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

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

 Author: Roberto Carlo Angelone

 Dataset DOI:

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

 Follow-up to:

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

ABSTRACT

 This dataset presents a fully nested 243 x 243 magic square whose 59,049 entries are all distinct prime numbers. It is a follow-up to the previously published 81 x 81 unique-prime magic square dataset:

 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 earlier 81 x 81 construction established the recursive 3 x 3 centre-shell expansion method and produced a fully nested 81 x 81 magic square with 6,561 globally distinct prime entries.

 The present construction extends that framework one recursive level further by assembling nine mutually disjoint 81 x 81 unique-prime magic-square blocks. The centres of the nine 81 x 81 blocks are themselves arranged as a 3 x 3 prime magic square. Each 81 x 81 block therefore functions as a macro-entry inside the larger 243 x 243 construction.

 The resulting 243 x 243 square has master centre 10,000,000,033 and magic constant 2,430,000,008,019 = 243 x 10,000,000,033.

 The square verifies at all aligned recursive levels: 3 x 3, 9 x 9, 27 x 27, 81 x 81, and the full 243 x 243 square. All 59,049 entries are prime and globally unique.

 This release is a constructional extension of the earlier 81 x 81 dataset. It does not claim a proof of an infinite family or a theorem about prime distribution. It demonstrates that the recursive centre-shell prime magic-square method can be scaled from 81 x 81 to 243 x 243 when the construction is organized through coordinated 81 x 81 macro-blocks with disjoint prime entries.

MAIN VERIFIED DATA

Square order: 243 x 243

 Total entries: 59,049

 All entries prime: yes

 All entries globally unique: yes

Master centre: 10,000,000,033

 Magic constant: 2,430,000,008,019

 Minimum entry: 9,081,748,723

 Maximum entry: 10,915,510,273

VERIFICATION METHOD

 The 243 x 243 square was independently verified from the uploaded CSV file.

 The verification checked:

 All 59,049 entries were tested for primality.

 All 59,049 entries were checked for global uniqueness.

 All 243 rows were checked against the global magic constant.

 All 243 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, and 81 x 81 sub-squares were checked recursively.

 The primality verification was performed computationally using exact integer primality testing as implemented in SymPy's isprime function. All nine 81 x 81 block centres were also individually checked as prime.

RECURSIVE VERIFICATION

 The square was checked at all aligned recursive levels:

 6,561 aligned 3 x 3 blocks checked, with 0 failures

 729 aligned 9 x 9 blocks checked, with 0 failures

 81 aligned 27 x 27 blocks checked, with 0 failures

 9 aligned 81 x 81 blocks checked, with 0 failures

Full 243 x 243 rows, columns, and both main diagonals checked, with 0 failures

RELATIONSHIP TO THE PREVIOUS 81 x 81 DATASET

 This dataset follows the earlier release:

 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 previous 81 x 81 dataset established the local recursive construction mechanism. In that construction, each 3 x 3 block is generated from a centre c and two integer displacements a and b using the centre-shell form:

 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 local 3 x 3 shell sums to 3c.

 The 81 x 81 construction used a nested 27 x 27 prime magic square as a grid of 729 prime centres, then expanded each centre into a disjoint 3 x 3 prime shell. That produced a fully nested 81 x 81 magic square with 6,561 distinct prime entries.

 The present 243 x 243 construction extends the same recursive principle, but at the block level. Instead of directly expanding 6,561 individual centres into local shells, the construction uses nine complete 81 x 81 prime magic-square blocks. These nine blocks are arranged as a 3 x 3 macro-square whose entries are the centres of the 81 x 81 blocks.

 Thus the relationship between the two constructions is:

 81 x 81 construction:

27 x 27 centre grid -> 729 local 3 x 3 prime shells

 243 x 243 construction:

3 x 3 macro-grid of 81 x 81 unique-prime magic-square blocks

 The 243 x 243 square is therefore a follow-up construction, not a replacement of the 81 x 81 dataset. It extends the same centre-shell grammar to the next recursive scale.

MACRO-CENTRE STRUCTURE

 The nine 81 x 81 block centres form the following 3 x 3 prime magic square:

 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 sums to:

 30,000,000,099 = 3 x 10,000,000,033

 The nine block centres correspond to 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

 All nine macro-centres were individually verified as prime by computational primality testing.

 This means the nine 81 x 81 blocks are not merely collected side by side. Their centres are harmonized through a higher-level 3 x 3 prime magic-square structure.

BLOCK-LEVEL CONSTRUCTION AND PRIME BANDS

 Each of the nine 81 x 81 blocks is itself a fully nested unique-prime magic square. The blocks are mutually disjoint: no prime entry occurs in more than one 81 x 81 block.

