This work brings different ways representing magic squares in prime numbers. Initial aim is represent in terms of blocks. Some of possibilities led us to concentric or single-layer bordered magic squares. It includes blocks of orders 3, 4, 5, etc. Sometimes these blocks are of equal or sometimes of different magic sums. This work is for the magic squares of orders 27, 28, 29, 30 and 31 with prime numbers entries.
Whenever, we write prime magic square, it means that the magic squares are composed of prime number entries mostly distinct, except in few cases, there are repetition of some fixed numbers,
This work is specially for distinct prime numbers, except in some cases there are repetition of internal values. For similar kind of work for sequential numbers entries, see the author’s work given in the reference list.
In each order we worked with different kinds of representations prime magic squares. Below are the examples considered in each order.
- Order 27: 07 Examples (01-07);
- Order 28: 12 Examples (08-19);
- Order 29: 03 Examples (20-22);
- Order 30: 13 Examples (23-35);
- Order 31: 03 Examples (11-14);
- Total Examples: 38
This work is also available online at the following link:
Prime Magic Squares of Order 27
Below are few examples of prime magic squares of order 27 with different representations. We know that 27=33 . It means that most of the blocks are composition prime magic squares of order 3 and 9
Example 1. (27-3d) Different Sums Blocks of Order 3
It is block-structured prime magic square of order 27. It is composed of different sums prime magic squares of order 3.
Example 2. (27-3e) Equal Sums Blocks of Order 3
It is block-structured prime magic square of order 27. It is composed of equal sums prime magic squares of order 3. In order to bring these equal sums we used the idea of fixed prime 1000003 in middle of each block of order 3. All other primes are distinct.
Example 3. (27-9cd) Different Sums Cocentric Prime Magic Squares
It is block-structured prime magic square of order 27. It is composed of different sums cocentric prime magic squares of order 9. The intenal blocks of orders 7, 5 and 3 are also of different sums magic squares.
Example 4. (27-9ce) Equal Sums Cocentric Prime Magic Squares
It is block-structured prime magic square of order 27. It is composed of equal sums cocentric prime magic squares of order 9. The intenal blocks of orders 7, 5 and 3 are also of different sums magic squares.
Example 5.27- (9d-3d) Block-Structured Prime Magic Square
It is block-structured prime magic square of order 27. It is composed of different sums prime magic square of order 9. Each block of order 9 is again formed by 9 different sum prime magic squares.
Example 6. (27-9e-3d) Block-Structured Prime Magic Square
It is block-structured prime magic square of order 27. It is composed of equal sums prime magic square of order 9. Each block of order 9 is again formed by 9 equal sums prime magic squares.
Example 7. 27c Concentric Prime Magic Square
It is concentric single-layer bordered prime magic square of order 27. The internal blocks of orders 25, 23, …., 3 are also prime magic squares studied in the previous work.
Prime Magic Squares of Order 28
Below are few examples of prime magic squares of order 28 with different representations. We know that 28=4×7. It means that most of the blocks are composition of prime magic squares of orders 4 and 7.
Example 8. 28-4d-1. Different Sums Blocks of Order 4
It is block-structured prime magic square of order 28. It is composed of different sums prime magic squares of order 4.
Example 9. 28-4e-1. Equal Blocks of Order 4
It is block-structured prime magic square of order 28. It is composed of equal sums prime magic squares of order 4.
Example 12. 28-7d-1. Blocks of Prime Magic Squares
It is block-structured prime magic square of order 28. It is composed of different sums prime magic squares of order 7.
Example 10. 28-7cd-1. Blocks of Concentric Prime Magic Squares
It is block-structured concentric prime magic square of order 28. It is composed of differnt sums prime magic squares of order 7. The internal blocks of orders 5 and 3 are also of different sums.
Example 11. 28-7ce-1 Blocks of Concentric Prime Magic Squares
It is block-structured concentric prime magic square of order 28. It is composed of equal sums prime magic squares of order 7. The internal blocks of orders 5 and 3 are also of equal sums. It has repeated prime number 1000003 in the middle of each block. All other primes are distinct.
