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.