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. Initially this work is for the orders 6 to 29. In case of order 7, 11, 13, etc, we don’t have division in blocks but except 7, the magic squares type 17, 19, 22, 23 etc. are block-bordered. This means there are one or two layers are single-border with internal part as borders.
Whenever, we write prime magic square, it means that the magic squares are composed of prime distinct number entries except only in few cases, there are repetition of numbers but this repepetition is cleary explained.
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 15: 5 Examples;
- Order 16: 4 Example;
- Order 17: 5 Examples;
- Order 18: 6 Examples;
- Order 19: 5 Examples;
- Order 20: 7 Examples
- Order 21: 6 Examples;
- Order 22: 5 Examples;
- Order 23: 5 Examples:
This work is also available online at the following link:
Prime Magic Squares of Order 15
Below are few examples of prime magic squares of order 15 with different representations.
Example 1. Different Sums Blocks of Order 3
It is block-bordered prime magic square of order 15. It is composed of different sums prime magic squares of order 3. See below few examples of order 3×3:
Example 2. Equal Sums Blocks of Order 3
It is block-bordered prime magic square of order 15. It is composed of equal sums prime magic squares of order 3. See below few examples of order 3×3:
Example 3. Different Sums Blocks of Cocentric Prime Magic Squares of Order 5
It is block-bordered prime magic square of order 15. It is composed of different sum cocentric prime magic squares of order 5. It contains prime magic squares of order 3. See below few examples of order 5×5:
Example 4. Equal Sums Blocks of Cocentric Prime Magic Squares of Order 5
It is block-bordered prime magic square of order 15. It is composed of equal sums cocentric prime magic squares of order 5. It contains prime magic squares of order 3. See below few examples of orders 5 and 3:
Example 5. Cocentric Single-Layer Bordered Prime Magic Square
It is cocentric single-layer bordered prime magic square of order 15. the internal layer are also prime magic squares of orders 13, 11, 9, 7, 5 and 3. See below:
Prime Magic Squares of Order 16
Below are few examples of prime magic squares of order 16 with different representations.
Example 1. Different Sums Prime Magic Squares of Order 4
It is block-bordered prime magic square of order 16. The internal blocks are different sums prime magic square of order 4. See below few examples of order 4:
Example 2. Equal Sums Prime Magic Squares of Order 4
It is block-bordered prime magic square of order 16. The internal blocks are equal sums prime magic square of order 4. See below few examples of order 4:
Example 3: Equal Sums Concentric Prime Magic Squares of Order 8
Example 4. Concentric Single-Layer Prime Magic Squares
t is cocentric single-layer bordered prime magic square of order 16. the internal layer are also prime magic squares of orders 14, 12, 10, 8, 6 and 4. The orders 4 to 14 is given before in another part on prime magic squares of order 14.
Prime Magic Squares of Order 17
Below are few examples of prime magic squares of order 17 with different representations.
Example 1. Block-Bordered Nested with Different Sums Blocks of Order 3
It is block-bordered prime magic square of order 17 embedded with a prime magic square of order 15 with different sums prime magic squares of order 3. The prime magic square of order 15 is the same as given in section on prime magic squares of order 15.
Example 2. Block-Bordered Nested with Equal Sums Blocks of Order 3
It is block-bordered prime magic square of order 17 embedded with a prime magic square of order 15 with equal sums prime magic squares of order 3. The prime magic square of order 15 is the same as given in section on prime magic squares of order 15.
Example 3. Block-Bordered Nested with Different Sums Blocks of Cocentric Blocks of Order 5
It is block-bordered prime magic square of order 17 embedded with a prime magic square of order 15 with different sums cocentric prime magic squares of order 5. The prime magic square of order 15 is the same as given in section on prime magic squares of order 15.
Example 4. Block-Bordered Nested with Equal Sums Blocks of Cocentric Blocks of Order 5
It is block-bordered prime magic square of order 17 embedded with a prime magic square of order 15 with equal sums cocentric prime magic squares of order 5. The prime magic square of order 15 is the same as given in section on prime magic squares of order 15.
Example 5. Concentric Single-Layer Prime Magic Squares
It is cocentric single-layer bordered prime magic square of order 17. the internal layer are also prime magic squares of orders 15, 13, 11, 9, 7, 5 and 3. It is the same as given in section on prime magic square of order 15,
Prime Magic Squares of Order 18
Below are few examples of prime magic squares of order 18 with different representations.
