A Fully Nested 729 x 729 Unique-Prime Magic Square Constructed from Nine Correlated 243 x 243 Prime Magic Blocks

A Fully Nested 729 x 729 Unique-Prime Magic Square Constructed from Nine Correlated 243 x 243 Prime Magic Blocks

Author: Roberto Carlo Angelone

 Dataset DOI: https://doi.org/10.5281/zenodo.20040831

Related datasets:

 A Fully Nested 81 x 81 Unique-Prime Magic Square Constructed by Recursive 3 x 3 Centre-Shell Expansion

DOI: https://doi.org/10.5281/zenodo.20005776

 A Fully Nested 243 x 243 Unique-Prime Magic Square Constructed from Nine Disjoint 81 x 81 Prime Magic Blocks

DOI: https://doi.org/10.5281/zenodo.20037509

ABSTRACT

 This dataset presents a fully nested 729 x 729 magic square whose 531,441 entries are all distinct prime numbers.

 The construction extends two previous releases: the fully nested 81 x 81 unique-prime magic square and the fully nested 243 x 243 unique-prime magic square. The present 729 x 729 construction is built from nine correlated 243 x 243 prime magic-square blocks. These nine 243 x 243 blocks are arranged as a 3 x 3 macro-square whose block centres themselves form a prime magic square.

 The resulting 729 x 729 square has master centre 10,000,000,033 and magic constant:

 7,290,000,024,057 = 729 x 10,000,000,033

 The square verifies at every aligned recursive level:

 3 x 3

9 x 9

27 x 27

81 x 81

243 x 243

729 x 729

 All 531,441 entries are prime, globally unique, and final-digit lane pure: every prime entry ends in the digit 3.

 This release is a computational construction and dataset. It does not claim a proof of an infinite family, a theorem about prime distribution, or optimality of the chosen entries. It demonstrates that the recursive centre-shell construction method can be scaled through the verified sequence 81 x 81, 243 x 243, and 729 x 729 when the construction is organized through correlated prime block-centres and globally disjoint prime entries.

MAIN VERIFIED DATA

 Square order: 729 x 729

 Total entries: 531,441

 All entries prime: yes

 All entries globally unique: yes

 All entries end in digit 3: yes

 Master centre: 10,000,000,033

 Magic constant: 7,290,000,024,057

 Minimum entry: 9,082,712,503

 Maximum entry: 10,917,195,103

INDEPENDENT VERIFICATION SUMMARY

 The uploaded CSV was independently checked from the data file.

 The actual 729 x 729 square was verified directly. If the CSV version includes marginal row or column sums, verification is performed on the top-left 729 x 729 data region only.

 The verification checked:

 All 531,441 entries were tested for primality.

 All 531,441 entries were checked for global uniqueness.

 All 729 rows were checked against the global magic constant.

 All 729 columns were checked against the global magic constant.

 Both main diagonals were checked against the global magic constant.

 All aligned 3 x 3, 9 x 9, 27 x 27, 81 x 81, and 243 x 243 sub-squares were checked recursively.

 All nested block centres were checked for primality and uniqueness at their respective aligned levels.

 The primality verification was performed computationally using exact integer primality testing as implemented in SymPy's isprime function.

FULL MAGIC-SQUARE VERIFICATION

 Centre:

 10,000,000,033

 Expected magic constant:

 729 x 10,000,000,033 = 7,290,000,024,057

 Rows verified: yes

 Columns verified: yes

 Main diagonal verified: yes

 Other main diagonal verified: yes

 Main diagonal sum:

 7,290,000,024,057

 Other main diagonal sum:

 7,290,000,024,057

 RECURSIVE NESTED VERIFICATION

 The square was checked at all aligned recursive levels:

 59,049 aligned 3 x 3 blocks checked, with 0 failures

 6,561 aligned 9 x 9 blocks checked, with 0 failures

 729 aligned 27 x 27 blocks checked, with 0 failures

 81 aligned 81 x 81 blocks checked, with 0 failures

 9 aligned 243 x 243 blocks checked, with 0 failures

 1 aligned 729 x 729 block checked, with 0 failures

 NESTED CENTRE VERIFICATION

 The aligned block centres were also checked.

 3 x 3 block centres: 59,049 distinct centres, all prime

 9 x 9 block centres: 6,561 distinct centres, all prime

 27 x 27 block centres: 729 distinct centres, all prime

 81 x 81 block centres: 81 distinct centres, all prime

 243 x 243 block centres: 9 distinct centres, all prime

 729 x 729 centre: 1 centre, prime

RELATIONSHIP TO PREVIOUS RELEASES

 This 729 x 729 construction follows two earlier datasets.

