This work bring magic square of order 1220×1220 in prime numbers. This composed of equal sums magic squares of order 4×4. In order to reach this order, we have passed through 7 levels. These steps are:
- Level 1: Block-structured prime magic square of order 120×120;
- Level 2: Block-structured prime magic square of order 160×160;
- Level 3: Block-structured prime magic square of order 324×324;
- Level 4: Block-structured prime magic square of order 484×484;
- Level 5: Block-structured prime magic square of order 656×656;
- Level 6: Block-structured prime magic square of order 896×896;
- Level 7: Block-structured prime magic square of order 1220×1220.
Let’s analyse each step seperately. Interesting part is that in each step the magic square sums is same for order 4×4. The difference is that the value of magic sum changes in each case. Moreover, all the primes are distinct in each case. Summarizing the table below give the magic sum of order 4×4 in each case. et’s see below details.
Level 1: Block-Structured Prime Magic Square of Order 120×120
Lets see the following four examples:
The four examples are considered randomly. The prime magic square of order 120×120 is composed of equal sum prime magic square of order 4×4. All the entries are distinct primes. The magic sums are:
S120×120:=120000360 and S4×4:=4000012
This magic sum is based on the internal four entries of each block of order 4, It has some interesting properties. See below:
Looking above, the four members of each color are of same sum as of magic square. These properties are the part of perfect magic square of order 4. Since there are much more properties to be a perfect magic square, we shall call them as semi-perfect prime magic squares.
It gives us 29 blocks of magic squares of orders 8, 12, 16, … ,120. All with equal sums distinct primes magic squares of order 4 with magic constant: C=S/4:=1000003.
This it the fundamental constant to bring prime magic square of order 120×120 with equal sums blocks of order 4×4 having distinct primes. This constant allow us to reach upto order 120×120. To get the results for higher orders we shall use another constant of higher value. Below is an example of prime magic square of order 20 extracted randomly from the order 120×120.
Level 2: Block-Structured Prime Magic Square of Order 160×160
Lets see the following four examples:
The four examples are considered randomly. The prime magic square of order 160×160 is composed of equal sum prime magic square of order 4×4. All the entries are distinct primes. The magic sums are:
S160×160:=32000480 and S4×4:=8000012
This magic sum is based on the internal four entries of each block of order 4, It has some interesting properties. See below:
Looking above, the four members of each color are of same sum as of magic square. These properties are the part of perfect magic square of order 4. Since there are much more properties to be a perfect magic square, we shall call them as semi-perfect prime magic squares.
It gives us 39 blocks of magic squares of orders 8, 12, 16, … ,160. All with equal sums distinct primes magic squares of order 4 with magic constant: C=S/4:=2000003.
This it the fundamental constant to bring prime magic square of order 160×160 with equal sums blocks of order 4×4 having distinct primes. This constant allow us to reach upto order 160×160. To get the results for higher orders we shall use another constant of higher value. Below is an example of prime magic square of order 20 extracted randomly from the order 160×160.
Level 3: Block-Structured Prime Magic Square of Order 324×324
Lets see the following four examples:
The four examples are considered randomly. The prime magic square of order 324×324 is composed of equal sum prime magic square of order 4×4. All the entries are distinct primes. The magic sums are:
S324×324:=3240006156 and S4×4:=40000076
This magic sum is based on the internal four entries of each block of order 4, It has some interesting properties. See below:
Looking above, the four members of each color are of same sum as of magic square. These properties are the part of perfect magic square of order 4. Since there are much more properties to be a perfect magic square, we shall call them as semi-perfect prime magic squares.
It gives us 80 blocks of magic squares of orders 8, 12, 16, … , 324. All with equal sums distinct primes magic squares of order 4 with magic constant C=S/4:=10000019.
This it the fundamental constant to bring prime magic square of order 324×324 with equal sums blocks of order 4×4 having distinct primes. This constant allow us to reach upto order 324×324. To get the results for higher orders we shall use another constant of higher value. Below is an example of prime magic square of order 20 extracted randomly from the order 324×324.
