----- Original Message ----- From: "David Wilson" <davidwwilson@comcast.net> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Thursday, September 22, 2005 3:47 PM Subject: Re: [math-fun] Multiplicative Magic Squares
So far, I have found the following criteria on the minimal 4x4 magic product m:
m <= 5040 m must have nonincreasing exponents in its standard prime factorization m must have >= 16 divisors d for which m/d is not of the form 1, p, p^2, p^3, p^4, p^5, p^6, p^7, pq, p^2q, p^3q, pqr for primes p,q,r.
To explain this last criterion. Let m be the magic product of a 4x4 square. Let d be a cell of that square. Let the row containing d have cells d,u,v,w. Let the column containing d have cells d,x,y,z. Then u,v,w,x,y,z are distinct positive integers with uvw = xyz = m/d. This is insoluble if m/d has any of the prime signatures listed above.
I believe these criteria reduce the candidates to:
1440 1680 1800 2160 2520 2880 3360 3456 3600 3840 4320 4608 5040
For these candidate magic products m, the allowable cell values are: m = 1440: d = (1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 30 40) m = 1680: d = (1 2 3 4 5 6 7 8 10 12 14 15 20 21 28 35) m = 1800: d = (1 2 3 4 5 6 8 9 10 12 15 18 20 25 30 50) m = 2160: d = (1 2 3 4 5 6 8 9 10 12 15 18 20 24 27 30 36 45 60) m = 2520: d = (1 2 3 4 5 6 7 8 9 10 12 14 15 18 20 21 28 30 35 42 70) m = 2880: d = (1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 30 32 36 40 48 60 80) m = 3360: d = (1 2 3 4 5 6 7 8 10 12 14 15 16 20 21 24 28 30 35 40 42 56 70) m = 3456: d= (1 2 3 4 6 8 9 12 16 18 24 32 36 48 72 96) m = 3600: d = (1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 25 30 36 40 45 50 60 75 100) m = 3840: d = (1 2 3 4 5 6 8 10 12 15 16 20 24 32 40 48 64 80) m = 4320: d = (1 2 3 4 5 6 8 9 10 12 15 16 18 20 24 27 30 36 40 45 48 54 60 72 90 120) m = 4608: d = (1 2 3 4 6 8 9 12 16 18 24 32 48 64 96 128) m = 5040: d = (1 2 3 4 5 6 7 8 9 10 12 14 15 16 18 20 21 24 28 30 35 36 40 42 45 56 60 63 70 84 105 140) The product of all the cells in the square is m^4. This means that for a given m, - m^4 must divide the product of all d values. - The product of the smallest 16 d values must be <= m^4. - The product of the largest 16 d values must be >= m^4. The only two m above that satisfy all these criteria are 4320 and 5040. So the smallest magic product of a 4x4 square of positive integers is either 4320 or 5040. Presumably someone can check whether 4320 is a magic product.