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How to explain the patterns in the matrix defined in François Brunault's answer to the question Freeness of a Z[x] module depicted below? --

Choosing colors according to the highest power of 2 which divides an entry, we can obtain for example the following pictures:

A picture of resolution 512 x 512 pixels depicting the entries of François Brunault's matrix

A picture of resolution 512 x 512 pixels depicting the entries of François Brunault's matrix

Taking instead highest power of 3 dividing an entry, we can obtain for example this picture:

A picture of resolution 512 x 512 pixels depicting the entries of François Brunault's matrix

Coloring zero entries white, negative entries red and positive entries blue yields the following picture:

A picture of resolution 512 x 512 pixels depicting the entries of François Brunault's matrix

And finally, marking non-zero entries black and leaving zero entries blank, the pattern looks like this:

A picture of resolution 512 x 512 pixels depicting the entries of François Brunault's matrix

Added: Assigning colors according to numbers of prime factors of the entries, we can obtain the following picture:

A picture of resolution 182 x 182 pixels depicting the entries of François Brunault's matrix

Added on June 15: Choosing colors depending on entries (mod 47), we can obtain the following picture (nonzero entries divisible by 47 are colored red):

A picture of resolution 512 x 512 pixels depicting the entries of François Brunault's matrix

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Have you tried plotting the number of prime factors in colors? My plot: s24.postimg.org/5rj4gkkat/rxdivisors.png – joro Jun 14 '13 at 12:43
    
@Joro: Thanks. Yes, I tried also this- Though computing such picture in the same resolution (512 x 512 pixels) as the others seems to take notably more time, so I computed only smaller ones. – Stefan Kohl Jun 14 '13 at 13:31
    
Factoring the matrix entries is not always easy -- the factors are sometimes not all very small. Until line 140 of the matrix, the largest second-largest factor which occurs is 278454866719523378494429. – Stefan Kohl Jun 14 '13 at 14:52
    
Primes in the matrix appear very rare. – joro Jun 14 '13 at 15:32
3  
Thank you for spotting these nice patterns! Not a precise explanation, but it may not be surprising that the entries of the matrix have nice $p$-adic properties. Indeed, any generalized polynomial map (in the sense of the question you link to) extends to a map $\mathbf{Z}_p \to \mathbf{Z}_p$ which is continuous (and in fact, 1-Lipschitz). – François Brunault Jun 15 '13 at 18:29

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