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. Questions: 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.
Solutions 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 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" *** 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 primemagic 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 (lowend pentiumII) with 64 MB of RAM. An unfinished sketch of his paper can be
seen here: *** On March 5, 2019 Bogdan Golunski announced:
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