Example 13. 28-14-12-3d. Block-Structured Prime Magic Square
It is block-structured prime magic square of order 28. It is composed of four equal sums prime magic squares of order 14. Again each block of order 14 is embedded with different sums prime magic squares of order 3 composing a prime magic square of order 12.
Example 14. 28-14-12-3e. Block-Structured Prime Magic Square
It is block-structured prime magic square of order 28. It is composed of four equal sums prime magic squares of order 14. Again each block of order 14 is embedded with equal sums prime magic squares of order 3 composing a prime magic square of order 12.
Example 15. 28-14-12-4d. Block-Structured Prime Magic Square
It is block-structured prime magic square of order 28. It is composed of four equal sums prime magic squares of order 14. Again each block of order 14 is embedded with diferent sums prime magic squares of order 4 composing a prime magic square of order 12.
Example 16. 28-14-12-4e. Block-Structured Prime Magic Square
It is block-structured prime magic square of order 28. It is composed of four equal sums prime magic squares of order 14. Again each block of order 14 is embedded with equal sums prime magic squares of order 4 composing a prime magic square of order 12.
Example 17. 28-14-12-6e. Block-Structured Prime Magic Square
It is block-structured prime magic square of order 28. It is composed of four equal sums prime magic squares of order 14. Again each block of order 14 is embedded with equal sums prime magic squares of order 6 composing a prime magic square of order 12.
Example 18. 28-14ce-1. Block-Structured Prime Magic Square
It is block-structured prime magic square of order 28. It is composed of four equal sums prime magic squares of order 14. Again each block of order 14 is a concentric single-layer prime magic squares. The internal blocks of order 12,10, 8, 6 and 4 are also prime magic squares.
Example 19. 28c-1. Concentric Single-Layer Prime Magic Square
It is concentric single-layer bordered prime magic square of order 28. The internal block of order 26 is also a concentric single-layer bordered prime magic square as studied in the previous work.
Prime Magic Squares of Order 29
Below are few examples of prime magic squares of order 29 with different representations.
Example 20. 29-27-9d. Block-Bordered Prime Magic Square
It is block-bordered prime magic square of order 29. It is composed of nine different sums prime magic squares of order 9 forming a prime magic square of order 27. These are again composed of different sums prime magic squares of order 3.
Example 21. 29-27-9cd. Block-Bordered Prime Magic Square
It is block-bordered prime magic square of order 29. It is composed of nine different sums concentric single-layer prime magic squares of order 9 forming a prime magic square of order 27.
Example 22. 29c. Concentric Single-Layer Prime Magic Square
It is concentric single-layer bordered prime magic square of order 29. The internal blocks of orders 27 is also a concentric single-layer bordered prime magic square studied in the first section.
Prime Magic Squares of Order 30
Below are few examples of prime magic squares of order 30 with different representations.
Example 23. 30-3d. Different Sums Blocks of Order 3
It is block-structured prime magic square of order 30 composed of different sums prime magic squares of order 30
Example 24. 30-3e. Equal Sums Blocks of Order 3
It is block-structured prime magic square of order 30 composed of equal sums prime magic squares of order 3. The blocks of order 3 are composed with a repeated prime number 1000003. This constant helps in bringing equal sums prime magic squares of order 3.
Example 25. 30-5d. Different Sums Blocks of Order 5
It is block-structured prime magic square of order 30 composed of different sums prime magic squares of order 5
Example 26. 30-5cd. Different Sums Blocks of Order 5
It is block-structured prime magic square of order 30 composed of different sums prime magic squares of order 5. The blocks of order 5 are again composed with inner part as prime magic squares of order 3 of different sums.
Example 27. 30-5ce. Equal Sums Blocks of Order 5
It is block-structured prime magic square of order 30 composed of equal sums prime magic squares of order 5. The blocks of order 5 are again composed with inner part as prime magic squares of order 3 of equal sums.
