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.基于9个关联243×243素数幻方块构建的全嵌套729×729唯一素数幻方 作者：罗伯托·卡洛·安杰洛内（Roberto Carlo Angelone） 数据集DOI：https://doi.org/10.5281/zenodo.20040831 相关数据集： 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 摘要 本数据集包含一个全嵌套729×729幻方（magic square），其531441个元素均为互不相同的素数。 本次构建基于两项既往发布成果：全嵌套81×81唯一素数幻方与全嵌套243×243唯一素数幻方。本729×729幻方由9个关联243×243素数幻方块构建而成，这9个243×243块按3×3宏方阵排列，其块中心本身构成一个素数幻方。 所得729×729幻方的主中心为10000000033，幻和（magic constant）为7290000024057，即729×10000000033。 该幻方在所有对齐递归层级均通过验证：3×3、9×9、27×27、81×81、243×243及729×729层级。 所有531441个元素均为素数、全局唯一且具备末位纯一特性：每个素数元素均以数字3结尾。 本发布成果为一项计算构建结果与数据集，并未声称证明了无穷多此类幻方的存在、素数分布相关定理，亦未声称所选元素具有最优性。本成果证明，当构建采用关联素数块中心与全局互不重叠的素数元素时，递归中心-壳层构建方法可按已验证的序列81×81、243×243、729×729进行扩展。 主要验证数据 方阵阶数：729×729 总元素数：531441 所有元素均为素数：是 所有元素全局唯一：是 所有元素末位均为3：是 主中心：10000000033 幻和：7290000024057 最小元素：9082712503 最大元素：10917195103 独立验证总结 上传的CSV文件已通过原始数据文件进行独立校验。实际729×729幻方已直接完成验证。若CSV版本包含边缘行或列和，则仅对左上角729×729数据区域进行验证。 本次验证涵盖以下内容： 所有531441个元素均进行了素性测试。 所有531441个元素均进行了全局唯一性校验。 针对全局幻和校验了全部729行的和。 针对全局幻和校验了全部729列的和。 针对全局幻和校验了两条主对角线的和。 递归校验了所有对齐的3×3、9×9、27×27、81×81及243×243子方阵。 在各自对齐层级上，校验了所有嵌套块中心的素性与唯一性。 素性验证采用SymPy库的isprime函数实现的精确整数素性测试完成。 完整幻方验证 中心：10000000033 预期幻和：729×10000000033 = 7290000024057 行和验证：通过 列和验证：通过 主对角线和验证：通过 副对角线和验证：通过 主对角线和：7290000024057 副对角线和：7290000024057 递归嵌套验证 该幻方在所有对齐递归层级均完成校验： 59049个对齐3×3块校验，无失败案例 6561个对齐9×9块校验，无失败案例 729个对齐27×27块校验，无失败案例 81个对齐81×81块校验，无失败案例 9个对齐243×243块校验，无失败案例 1个对齐729×729块校验，无失败案例 嵌套中心验证 对齐块中心亦完成校验。 3×3块中心：59049个不同中心，均为素数 9×9块中心：6561个不同中心，均为素数 27×27块中心：729个不同中心，均为素数 81×81块中心：81个不同中心，均为素数 243×243块中心：9个不同中心，均为素数 729×729中心：1个中心，为素数 与既往发布成果的关系 本729×729构建基于两项早期数据集。 第一项发布成果为81×81全嵌套唯一素数幻方： 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 第二项发布成果将该方法扩展至243×243阶： 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 本729×729数据集将同一递归构建框架再扩展一个层级，因此已验证的构建序列为： 81×81 → 243×243 → 729×729 本729×729幻方并非对早期数据集的替代，而是下一递归尺度下的后续构建成果。 局部中心-壳层构建规则 基本局部单元为3×3中心-壳层幻方形式。 给定中心c与两个整数偏移量a和b，局部壳层为： c + a c - a - b c + b c - a + b c c + a - b c - b c + a + b c - a 该3×3壳层的每行、每列及对角线和均为：3c 在本数据集中，每个壳层的元素均被选为素数。随后构建过程将以3的幂次对齐的方式，重复应用同一中心-壳层构建规则。 仿射3×3位置语法 局部中心-壳层形式可视为自然以(Z/3Z)^2网格结构索引的仿射3×3位置语法。 中心占据零位置，其余8个周边位置以成对相反的方式分配相对中心的偏移量。该仿射3×3语法可解释为何构建过程可自然按3的幂次扩展： 3×3 → 9×9 → 27×27 → 81×81 → 243×243 → 729×729 该结论涉及构建的位置与递归语法，而素性与全局唯一性要求仍为需通过计算满足的独立算术约束。 243块宏中心结构 729×729幻方由9个243×243素数幻方块按3×3排列组装而成。 这9个243×243块的中心为： 10216927153 9095500783 10687572163 10470645043 10000000033 9529355023 9312427903 10904499283 9783072913 该3×3宏中心方阵的每行、每列及对角线和均为：30000000099，即3×10000000033。 