[math-fun] sum-product matrices
A matrix of distinct positive integers is sum-product if its rows have equal sum and columns have equal (possibly distinct) product. Any 1 x 1 matrix is trivially sum-product. No 1 x n or n x 1 matrix is sum-product. No 2 x 2 matrix is sum-product. In a sum-product matrix a b c d we must have a+b = c+d and ac = bd. But then ac+bc = bc+bd => (a+b)c = b(c+d) => c = b, and the elements are not distinct. There are an infinitude of 2 x 3 sum-product matrices. I believe the smallest, both in row sum and colum product, is 4 10 12 15 6 5 with row sum 26 and column product 60. I believe the following is true: For positive integers x, y, z, let u = x+y+z and v = xy+xz+yz. Then the matrix M(x, y, z) = vx vy vz uyz uxz uxy has equal row sums uv and equal column products xyzuv. If you choose x < y < z with xz < y^2 and gcd(x, y, z) = 1, form M(x, y, z), and divide this by the gcd of its elements, then I believe you will generate the "primitive" 2 x 3 sum-product matrices. I'm pretty sure that apart from 1 x 1 matrices, there are no square sum-product matrices. ? But I was pleased to find that there are indeed sum-product matrices with equal row sum and column product. The following are (all?) "primitive" 14 x 2 sum-product matrices with sum = product = 1890. By "primitive" I mean distinct up to row and column permutation. 2 3 15 45 63 70 90 105 135 189 210 270 315 378 945 630 126 42 30 27 21 18 14 10 9 7 6 5 2 5 7 21 30 45 70 105 126 135 189 210 315 630 945 378 270 90 63 42 27 18 15 14 10 9 6 3 2 5 9 14 27 45 63 90 105 126 189 270 315 630 945 378 210 135 70 42 30 21 18 15 10 7 6 3 2 5 14 15 18 27 42 63 90 189 210 270 315 630 945 378 135 126 105 70 45 30 21 10 9 7 6 3 2 6 7 18 27 30 42 90 126 135 189 210 378 630 945 315 270 105 70 63 45 21 15 14 10 9 5 3 2 7 9 15 18 27 30 45 90 135 189 315 378 630 945 270 210 126 105 70 63 42 21 14 10 6 5 3
I'm pretty sure that apart from 1 x 1 matrices, there are no square sum-product matrices.
1 10 15 12 6 8 20 4 2
But I was pleased to find that there are indeed sum-product matrices with equal row sum and column product. The following are (all?) "primitive" 14 x 2 sum-product matrices with sum = product = 1890.
there is a 10x2 sum-product matrix with equal row sum 840 and column product 840: 2 4 7 24 40 70 105 140 168 280 420 210 120 35 21 12 8 6 5 3 you can find more info on sum-product matrices here: http://www2.stetson.edu/~efriedma/mathmagic/1102.html erich
Erich, I should have known you would be all over this. My name was clearly original. I should never even have an idea without consulting you first. :-) ----- Original Message ----- From: "Erich Friedman" <efriedma@stetson.edu> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Wednesday, January 20, 2010 7:54 PM Subject: Re: [math-fun] sum-product matrices
I'm pretty sure that apart from 1 x 1 matrices, there are no square sum-product matrices.
1 10 15 12 6 8 20 4 2
But I was pleased to find that there are indeed sum-product matrices with equal row sum and column product. The following are (all?) "primitive" 14 x 2 sum-product matrices with sum = product = 1890.
there is a 10x2 sum-product matrix with equal row sum 840 and column product 840:
2 4 7 24 40 70 105 140 168 280 420 210 120 35 21 12 8 6 5 3
you can find more info on sum-product matrices here:
http://www2.stetson.edu/~efriedma/mathmagic/1102.html
erich
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-------------------------------------------------------------------------------- No virus found in this incoming message. Checked by AVG - www.avg.com Version: 8.5.432 / Virus Database: 271.1.1/2634 - Release Date: 01/20/10 09:12:00
Coincidence on our subjects! The additive-multiplicative magic squares n x n, mentioned in my previous email on multimagie.com, are sum-product matrices m x n ... of course with m = n... but with supplemental properties: -their n rows AND their n columns (and also their 2 diagonals) have the same sum S -their n rows AND their n columns (and also their 2 diagonals) have the same product P Christian. -----Message d'origine----- De : math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] De la part de David Wilson Envoyé : jeudi 21 janvier 2010 10:15 À : math-fun Objet : Re: [math-fun] sum-product matrices Erich, I should have known you would be all over this. My name was clearly original. I should never even have an idea without consulting you first. :-) ----- Original Message ----- From: "Erich Friedman" <efriedma@stetson.edu> To: "math-fun" <math-fun@mailman.xmission.com> Sent: Wednesday, January 20, 2010 7:54 PM Subject: Re: [math-fun] sum-product matrices
I'm pretty sure that apart from 1 x 1 matrices, there are no square sum-product matrices.
1 10 15 12 6 8 20 4 2
But I was pleased to find that there are indeed sum-product matrices with equal row sum and column product. The following are (all?) "primitive" 14 x 2 sum-product matrices with sum = product = 1890.
there is a 10x2 sum-product matrix with equal row sum 840 and column product 840:
2 4 7 24 40 70 105 140 168 280 420 210 120 35 21 12 8 6 5 3
you can find more info on sum-product matrices here:
http://www2.stetson.edu/~efriedma/mathmagic/1102.html
erich
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
---------------------------------------------------------------------------- ---- No virus found in this incoming message. Checked by AVG - www.avg.com Version: 8.5.432 / Virus Database: 271.1.1/2634 - Release Date: 01/20/10 09:12:00 _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (3)
-
Christian Boyer -
David Wilson -
Erich Friedman