Example 28. 30-6cd. Different Sums Blocks of Order 6
It is block-structured prime magic square of order 30 composed of different sums prime magic squares of order 6. The blocks of order 6 are again composed with inner part as prime magic squares of order 4 of different sums.
Example 29. 30-6ce. Equal Sums Blocks of Order 3
It is block-structured prime magic square of order 30 composed of equal sums prime magic squares of order 6. The blocks of order 6 are again composed with inner part as prime magic squares of order 4 of equal sums.
Example 30. 30-10d. Different Sums Blocks of Order 10
It is block-structured prime magic square of order 30 composed of different sums prime magic squares of order 10
Example 31. 30-10cd. Different Sums Blocks of Order 10
It is block-structured prime magic square of order 30 composed of different sums prime magic squares of order 10. The blocks of order 10 are a concentric single-layer prime magic square of order 10.
Example 32. 30-10ce. Equal Sums Blocks of Order 10
It is block-structured prime magic square of order 30 composed of equal sums prime magic squares of order 10. The blocks of order 10 are a concentric single-layer prime magic square of order 10.
Example 33. 30-10e-8e-4e. Equal Sums Blocks of Order 10
It is block-structured prime magic square of order 30 composed of equal sums prime magic squares of order 10. The blcoks of order 10 block-borderd with inner part as equal sums prime magic squares of order 4 formng magic squares of order 8.
Example 34. 30-15ce. Equal Sums Blocks of Order 15
It is block-structured prime magic square of order 30 composed of four equal sums concentric single-layer prime magic squares of order 15
Example 35. 30c. Concentric Single-Layer Prime Magic Square
It is cocentric single-layer bordered prime magic square of order 30. The internal prime magic squares of orders 28 is also single-layer bordered prime magic square of order 28 as studied above.
Prime Magic Squares of Order 31
Below are few examples of prime magic squares of order 31 with different representations.
Example 36. 31-29-27-9d. Block-Bordered Prime Magic Square
It is block-bordered prime magic square of order 31. It is composed of nine different sums prime squares of order 9 forming a prime magic square of order 27. Each block of order 9 is again composed of 9 different sums prime magic squares of order 3.
Example 37. 31-29-27-9cd. Block-Bordered Prime Magic Square
It is block-bordered prime magic square of order 31. It is composed of nine different sums concentric single-layer prime magic squares of order 9 forming a prime magic square of order 27
Example 38. 31c. Concentric Single-Layer Prime Magic Square
It is concentric single-layer bordered prime magic square of order 31. The internal blocks of orders 29 is also a concentric single-layer bordered prime magic square studied before.
References
- H. White, Magic Squares of Prime Numbers, https://budshaw.ca/PrimeMagicSquares.html.
- Heinz, Harvey, Prime Numbers Magic Squares, http://recmath.org/Magic%20Squares/primesqr.htm.
- Makarova, Natalia, Concentric magic squares of primes, http://primesmagicgames.altervista.org/wp/forums/topic/concentric-magic-squares-of-primes/.
- Roberto C. Angelone, A Fully Nested 729 x 729 Unique-Prime Magic Square Constructed from Nine Correlated 243 x 243 Prime Magic Blocks, https://zenodo.org/records/20098521.
- Inder J. Taneja, Single-Layer Bordered Even and Odd Orders Primes Magic Squares: Orders 120 and 121, Zenodo, May June 16, 2026, pp. 1-36, https://doi.org/10.5281/zenodo.20723312
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- Inder J. Taneja, Block-Structured Prime Magic Squares: Orders 15 to 23, Zenodo, June 16, 2026, pp. 1-76, https://doi.org/10.5281/zenodo.20723291.
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- Inder J. Taneja, Block-Structured Prime Magic Squares: Orders 24, 25 and 26, Zenodo, June 19, 2026,
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