 The first release was an 81 x 81 fully nested unique-prime magic square:

 A Fully Nested 81 x 81 Unique-Prime Magic Square Constructed by Recursive 3 x 3 Centre-Shell Expansion

DOI: https://doi.org/10.5281/zenodo.20005776

 The second release extended the method to 243 x 243:

 A Fully Nested 243 x 243 Unique-Prime Magic Square Constructed from Nine Disjoint 81 x 81 Prime Magic Blocks

DOI: https://doi.org/10.5281/zenodo.20037509

 The present 729 x 729 dataset extends the same recursive construction framework one further level. The verified ladder is therefore:

 81 x 81

243 x 243

729 x 729

 The 729 x 729 square is not a replacement for the earlier datasets. It is a follow-up construction at the next recursive scale.

 LOCAL CENTRE-SHELL CONSTRUCTION RULE

 The basic local unit is the 3 x 3 centre-shell magic-square form.

 Given a centre c and two integer displacements a and b, the local shell is:

 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 of this 3 x 3 shell sums to:

 3c

 In this dataset, the entries in each shell are chosen so that all shell values are prime. The construction then repeats the same centre-shell grammar through aligned powers of 3.

 AFFINE 3 x 3 POSITIONAL GRAMMAR

 The local centre-shell form can be viewed as an affine 3 x 3 positional grammar naturally indexed by the grid structure of (Z/3Z)^2.

 The centre occupies the zero position. The eight surrounding positions are assigned centre-relative displacements in opposite-pair balance. This affine 3 x 3 grammar helps explain why the construction naturally scales through powers of 3:

 3 x 3

9 x 9

27 x 27

81 x 81

243 x 243

729 x 729

 This observation concerns the positional and recursive grammar of the construction. The primality and global uniqueness requirements remain separate arithmetic constraints that must be satisfied computationally.

 MACRO 243-BLOCK CENTRE STRUCTURE

 The 729 x 729 square is assembled as a 3 x 3 arrangement of nine 243 x 243 prime magic-square blocks.

 The nine 243 x 243 block centres are:

 10,216,927,153     9,095,500,783    10,687,572,163

10,470,645,043    10,000,000,033     9,529,355,023

 9,312,427,903    10,904,499,283     9,783,072,913

 Each row, column, and diagonal of this 3 x 3 macro-centre square sums to:

 30,000,000,099

 This equals:

 3 x 10,000,000,033

 Thus the nine 243 x 243 blocks are not merely placed side by side. Their centres are coordinated through a higher-level 3 x 3 prime magic-square structure.

MACRO-CENTRE DISPLACEMENT FORM

 The nine macro-centres follow the same centre-shell form:

 C + A       C - A - B     C + B

C - A + B   C             C + A - B

C - B       C + A + B     C - A

 where:

 C = 10,000,000,033

 A = 216,927,120

 B = 687,572,130

 This gives the nine 243 x 243 block centres listed above.

 243 x 243 BLOCK CONSTANTS

 Since each 243 x 243 block has magic constant equal to 243 times its centre, the nine block magic constants are:

Block 1 centre: 10,216,927,153

Block 1 magic constant: 2,482,713,298,179

 Block 2 centre: 9,095,500,783

Block 2 magic constant: 2,210,206,690,269

 Block 3 centre: 10,687,572,163

Block 3 magic constant: 2,597,080,035,609

 Block 4 centre: 10,470,645,043

Block 4 magic constant: 2,544,366,745,449

 Block 5 centre: 10,000,000,033

Block 5 magic constant: 2,430,000,008,019

 Block 6 centre: 9,529,355,023

Block 6 magic constant: 2,315,633,270,589

 Block 7 centre: 9,312,427,903

Block 7 magic constant: 2,262,919,980,429

 Block 8 centre: 10,904,499,283

Block 8 magic constant: 2,649,793,325,769

 Block 9 centre: 9,783,072,913

Block 9 magic constant: 2,377,286,717,859

WHY THE 729 x 729 MAGIC CONSTANT FOLLOWS

 Each 243 x 243 block has row and column sums equal to 243 times its own centre.

 At the 729 x 729 level, each global row passes through three 243 x 243 blocks. The sum of the three relevant block centres in any macro-row is:

 30,000,000,099

 Therefore each full 729-entry row has sum:

 243 x 30,000,000,099 = 7,290,000,024,057

 Equivalently:

 729 x 10,000,000,033 = 7,290,000,024,057

 The same reasoning applies to columns and the two main diagonals.

