful centre"→"block-compatible prime set".
11.2 The magic square acts as a coordination sieve
The construction behaves like a sieve, but not merely a divisibility sieve. It filters primes by their ability to participate in coordinated symmetric structures.
A centre cmust support four simultaneous complementary prime-pair distances:
a,b,a+b,∣a-b∣.
At larger scales, an entire block must also cooperate with eight other blocks in a macro-square. This is the block-level version of coordination capacity.
11.3 Modulo 30 is structural, not decorative
Modulo 30appears because primes greater than 5must avoid divisibility by 2, 3, and 5. For symmetric pairs around a centre, displacement residues determine whether both partners remain prime-compatible.
The primary lane:
d≡0(mod30)
preserves centre residue and is therefore especially stable.
The 243×243and 729×729constructions show that full-object lane purity in 13(mod30)is compatible with very large nested unique-prime constructions.
11.4 Global uniqueness dominates local fertility
A centre with many locally valid shells can become unusable if its available shells collide with primes used elsewhere. Similarly, a valid block can become unusable if it overlaps with another block’s prime set.
This distinction between local fertility and usable coordination capacity is essential.
11.5 Recursion repeats grammar, not vocabulary
The failed template reuse at the 243×243level showed that exact displacement values do not scale reliably. The repeatable object is the centre-shell grammar, not the literal displacement list.
The successful larger constructions show the parallel block-level principle:
recursion repeats block grammar, not block content.
The blocks must be disjoint realizations of the same structural role, not repeated copies.
11.6 Ordinary recursive skeletons are scaffolds, not novelty
The ordinary sequential 3^nmagic-square skeleton is useful because it provides block roles, centre hierarchy, and recursive alignment. However, it is not the new object. The contribution lies in prime-shell sourcing, global uniqueness, and disjoint recursive block assembly under that skeleton.
________________________________________
12. Improved Future Search Strategy
A stronger construction pipeline would use the following stages.
Stage 1: centre preselection
Compute F_R (c)for a large pool of prime centres and retain centres with high local fertility.
Stage 2: residue classification
Classify candidate centres by residue modulo 30, then possibly modulo 210, 2310, or higher primorial wheels.
Stage 3: shell-library generation
For each centre, generate a library of admissible (aⓜ,b)pairs and classify each shell by:
radius;
primary or secondary residue lane;
displacement ratio;
overlap risk;
local shell count;
compatibility with neighbouring centre roles.
Stage 4: global ledger scoring
Before accepting a shell, score it against the global ledger. Penalize shells that consume primes appearing in many other candidate shell libraries.
A practical heuristic is:
"cost"(S)=∑_(p∈S)▒1/(A(p)),
where Sis the shell and A(p)is the number of available future shells in which prime pappears. A shell containing rare-but-needed primes should be more expensive than a shell using abundant alternatives.
Stage 5: recursive assembly
Build or select the centre grid before locking shells. The centre grid determines many future constraints, so shell choices should remain flexible until enough global context is known.
Stage 6: disjoint block assembly
For scales above 81×81, generate mutually disjoint block libraries. A candidate block must be evaluated not only by its internal verification, but also by:
overlap with the global forbidden ledger;
compatibility of its centre with macro-square placement;
residue-lane compatibility;
future ability to serve as a block in the next scale.
Stage 7: repair search
When a collision occurs, first replace local shell completions or block choices before disturbing the higher-level centre grid. Centre-grid repair should be reserved for deeper deadlocks.
Stage 8: audit and export
Each completed object should be exported with:
CSV data;
manifest;
SHA-256 checksums;
verification script;
verification summary;
residue-lane audit;
shell-displacement audit;
repetition report if repetitions exist.
________________________________________
13. Limitations
This report does not claim:
a proof of an infinite family;
a theorem about prime distribution;
that modulo-30 primary-lane behaviour is necessary in all constructions;
that ordinary recursive 3^nmagic-square construction is new;
that centre-shell fertility alone guarantees large-scale success;
that disjoint block assembly must continue indefinitely.
The existence of verified 81×81, 243×243, and 729×729objects does not by itself imply abundance, density, or infinite extendability of such objects.