由此可见，这9个243×243块并非简单并排摆放，其中心通过更高层级的3×3素数幻方结构进行关联协调。 宏中心偏移形式 9个宏中心遵循同一中心-壳层形式： C + A C - A - B C + B C - A + B C C + A - B C - B C + A + B C - A 其中： C = 10000000033 A = 216927120 B = 687572130 由此得到上述列出的9个243×243块中心。 243×243块幻和 由于每个243×243块的幻和等于其中心值的243倍，因此9个块的幻和分别为： 块1中心：10216927153，块1幻和：2482713298179 块2中心：9095500783，块2幻和：2210206690269 块3中心：10687572163，块3幻和：2597080035609 块4中心：10470645043，块4幻和：2544366745449 块5中心：10000000033，块5幻和：2430000008019 块6中心：9529355023，块6幻和：2315633270589 块7中心：9312427903，块7幻和：2262919980429 块8中心：10904499283，块8幻和：2649793325769 块9中心：9783072913，块9幻和：2377286717859 729×729幻和的由来 每个243×243块的行和与列和均等于其自身中心值的243倍。 在729×729层级中，每条全局行穿过3个243×243块。任意宏行中3个相关块中心的和为：30000000099。 因此，每条完整的729元素行的和为：243×30000000099 = 7290000024057，等价于729×10000000033。 该推导同样适用于列与两条主对角线。 末位纯一特性 本构建的另一特性为末位纯一特性。 全部531441个素数元素均以数字3结尾。 这并非幻方条件本身的要求，而是源于所选的中心-壳层偏移语法：中心值模10余3，且壳层偏移量均为10的倍数。因此所有形如c+a、c-a、c+b、c-b、c+a+b、c-a-b、c+a-b、c-a+b的值均模10余3。 由于所有元素均为大于5的素数，该末位特性与合法素数剩余类兼容。本说明仅声称末位纯一特性，并未声称所有元素均属于模30的单一剩余类，除非该模30分布已得到显式验证与记录。 构建方法 本构建采用递归块作为元素的策略。 首先，选择一个3×3素数宏中心壳层，其9个元素将作为9个243×243块的所需中心。 其次，将每个243×243块构建为9个81×81唯一素数幻方块按3×3排列的形式。 再次，每个81×81块采用早期81×81与243×243发布成果中确立的递归中心-壳层方法进行构建。 随后，所有素数元素均通过共享的全局唯一性账本进行管理，确保最终729×729幻方中无素数元素重复出现。 最后，对完成的729×729幻方进行独立验证，包括素性、全局唯一性、行和、列和、对角线和，以及所有嵌套层级的对齐递归块和。 本构建的证明意义 本构建证明，当采用关联块中心与全局互不重叠的素数元素进行组织时，递归唯一素数幻方方法至少可扩展至729阶。 当前已验证的构建序列包括： 81×81阶，包含6561个不同素数元素 243×243阶，包含59049个不同素数元素 729×729阶，包含531441个不同素数元素 因此，本729×729构建并非仅为更大规模的数组，而是一个六层对齐的递归结构，由素数元素构建而成，且在所有相关尺度上均保留幻方和与全局唯一性。 推测展望 已验证的81、243及729阶案例可引出如下推测：全嵌套唯一素数幻方可存在于更高阶的3的幂次中。 本数据集并未证明该推测。 本成果应被视为一项已验证的计算构建结果，同时证明：结合关联块中心规划与全局素数互不重叠管理的中心-壳层方法，可超越此前发表的81×81与243×243案例实现扩展。 范围与局限性 本数据集声称： 一个已验证的729×729幻方 531441个全局唯一的素数元素 所有元素均为素数 所有元素全局唯一 所有元素末位均为3 在3×3、9×9、27×27、81×81、243×243及729×729层级均具备完整对齐递归嵌套性 对早期81×81与243×243中心-壳层框架的构建性扩展 本数据集未声称： 无穷多此类幻方存在的证明 所有3^n阶均可通过该方法构建的证明 素数分布相关定理 该方法的唯一性 素数元素的最小性 所选中心或偏移量的最优性 该方法必然可持续扩展的形式证明 AI辅助探索声明 本数据集通过人类主导、AI辅助的数学探索完成制作。 人类贡献者主导了构建策略，确定了递归中心-壳层框架，提出将规模从81×81扩展至243×243，再扩展至729×729，并指导了关联块中心的构建方法。 AI工具被用于辅助计算搜索、构建管理、验证、故障分析、文档编写与红队审查。 最终数据集以计算构建结果与显式验证数据的形式呈现，而非作为自动证明或针对所有可能情况的定理。 通俗总结 本数据集包含一个完全由素数构成的729×729幻方。 该幻方共有531441个元素，每个元素均为素数，无重复素数，且所有元素均以数字3结尾。 该幻方同时具备嵌套特性：其内部所有对齐的子方阵同样为幻方，这在以下层级均成立： 3×3 → 9×9 → 27×27 → 81×81 → 243×243 → 729×729 本构建扩展了两项早期数据集：81×81唯一素数幻方与243×243唯一素数幻方。 本729×729幻方由9个关联243×243素数幻方块构建而成，这9个块的中心本身构成一个3×3素数幻方。 最终得到一个已验证的递归素数幻方结构，包含531441个全局唯一的素数元素。