Example 1. Different Sums Blocks of Order 3
It is block-structured prime magic square of order 18. The internal blocks are different sums prime magic squares of order 3. See below few examples of order 3:
Example 2. Equal Sums Blocks of Order 3
It is block-structured prime magic square of order 18. The internal blocks are equal sums prime magic squares of order 3. See below few examples of order 3:
Example 3. Different Sums Cocentri Blocks of Order 6
It is block-structured prime magic square of order 18. The internal blocks are different sums cocentric prime magic squares of order 6. See below few examples:
Example 4. Four Equal Sums Blocks of Order 9 with Blocks of Order 3
It is block-structured prime magic square of order 18. The internal blocks are four equal sums prime magic squares of order 9. These are again composed of different sums prime magic squares oforder 3.
Example 5. Four Equal Sums Cocentric Blocks of Order 9
It is block-structured prime magic square of order 18. The internal blocks are four equal sums cocentric prime magic squares of order 9.
Example 6. Cocentric Single-Layer Bordered Prime Magic Square
It is cocentric single-layer bordered prime magic square of order 18. The internal part of order 16 is the same as given above in the section of prime magic square of order 16.
Prime Magic Squares of Order 19
Below are few examples of prime magic squares of order 19 with different representations.
Example 1. Block-Bordered Nested with Different Sums Blocks of Order 3
It is block-bordered prime magic square of order 19 embedded with a prime magic square of order 15 with different sums prime magic squares of order 3. The prime magic square of order 15 is the same as given in section on prime magic squares of order 15.
Example 2. Block-Bordered Nested with Equal Sums Blocks of Order 3
It is block-bordered prime magic square of order 19 embedded with a prime magic square of order 15 with equal sums prime magic squares of order 3. The prime magic square of order 15 is the same as given in section on prime magic squares of order 15.
Example 3. Block-Bordered Nested with Different Sums Blocks of Cocentric Blocks of Order 5
It is block-bordered prime magic square of order 19 embedded with a prime magic square of order 15 with different sums cocentric prime magic squares of order 5. The prime magic square of order 15 is the same as given in section on prime magic squares of order 15.
Example 4. Block-Bordered Nested with Equal Sums Blocks of Cocentric Blocks of Order 5
It is block-bordered prime magic square of order 19 embedded with a prime magic square of order 15 with equal sums cocentric prime magic squares of order 5. The prime magic square of order 15 is the same as given in section on prime magic squares of order 15.
Example 5. Concentric Single-Layer Prime Magic Squares
It is cocentric single-layer bordered prime magic square of order 19. the internal layer are also prime magic squares of orders 17, 15, 13, 11, 9, 7, 5 and 3. It is the same as given in section on prime magic square of order 17,
Prime Magic Squares of Order 20
Below are few examples of prime magic squares of order 20 with different representations.
Example 1. Different Sums Blocks of Order 4
It is block-structured prime magic square of order 20. The internal blocks are different sums prime magic squares of order 3. See below few examples of order 4:
Example 2. Equal Sums Blocks of Order 4
It is block-structured prime magic square of order 20. The internal blocks are equal sums prime magic squares of order 4. See below few examples of order 4:
Example 3. Different Sums Cocentric Blocks of Order 5
It is block-structured prime magic square of order 20. The internal blocks are different sums cocentric prime magic squares of order 5. See below few examples:
Example 4. Equal Sums Cocentric Blocks of Order 5
It is block-structured prime magic square of order 20. The internal blocks are equal sums prime magic squares of order 5. See below few examples:
Example 5. Four Equal Sums Blocks of Order 10
It is block-structured prime magic square of order 20. It is composed of 4 equal sums block-bordered primes magic squares of order 10.
Example 6. Four Equal Sums Cocentric Blocks of Order 10
It is block-structured prime magic square of order 20. The internal blocks are four equal sums cocentric prime magic squares of order 10.
Example 7. Cocentric Single-Layer Bordered Prime Magic Square
It is cocentric single-layer bordered prime magic square of order 20. The internal part of order 18 is the same as given above in the section of prime magic square of order 18.
Prime Magic Squares of Order 21
Below are few examples of prime magic squares of order 21 with different representations.