It claims only:
Fully nested unique-prime magic squares were constructed and verified at orders 81×81, 243×243, and 729×729.
The 81×81construction used 3×3centre-shell prime expansion.
The 243×243and 729×729constructions used disjoint recursive block assembly.
Prime sourcing was governed by shell fertility, residue admissibility, recursive centre grids, block compatibility, and global uniqueness pressure.
Failed extensions produced useful information about scaling limitations.
The best current interpretation is that recursive prime magic-square construction repeats a sourcing grammar, not fixed displacement templates.
________________________________________
14. Conclusion
The primes in the recursive 3^nprime magic-square constructions were sourced by treating each candidate centre or block as a possible participant in a larger symmetric structure. At the local level, the shell condition required the eight shifted values
c±a,c±b,c±(a+b),c±(a-b)
to be prime and globally non-repeating.
At higher levels, the successful 243×243and 729×729constructions show that the same principle can operate through disjoint block assembly. Blocks replace individual entries as macro-objects, but the same constraints remain: magic compatibility, centre consistency, primality, residue discipline, and global non-reuse.
The central methodological conclusion is:
recursive prime magic-square construction repeats grammar, not numerical templates.
The ordinary recursive 3^nmagic-square skeleton supplies the role field. The prime-shell search and block-disjoint assembly supply the content. The verified 81×81, 243×243, and 729×729constructions show that this grammar can operate at substantial scale.
This is therefore best understood as a computational construction report and a methodology proposal: a way of sourcing primes by structural cooperation under recursive magic-square constraints.
________________________________________
15. Reproducibility Package
A publication or archive version of this report should be accompanied by reproducibility packages. The purpose of these packages is to make the constructions independently checkable without relying on trust in the narrative.
The minimum package for each verified object should contain:
the completed CSV dataset;
a manifest recording order, centre value, magic constant, entry count, and file checksum;
a verifier script;
a verification output file;
a checksum file for all submitted artifacts;
the companion report in PDF or LaTeX form.
The verifier should check, at minimum:
the grid has the stated shape;
every entry is an integer;
every entry is prime;
all entries are globally distinct;
all row sums equal the relevant magic constant;
all column sums equal the relevant magic constant;
both main diagonal sums equal the relevant magic constant;
all aligned lower-order blocks verify as magic;
each aligned block has local magic constant equal to block order times its centre value;
optional residue-lane properties, such as all entries congruent to 13(mod30).
The verification output should be plain and machine-readable, preferably JSON, with each test recorded as passed or failed. A human-readable summary may also be included.
A suitable data-availability statement is:
The completed 81×81unique-prime magic square dataset is available at Zenodo under DOI https://doi.org/10.5281/zenodo.20005776
. The 243×243and 729×729verifier packages should be archived with the same project record or a linked follow-up record. The submitted archive includes CSV grids, checksum data, and verification materials sufficient to reproduce the primality, uniqueness, residue-lane, and nested magic-square checks.
This section is included to separate the mathematical report from the computational evidence. The paper explains the construction; the reproducibility packages allow the objects to be audited.
________________________________________
16. Submission Positioning
This work is best submitted as a construction report in recreational mathematics or computational combinatorial number theory. It should not be framed as a proof of an infinite family, nor as the invention of nested magic squares or prime magic squares. The defensible claim is narrower and stronger:
Verified fully nested unique-prime magic squares of orders 81×81, 243×243, and 729×729were constructed using centre-shell sourcing and disjoint recursive block assembly, and the report formalizes the constructional concepts of prime-shell fertility and coordination capacity under global uniqueness constraints.
Suitable submission categories include:
recreational mathematics;
computational number theory;
combinatorial constructions;
magic-square theory;
prime-pattern constructions.
The strongest submission version should include both the report and the verifier packages. Without the verifier, the paper is a plausible construction narrative. With the verifier, it becomes an auditable mathematical artifact.
________________________________________
17. Open Problems and Future Questions
The construction suggests several natural problems.
