Problems & Puzzles: Puzzles

Puzzle 68.- The largest prime magic squares at the end of this millenium (a puzzle suggested by Jaime Ayala, and previously - 3/6/99 - asked by Warut Roonguthai also)

As far as I know, trough the splendid pages "Material from REC" of Harvey Heinz, the largest magic squares composed by prime numbers are:

1) A magic 9x9 square, composed of 81 consecutive primes, from 43 to 491.

2) A magic 16x16 square, composed of 256 distinct primes, from 11 to 2633.

Both magic squares are original from Alan W. Johnson, Jr., published by REC at 1993 and 1991 respectively.


1. Is this 9x9 consecutive primes magic square the least of his order?

2. Is this 16x16 prime magic square the least of his order?

3. Are these magic squares going to remain as the largest magic squares with prime numbers of the current millenium?

Hint: Since today (20/09/99) you have the use of 102 days (or 102+366 days) to beat them (depending what do you believe about the end of the current millenium)

See below images of the alluded magic square, taken from the pages of Harvey Heinz, after his kind permission:

1.- Magic square 9x9, consecutive primes

2.- Magic square 16x16, composed by prime numbers.


Well, it happens everyday, everywhere... Surfing through the very organized and complete pages of Mutsumi Suzuki I have found a smaller and older matrix than the 9x9 magic square of Alen W. Johnson. This older one is composed of nothing more but 81 consecutive matrix (from 37 to 479), and was discovered by Akio Suzuki. According to Mutsumi Suzuki "The squares by continuous prime numbers in this page are from Abe G. 's book 'Study of Magic Suares'. They were created by G. Abe and A. Suzuki in 1957"

173 97 191 163 149 383 257 389 409
181 431 179 113 277 251 317 419 43
479 199 193 131 137 139 379 271 283
211 67 449 241 349 233 157 37 467
457 433 47 337 239 71 59 401 167
439 313 463 223 359 227 53 61 73
83 461 127 263 151 331 311 443 41
109 103 293 373 197 229 397 89 421
79 107 269 367 353 347 281 101 307

Mutsumi Suzuki, asked by me about the Akio and Abe identities, kindly responded this:

"Dear Carlos; 1) Mr. Akio Suzuki is an amateur (not mathematicain but an owner of now old book shop).  He is not from my family. 2) Mr. G. Abe's book is written in Japanese and no English translation is available. Mr. Abe is also an amateur. He makes and sells Japanese lacker wares"

Then I would say that, attending to its historical primacy and its magic smaller sum, this is the matrix to beat or not to beat...


John L. Miller has broken all the shown records before existed about prime magical squares!... according to his communication he got a 41x41 one "at the end of the 80's ... in collaboration with Channon Price...". Now, in order to collaborate with this pages John has sent (23/02/2000) a new and larger magic & layered square composed by 99x99 distinct primes not consecutive!!!...

Please click here to download an Excel workbook that contains this prime-magic monster square sent by John L. Miller and my verfifications.

John wrote: "I wrote the code in 'C' on Win98. It's approximately 1500 lines of code. The square I gave you was the result of about 10 minutes of CPU time on a Celeron 433 (low-end pentium-II) with 64 MB of RAM. One special property of this square is that it's layered. In other words, remove the outer layer to make it a 97x97, and you'll still have a prime magic square. I will probably update the largesquare paper, though not for a week or two."

An unfinished sketch of his paper can be seen here: (detected broken 1/9/01)


On March 5, 2019 Bogdan Golunski announced:

"I made a borded prime magic square of order 503 with center prime number 45,367,123 and with 253,009 prime numbers. This monster bordered prime magic square is on my website in left link „news.


Carlos Rivera has analyzed this magic square and has confirmed the magicity of this prime square. The 253009 primes used are not consecutive. The smallest prime is 247519 and the largest one is 90486727 (both in the innermost 3x3 magic square). Confirmed that no repeated primes are used.




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