Example 1. Different Sums Blocks of Order 3
It is block-structured prime magic square of order 21. The internal blocks are different sums prime magic squares of order 3. See below few examples of order 3:
Example 2. Equal Sums Blocks of Order 3
It is block-structured prime magic square of order 21. The internal blocks are equal sums prime magic squares of order 3. See below few examples of order 3:
Example 3. Different Sums Cocentric Blocks of Order 7
It is block-structured prime magic square of order 21. The internal blocks are different sums cocentric prime magic squares of order 7. See below few examples:
Example 4. Equal Sums Blocks of Order 7 with Blocks of Order 3
It is block-structured prime magic square of order 21. The internal blocks are equal sums cocentric prime magic squares of order 7. See below few examples:
Example 6. Cocentric Single-Layer Bordered Prime Magic Square
It is cocentric single-layer bordered prime magic square of order 21. The internal part of order 19 is the same as given above in the section on prime magic square of order 19.
Prime Magic Squares of Order 22
Below are few examples of prime magic squares of order 22 with different representations.
Example 1. Block-Bordered Nested with Different Sums Blocks of Order 4
It is block-bordered prime magic square of order 22 embedded with a prime magic square of order 20 with different sums prime magic squares of order 4. The prime magic square of order 20 is the same as given in section on prime magic squares of order 20.
Example 2. Block-Bordered Nested with Equal Sums Blocks of Order 4
It is block-bordered prime magic square of order 22 embedded with a prime magic square of order 20 with equal sums prime magic squares of order 4. The prime magic square of order 20 is the same as given in section on prime magic squares of order 20.
Example 3. Block-Bordered Nested with Different Sums Blocks of Order 5
It is block-bordered prime magic square of order 22 embedded with a prime magic square of order 20 with different sums cocentric prime magic squares of order 5. The prime magic square of order 20 is the same as given in section on prime magic squares of order 20.
Example 4. Block-Bordered Nested with Equal Sums Blocks of Order 5
It is block-bordered prime magic square of order 22 embedded with a prime magic square of order 20 with equal sums cocentric prime magic squares of order 5. The prime magic square of order 20 is the same as given in section on prime magic squares of order 20.
Example 5. Block-Bordered Nested with Four Equal Sums Blocks of Order 10
It is block-bordered prime magic square of order 22 embedded with a prime magic square of order 20 with four equal sums block-bordered prime magic squares of order 10. These are again composed of four equal sums prime magic squares of order 4. The prime magic square of order 20 is the same as given in section on prime magic squares of order 20.
Example 6. Block-Bordered Nested with Four Equal Sums Cocentric Blocks of Order 10
It is block-bordered prime magic square of order 22 embedded with four equal sums cocentric prime magic square of order 10. The prime magic square of order 20 is the same as given in section on prime magic squares of order 20.
Example 7. Cocentric Single-Layer Bordered Prime Magic Square
It is cocentric single-layer bordered prime magic square of order 22. The internal part of order 20 is the same as given above in the section of prime magic square of order 20.
Prime Magic Squares of Order 23
Below are few examples of prime magic squares of order 23 with different representations.
Example 1. Block-Bordered Nested with Different Sums Blocks of Order 3
It is block-bordered prime magic square of order 23 embedded with a prime magic square of order 21 with different sums prime magic squares of order 3. The prime magic square of order 21 is the same as given in section on prime magic squares of order 21.
Example 2. Block-Bordered Nested with Equal Sums Blocks of Order 3
It is block-bordered prime magic square of order 23 embedded with a prime magic square of order 21 with equal sums prime magic squares of order 3. The prime magic square of order 21 is the same as given in section on prime magic squares of order 21.
Example 3. Block-Bordered Nested with Different Sums Blocks of Order 7
It is block-bordered prime magic square of order 23 embedded with a prime magic square of order 21 with different sums prime magic squares of order 7. The prime magic square of order 21 is the same as given in section on prime magic squares of order 21.
Example 4. Block-Bordered Nested with Equal Sums Blocks of Order 7
It is block-bordered prime magic square of order 23 embedded with a prime magic square of order 21 with equal sums prime magic squares of order 7. The prime magic square of order 21 is the same as given in section on prime magic squares of order 21.
Example 5.
It is cocentric single-layer bordered prime magic square of order 23. The internal part of order 21 is the same as given above in the section of prime magic square of order 21.
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 Prime Magic Squares of Even Orders.
- Inder J. Taneja, Single-Layer Bordered Prime Magic Square of Odd Orders.
- Inder J. Taneja, Block-Structured Prime Magic Square of Order 1220×1220.
- Inder J. Taneja, Block-Structured Prime Magic Squares of Orders 6 to 14.
- Inder J. Taneja, Block-Structured Prime Magic Squares of Orders 15 to 23.