 The blocks were generated in separated prime bands around their assigned macro-centres. This band separation helped prevent collisions between prime entries in different 81 x 81 blocks while preserving the recursive centre-shell structure inside each block.

 The nine block centres, approximate value ranges, and 81 x 81 magic constants are:

Block 1

Centre: 10,216,927,153

Minimum entry: 10,205,591,833

Maximum entry: 10,227,869,173

81 x 81 magic constant: 827,571,099,393

Block 2

Centre: 9,095,500,783

Minimum entry: 9,081,748,723

Maximum entry: 9,107,397,283

81 x 81 magic constant: 736,735,563,423

Block 3

Centre: 10,687,572,163

Minimum entry: 10,675,325,533

Maximum entry: 10,698,944,743

81 x 81 magic constant: 865,693,345,203

Block 4

Centre: 10,470,645,043

Minimum entry: 10,454,647,363

Maximum entry: 10,483,680,643

81 x 81 magic constant: 848,122,248,483

Block 5

Centre: 10,000,000,033

Minimum entry: 9,985,393,183

Maximum entry: 10,014,797,323

81 x 81 magic constant: 810,000,002,673

Block 6

Centre: 9,529,355,023

Minimum entry: 9,518,038,093

Maximum entry: 9,542,331,973

81 x 81 magic constant: 771,877,756,863

Block 7

Centre: 9,312,427,903

Minimum entry: 9,298,776,853

Maximum entry: 9,326,088,703

81 x 81 magic constant: 754,306,660,143

Block 8

Centre: 10,904,499,283

Minimum entry: 10,893,538,723

Maximum entry: 10,915,510,273

81 x 81 magic constant: 883,264,441,923

Block 9

Centre: 9,783,072,913

Minimum entry: 9,769,708,423

Maximum entry: 9,796,896,673

81 x 81 magic constant: 792,428,905,953

 Each block contains 6,561 entries, all prime and all distinct within the block. Across the nine blocks, the combined 59,049 entries are also globally distinct.

 Because the nine block centres form a 3 x 3 magic square, the assembled 243 x 243 square has global magic constant:

 81 x 30,000,000,099 = 2,430,000,008,019

 Equivalently:

 243 x 10,000,000,033 = 2,430,000,008,019

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 81 x 81 blocks.

 Second, for each macro-centre, a full 81 x 81 unique-prime magic square was generated using recursive 3 x 3 centre-shell expansion. Each block was generated in a separate prime band so that its entries would not collide with entries in the other blocks.

 Third, the nine 81 x 81 blocks were assembled according to the 3 x 3 macro-centre shell. Because each 81 x 81 block has magic constant equal to 81 times its centre, and because the nine centres form a 3 x 3 magic square, the assembled 243 x 243 square inherits the correct row, column, and diagonal sums at the macro level.

 Fourth, the completed 243 x 243 square was verified directly for primality, global uniqueness, row sums, column sums, diagonals, and aligned recursive block sums at 3 x 3, 9 x 9, 27 x 27, and 81 x 81 levels.

WHAT THIS CONSTRUCTION DEMONSTRATES

 This construction demonstrates that the previous 81 x 81 recursive prime magic-square method can be scaled to 243 x 243 when the construction is organized at the block level.

 The key scaling issue is global uniqueness. Directly expanding a large centre grid into thousands of local prime shells creates severe collision pressure between repeated prime entries. The present construction avoids that by coordinating nine independently generated 81 x 81 blocks through a prime macro-centre shell and by ensuring that the prime entries of all nine blocks are mutually disjoint.

 The result supports the working framework introduced in the previous 81 x 81 companion note:

prime-shell fertility

 centre-shell expansion

 residue-lane admissibility

 scale-separated displacement structure

 global uniqueness pressure

 recursive block-centre coordination

 The construction does not claim a proof of an infinite family or a theorem about prime distribution. It is a verified computational construction and a further data point for studying recursive prime magic-square structures.

SCOPE AND LIMITATIONS

 This dataset claims:

 a verified 243 x 243 magic square

 59,049 globally distinct prime entries

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

 a constructional extension of the earlier 81 x 81 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

PLAIN-ENGLISH SUMMARY

 This dataset gives a 243 x 243 magic square made entirely from prime numbers, with no prime repeated anywhere in the square.

 It extends a previous 81 x 81 prime magic-square construction by using nine separate 81 x 81 prime magic squares as blocks. The centres of those nine blocks are arranged as a smaller 3 x 3 prime magic square, allowing the whole 243 x 243 structure to remain magic and recursively nested.

 The square contains 59,049 distinct prime entries and has been checked at every aligned nested level: 3 x 3, 9 x 9, 27 x 27, 81 x 81, and 243 x 243.