Problems & Puzzles: Puzzles
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Problems & Puzzles: Puzzles
 From June 26 to July 4, contributions came from Michael Branicky, Paul Cleary, Roberto Angelone *** Michael wrote:
All 729x729 elements in the MS are prime. I also verified
that all row sums, columns sums, and diagonal and
anti-diagonal sums are the same: 7290000024057
For ease of future studies, I attach a clean .csv file
with 729 rows and 729 columns where each entry is a
plaintext integer (with no comma-for-thousands
separators). It is 5.9MB.
Michael added: For reference, attached is a short Python program to perform the verification on the clean csv file. (Note by CR: My plan is not to upload both files sent by Michael. I will keep just in my local site, ready to send it to anyone who ask for it, on request by email.) Paul wrote:
Q1. I found this if you could call it published..
Q2. All entries are distinct prime numbers.
*** Roberto Angelone wrote: On July 2, 2026, he announced larger prime MS, and perhapas a new record in this subject: 6561x6561 Consecutive primes found on: https://zenodo.org/records/20417121. You need to scroll down to the specific file called "07-6561X6561-CONSECUTIVE-PRIMES.csv". and 2187x2187 3^n Nested Primes found on: https://zenodo.org/records/20385047 and similarly scroll down to the either the file called "MASTER BLOCK-MATHS_2187x2187_assembled_unique_prime_magic_square.csv" or the one below it called "MASTER BLOCK-MATHS_2187x2187_assembled_unique After downloading his Excel version of the MS 6561x6561, this is what I could verify:
But the most intriguing feature is that 6561 x the value of the Central Cell is (2649325114341) not equal to the MS (2678155179653), Is this correct or incorrect? *** M.F. Hasler wrote on July 9, 2026: I have looked at https://www.primepuzzles.net/puzzles/puzz_1273.htm First, I have downloaded the 729 x 729 CSV file. I can confirm that it is made of 729² distinct primes. I have also downloaded "06-2187X2187-CONSECUTIVE-PRIMES.csv" from https://zenodo.org/records/20417121 I have checked that all rows have distinct primes in it, row, column & diagonal sums are 87047577443, smallest and largest primes are p = 24419 = prime(2711), and q = 82113467 = prime(4785679). Since primepi(q) - primepi(p) = 4782968 = 729² - 1 it is indeed most likely that the matrix is made of exactly all consecutive primes from p to q. Then this is indeed a new record magic square, until the one of size 6561² is verified. I did not yet verify the 6561² square, because I hesitate to download the 424 MB file [and then process it ...]. But looking at your comments: you are surprised that Product if the Central Cell by 6551 = different from row sums. Note, there are two typos :"if" => "of", "6551" => 6561. But, the fact is not surprising. For example, in the 2187 x 2187 magic square, the central value is 58061, which multiplied by 2187 gives only 126979407 while all row & column sums are 87047577443. - Maximilian /* PARI/GP code, for my records */ check(fn="/Users/mh/Downloads/06-2187X2187-CONSECUTIVE-PRIMES.csv")={ my(f=fileopen(fn), m=oo, M=0/*min & max*/, diagsum=0, antids=0, rows=0, rowsum, colsum, row); while( row=filereadstr(f), print1("."); if( !row=eval(Str("["row"]")),/*empty row*/ next, rows++&& !M,/*initialize*/ rowsum=vecsum(colsum=row), rowsum != vecsum(row), error("bad sum in row "rows), colsum += row /*=> error if different length*/)/*end if*/; diagsum += row[rows]; antids += row[#row-rows+1]; if(#row != #row=Set(row), error("duplicates in row "rows); /* update min / max */ row[1]<m, m=row[1] );/*end if*/ row[#row]>M && M=row[#row];/*both can happen in the same row!*/ foreach(v, p, ispseudoprime(p)|| error("Nonprime "p" in row "rows)); )/*end while*/; fileclose(f); [m, M, rowsum, rowsum-diagsum, rowsum-antids, Set(colsum)-[rowsum]] } ***
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