Existence at the next scale. Given verified 81×81, 243×243, and 729×729unique-prime constructions, does there exist a fully nested globally unique-prime 2187×2187magic square under the same recursive grammar?
Asymptotic coordination capacity. Can the expected size of K_R (c;C)be estimated for large centres cunder realistic global ledger constraints?
Minimality. What is the smallest possible master centre, magic constant, or maximum entry for a fully nested 81×81, 243×243, or 729×729unique-prime magic square of this type?
Residue-lane optimization. Are primary-lane shells and 13(mod30)block assemblies merely convenient, or do successful large-scale constructions require a dominant lane-preserving component?
Scalable repair. Can shell selection and block selection be formulated as exact-cover, matching, or hypergraph-packing problems, allowing systematic repair instead of greedy rejection?
Beyond powers of three. Can analogous centre-shell fertility methods be developed for recursive constructions based on other odd orders, such as 5^n, p^nfor odd primes p, or composite odd orders? Or is the 3^nhierarchy unusually natural because of the unique 3×3centre-shell primitive?
Rectangular or mixed-scale variants. Can related methods produce prime-valued magic rectangles, semimagic arrays, or nested 3^nby 3^mstructures with compatible line sums?
Independent block libraries. The 243×243construction demonstrates that nine mutually disjoint 81×81blocks can be assembled successfully. The 729×729construction demonstrates the same principle one level higher using 243×243blocks. How large can such mutually disjoint block libraries become, and can sufficiently many disjoint 729×729blocks be generated for the 2187×2187scale?
These questions separate computational next steps from theorem-level ambitions. The immediate constructive target has moved beyond the globally unique 243×243case. The next constructive target is the 2187×2187scale, while the theoretical target remains a usable estimate of coordination capacity under global constraints.
________________________________________
References
Dudeney, H. E. Amusements in Mathematics. Thomas Nelson and Sons, 1917. Relevant for early recreational prime magic-square problems.
Madachy, J. S. Madachy’s Mathematical Recreations. Dover Publications, 1979. Includes discussion of recreational magic-square constructions and prime-square examples, including Ondrejka-related material.
Weisstein, E. W. “Prime Magic Square.” MathWorld--A Wolfram Web Resource. Provides a concise summary of prime magic-square examples and historical notes.
Johnson, A. W. Jr. “Consecutive-Prime Magic Squares.” Journal of Recreational Mathematics, vol. 15, no. 1, 1982–1983, pp. 17–18.
Johnson, A. W. Jr. “A Bordered Prime Magic Square.” Journal of Recreational Mathematics, vol. 15, no. 2, 1982–1983, p. 84.
Skelt, A. L.; Perkins, S.; Roach, P. A. “Prime Strictly Concentric Magic Squares of Odd Order.” Mathematics, vol. 13, no. 8, article 1261, 2025. DOI: 10.3390/math13081261.
Skelt, A. L.; Perkins, S.; Roach, P. A. “Critical Sets and Unavoidable Sets of Strictly Concentric Magic Squares of Odd Order and Their Application to Prime Strictly Concentric Magic Squares of Order 5.” Axioms, vol. 14, no. 8, article 607, 2025. DOI: 10.3390/axioms14080607.
Pickover, C. A. The Zen of Magic Squares, Circles, and Stars. Princeton University Press, 2002. General background on magic squares and related recreational structures.
Andrews, W. S. Magic Squares and Cubes. Open Court Publishing, 1917. Classic early source on magic-square theory.
Guy, R. K. Unsolved Problems in Number Theory. Springer. Relevant background for prime patterns and number-theoretic caution, though not specifically a magic-square construction source.
Tao, T.; Vu, V. Additive Combinatorics. Cambridge University Press, 2006. Useful background for additive constraints and structured number sets, included as conceptual context rather than direct prior art.
Green, B.; Tao, T. “The primes contain arbitrarily long arithmetic progressions.” Annals of Mathematics, vol. 167, no. 2, 2008, pp. 481–547. Relevant as distant context for structured additive patterns in primes, not as direct magic-square prior art.
The final publication version should verify all bibliographic details against primary sources. The present list is a working bibliography designed to replace informal prior-art pointers with citable anchors.