Level 4: Block-Structured Prime Magic Square of Order 484×484
Lets see the following four examples:
The four examples are considered randomly. The prime magic square of order 484×484 is composed of equal sum prime magic square of order 4×4. All the entries are distinct primes. The magic sums are:
S484×484:=12100004356 and S4×4:=100000036
This magic sum is based on the internal four entries of each block of order 4, It has some interesting properties. See below:
Looking above, the four members of each color are of same sum as of magic square. These properties are the part of perfect magic square of order 4. Since there are much more properties to be a perfect magic square, we shall call them as semi-perfect prime magic squares.
It gives us 120 blocks of magic squares of orders 8, 12, 16, … , 484. All with equal sums distinct primes magic squares of order 4 with magic constant C=S/4:=25000009.
This it the fundamental constant to bring prime magic square of order 484×484 with equal sums blocks of order 4×4 having distinct primes. This constant allow us to reach upto order 484×484. To get the results for higher orders we shall use another constant of higher value. Below is an example of prime magic square of order 20 extracted randomly from the order 484×484.
Level 5: Block-Structured Prime Magic Square of Order 656×656
Lets see the following four examples:
The four examples are considered randomly. The prime magic square of order 656×656 is composed of equal sum prime magic square of order 4×4. All the entries are distinct primes. The magic sums are:
S656×656:=32800011152 and S4×4:=200000068
This magic sum is based on the internal four entries of each block of order 4, It has some interesting properties. See below:
Looking above, the four members of each color are of same sum as of magic square. These properties are the part of perfect magic square of order 4. Since there are much more properties to be a perfect magic square, we shall call them as semi-perfect prime magic squares.
It gives us 163 blocks of magic squares of orders 8, 12, 16, … , 656. All with equal sums distinct primes magic squares of order 4 with magic constant C=S/4:=50000017.
This it the fundamental constant to bring prime magic square of order 656×656 with equal sums blocks of order 4×4 having distinct primes. This constant allow us to reach upto order 656×656. To get the results for higher orders we shall use another constant of higher value. Below is an example of prime magic square of order 20 extracted randomly from the order 656×656.
Level 6: Block-Structured Prime Magic Square of Order 896×896
Lets see the following four examples:
The four examples are considered randomly. The prime magic square of order 896×896 is composed of equal sum prime magic square of order 4×4. All the entries are distinct primes. The magic sums are:
S896:=89600006272 and S4×4:=400000028
This magic sum is based on the internal four entries of each block of order 4, It has some interesting properties. See below:
Looking above, the four members of each color are of same sum as of magic square. These properties are the part of perfect magic square of order 4. Since there are much more properties to be a perfect magic square, we shall call them as semi-perfect prime magic squares.
It gives us 223 blocks of magic squares of orders 8, 12, 16, … , 896. All with equal sums distinct primes magic squares of order 4 with magic constant C=S/4:=100000007.
This it the fundamental constant to bring prime magic square of order 896×896 with equal sums blocks of order 4×4 having distinct primes. This constant allow us to reach upto order 896×896. To get the results for higher orders we shall use another constant of higher value. Below is an example of prime magic square of order 20 extracted randomly from the order 1220×1220.
Level 7: Block-Structured Prime Magic Square of Order 1220×1220
Lets see the following four examples:
The four examples are considered randomly. The prime magic square of order 1220×1220 is composed of equal sum prime magic square of order 4×4. All the entries are distinct primes. The magic sums are:
S1220×1220:=243999989020 and S4×4:=799999964
This magic sum is based on the internal four entries of each block of order 4, It has some interesting properties. See below:
Looking above, the four members of each color are of same sum as of magic square. These properties are the part of perfect magic square of order 4. Since there are much more properties to be a perfect magic square, we shall call them as semi-perfect prime magic squares.
It gives us 304 blocks of magic squares of orders 8, 12, 16, … , 1220. All with equal sums distinct primes magic squares of order 4 with magic constant C=S/4:=199999991.
This it the fundamental constant to bring prime magic square of order 1220×1220 with equal sums blocks of order 4×4 having distinct primes. This constant allow us to reach upto order 1220×1220.
The excel files of complete work are attached in work given in Zenodo.
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 Carlo 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