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 24, 25 and 26 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 24: 10 Examples (01-10);
- Order 25: 03 Examples (11-14);
- Order 26: 11 Examples (14-24);
This work is also available online at the following link:
Prime Magic Squares of Order 24
Below are few examples of prime magic squares of order 24 with different representations.
Example 1. Equal Sums Blocks of Order 12 Composed of Different Sums of Order 3
It is block-structured prime magic square of order 24. It is composed of four equal sum prime magic square 12. Each block of order 12 is again composed of 16 different sums prime magic squares of order 3.
Example 2. Equal Sums Blocks of Order 3
It is block-structured prime magic square of order 24 composed of all equal sums blocks of order 3. In order to bring these equal sums we used the idea of fixed prime 1000003 in each block of order 3.
Example 3. Equal Sums Blocks of Order 12 Composed of Different Sums of Order 4
It is block-structured prime magic square of order 24. It is composed of four equal sum prime magic square 12. Each block of order 12 is again composed of 9 different sums prime magic squares of order 4.
Example 4. Equal Sums Blocks of Order 4
It is block-structured prime magic square of order 24 composed of all equal sums blocks of order 4.
Example 5. Different Sums Blocks of Order 6 with Different Sums Blocks of Order 4
It is block-structured prime magic square of order 24. It is composed of different sums concentric prime magic square of order 6. The internal blocks of order 4 are also prime magic squares of order 4 with different sums
Example 6. Equal Sums Blocks of Order 6 with Equal Sums Blocks of Order 4
It is block-structured prime magic square of order 24. It is composed of equal sums concentric prime magic square of order 6. The internal blocks of order 4 are also prime magic squares of order 4 with equal magic sums.
Example 7. Different Sums Blocks of Order 8 with Different Sums Blocks of Orders 6 and 4
It is block-structured prime magic square of order 24. It is composed of different sums concentric prime magic square of order 8. The internal blocks of orders 6 and 4 are also of different sums prime magic squares.
Example 8. Equal Sums Blocks of Order 8 with Equal Sums Blocks of Orders 6 and 4
It is block-structured prime magic square of order 24. It is composed of equal sums concentric prime magic square of order 8. The internal blocks of orders 6 and 4 are also of equal sums prime magic squares.
Example 9. Equal Sums Blocks of Order 8 with Equal Sums Blocks of Orders 6 and 4
It is block-structured prime magic square of order 24. It is composed of four equal sums concentric prime magic square of order 12. The internal blocks of orders 10, 8, 6 and 4 are also of equal sums prime magic squares.
Example 10. Single-Layer Bordered Prime Magic Square
It is cocentric single-layer bordered prime magic square of order 24. The internal prime magic squares of orders 22, 22, …. ,6,4 are all prime magic squares as given in an previous work on prime magic square of order 22.
Prime Magic Squares of Order 25
Below are few examples of prime magic squares of order 25 with different representations.
Example 11. Different Sums Blocks of Order 5
It is block-structured prime magic square of order 25. It is composed of different sums prime magic squares of order 5. The internal blocks of order 3 are also different sums prime magic squares of order 3.
Example 12. Equal Sums Blocks of Order 5
It is block-structured prime magic square of order 25. It is composed of equal sums prime magic squares of order 5. The internal blocks of order 3 are also of equal sums prime magic squares of order 3.
Example 13. Single-Layer Bordered Prime Magic Square
It is cocentric single-layer bordered prime magic square of order 25. The internal prime magic squares of orders 23, 21, …, 3 are all prime magic squares as given in an previous work on prime magic squares of order 23.
Prime Magic Squares of Order 26
Below are few examples of prime magic squares of order 26 with different representations. It based on the examples of prime magic squares of order 24.
Example 14. Block-Bordered Embedded Prime Magic Square
It is block-bordered prime magic square of order 26 embedded with four equal sums prime magic squares of order 12. Each block of order 12 is composed of 16 different sums prime magic squares of order 3.
Example 15. Block-Bordered Embedded Prime Magic Square
It is block-bordered prime magic square of order 26 embedded with prime magic square of order 24. The prime magic square of order 24 is composed equal sums prime magic squares of order 3. It order to bring these magic squares of order 3 to becom equal sum a repeated prime 100003 is considere in middle of each block of order 3.
Example 16. Block-Bordered Embedded Prime Magic Square
It is block-bordered prime magic square of order 26 embedded with four equal sums prime magic squares of order 12. Each block of order 12 is composed of 9 different sums prime magic squares of order 4.
Example 17. Block-Bordered Embedded Prime Magic Square
It is block-bordered prime magic square of order 26 embedded with prime magic square of order 24. The prime magic square of order 24 is composed equal sums prime magic squares of order 4.
Example 18. Block-Bordered Embedded Prime Magic Square
It is block-bordered prime magic square of order 26 embedded with prime magic square of order 24. The prime magic square of order 24 is composed different sums concentric magic square of order 6. The internal blocks of order 4 are also prime magic squares with different magic sums.
Example 19. Block-Bordered Embedded Prime Magic Square
This example shall be give later on.
It is block-bordered prime magic square of order 26 embedded with prime magic square of order 24. The prime magic square of order 24 is composed equal sums concentric magic square of order 6. The internal blocks of order 4 are also prime magic squares with equal magic sums.
Example 20. Block-Bordered Embedded Prime Magic Square
It is block-structured prime magic square of order 26 embedded with prime magic square of order 24. It is composed of different sums concentric prime magic square of order 8. The internal blocks of orders 6 and 4 are also of different sums prime magic squares.
Example 21. Block-Bordered Embedded Prime Magic Square
It is block-structured prime magic square of order 26 embedded with prime magic square of order 24. It is composed of equal sums concentric prime magic square of order 8. The internal blocks of orders 6 and 4 are also of equal sums prime magic squares.
Example 22. Block-Bordered Embedded Prime Magic Square
It is block-structured prime magic square of order 24. It is composed of four equal sums concentric prime magic square of order 12. The internal blocks of orders 10, 8, 6 and 4 are also of equal sums prime magic squares.
Example 23. Block-Structured Magic Square
It is block-structured prime magic square of order 26 composed of four equal sums single-layer bordered prime magic squares of order 13.
Example 24. Cocentric Single-Layer Bordered Prime Magic Square
It is cocentric single-layer bordered prime magic square of order 26. The internal prime magic squares of orders 24 is also single-layer bordred of order 24 as studied above.
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
- Web-site Link: Single-Layer Bordered Prime Magic Square of Odd Orders
- Web-site Link: Single-Layer Bordered Prime Magic Squares of Even Orders.
- Inder J. Taneja, Higher Order Block-Structured Prime Magic Squares: Order 1220 Multiples of 4, Zenodo, June 16, 2026, pp. 1-27, http://doi.org/10.5281/zenodo.20723328.
- Web-site Link: Block-Structured Prime Magic Square of Order 1220×1220.
- Inder J. Taneja, Block-Structured Prime Magic Squares: Orders 6 to 14, Zenodo, June 16, 2026, pp. 1-41, https://doi.org/10.5281/zenodo.20723272
- Web-site Link: Block-Structured Prime Magic Squares of Orders 6 to 14.
- 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.
- Web-site Link: Block-Structured Prime Magic Squares of Orders 15 to 23.
- Inder J. Taneja, Block-Structured Prime Magic Squares: Orders 24, 25 and 26, Zenodo, June 18, 2026,
- Web-site Link: Block-Structured Prime Magic Squares of Orders 24, 25 and 26