FINAL-DIGIT LANE PURITY

 An additional feature of the construction is final-digit lane purity.

 Every one of the 531,441 prime entries ends in the digit 3.

This is not required by the magic-square condition alone. It follows from the chosen centre-shell displacement grammar: the centres are congruent to 3 modulo 10 and the shell displacements are multiples of 10. Therefore all values of the form:

 c + a

c - a

c + b

c - b

c + a + b

c - a - b

c + a - b

c - a + b

 remain congruent to 3 modulo 10.

 Since all entries are prime and greater than 5, this final-digit lane is compatible with admissible prime residue classes. This note claims final-digit purity. It does not require the stronger claim that all entries lie in a single residue class modulo 30 unless that separate mod-30 distribution is explicitly verified and recorded.

 CONSTRUCTION METHOD

 The construction uses a recursive block-as-entry strategy.

 First, a 3 x 3 prime macro-centre shell was selected. Its nine entries became the required centres of the nine 243 x 243 blocks.

 Second, each 243 x 243 block was constructed as a 3 x 3 arrangement of nine 81 x 81 unique-prime magic-square blocks.

 Third, each 81 x 81 block was constructed using the recursive centre-shell method established in the earlier 81 x 81 and 243 x 243 releases.

 Fourth, all prime entries were managed under a shared global uniqueness ledger so that no prime entry occurred more than once anywhere in the final 729 x 729 square.

 Fifth, the completed 729 x 729 square was independently verified for primality, global uniqueness, row sums, column sums, diagonal sums, and aligned recursive block sums at all nested levels.

 WHAT THIS CONSTRUCTION DEMONSTRATES

 This construction demonstrates that the recursive unique-prime magic-square method can be scaled at least through order 729 when the construction is organized through correlated block-centres and globally disjoint prime entries.

 The verified construction sequence now includes:

 81 x 81 with 6,561 distinct prime entries

 243 x 243 with 59,049 distinct prime entries

 729 x 729 with 531,441 distinct prime entries

 The 729 x 729 construction is therefore not just a larger array. It is a six-level aligned recursive structure built from prime entries while preserving magic-square sums and global uniqueness at every relevant scale.

 CONJECTURAL OUTLOOK

 The verified cases at orders 81, 243, and 729 suggest the conjecture that fully nested unique-prime magic squares may exist for further powers of 3.

 This dataset does not prove that conjecture.

 The result should be understood as a verified computational construction and as evidence that the centre-shell method, combined with correlated block-centre planning and global prime-disjointness management, can scale beyond the previously published 81 x 81 and 243 x 243 cases.

SCOPE AND LIMITATIONS

 This dataset claims:

 a verified 729 x 729 magic square

 531,441 globally distinct prime entries

 all entries prime

 all entries globally unique

 all entries ending in digit 3

 full aligned recursive nesting at 3 x 3, 9 x 9, 27 x 27, 81 x 81, 243 x 243, and 729 x 729 levels

 a constructional extension of the earlier 81 x 81 and 243 x 243 centre-shell framework

This dataset does not claim:

 a proof of infinitely many such squares

 a proof that every order 3^n can be constructed in this way

 a theorem about prime distribution

 uniqueness of the method

 minimality of the prime entries

 optimality of the chosen centres or displacements

 a formal proof that the method must always scale

AI-ASSISTED EXPLORATION STATEMENT

 This dataset was produced through human-directed, AI-assisted mathematical exploration.

 The human contributor directed the construction strategy, identified the recursive centre-shell framework, requested the scale-up from 81 x 81 to 243 x 243 and then to 729 x 729, and guided the correlated block-centre approach.

 AI tools were used to assist with computational search, construction management, verification, failure analysis, documentation, and red-team review.

 The final dataset is presented as a computational construction with explicit verification data, not as an automated proof or as a theorem about all possible cases.

PLAIN-ENGLISH SUMMARY

 This dataset contains a 729 x 729 magic square made entirely from prime numbers.

 It has 531,441 entries. Every entry is prime. No prime is repeated. Every entry ends in the digit 3.

 The square is also nested. Smaller aligned squares inside it are magic squares too. This holds at the following levels:

 3 x 3

9 x 9

27 x 27

81 x 81

243 x 243

729 x 729

 The construction extends two earlier datasets: an 81 x 81 unique-prime magic square and a 243 x 243 unique-prime magic square.

 The 729 x 729 square is built from nine correlated 243 x 243 prime magic-square blocks. The centres of those nine blocks themselves form a 3 x 3 prime magic square.

 The result is a verified recursive prime magic-square structure with 531,441 globally unique prime entries.