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This is a series of questions inspired by the MathOverflow question Find the least prime so that p-1 has two factors greater than $m$ and $n$ posted by Aaron Sterling.

I suggested plotting primes by marking the status of the number $(nm+1)$ at coordinate $(n,m)$. Using commutativity, I have combined two ASCII art plots for $1 \leq n,m \leq 50$ in a figure-ground contrast. (Perhaps Joseph O'Rourke will be inspired to provide some similar but nicer looking plots for other ranges of $n$ and $m$.) In the plots below, + indicates $nm+1$ is prime, and other characters (after a shift in one coordinate) indicates whether $nm + 1$ has a factor of two, three or five.

+  .o.Oo .oO o .O. o Oo. oO.o. O .o.Oo .oO o .O. o Oo
++  O o  O  o Oo  oO o  O  o Oo  oO o  O  o Oo  oO o
 +   O . .O. . O . .O. . O . .O. . O . .O. . O . .O.
+ ++   oO o  O  o Oo  oO o  O  o Oo  oO o  O  o Oo  o
 +     . o .o. o .o. o .o. o .o. o .o. o .o. o .o. o
+++ ++     O    O    O    O    O    O    O    O    O
   + +   Oo. oO.o. O .o.Oo .oO o .O. o Oo. oO.o. O .o
 +  +     O o  O  o Oo  oO o  O  o Oo  oO o  O  o Oo
 + +   +   . O . .O. . O . .O. . O . .O. . O . .O. .
+ ++ ++  +   o  o  o  o  o  o  o  o  o  o  o  o  o  o
 +   + +     . oO.o. O .o.Oo .oO o .O. o Oo. oO.o. O
+ + ++ ++     O    O    O    O    O    O    O    O
   + +   + +   Oo. oO.o. O .o.Oo .oO o .O. o Oo. oO.o
 ++ +  ++    +    O  o Oo  oO o  O  o Oo  oO o  O  o
 + +     + + +   . . . . . . . . . . . . . . . . . .
+    ++    +  ++   o Oo  oO o  O  o Oo  oO o  O  o Oo
     + +     +     O o .O. o Oo. oO.o. O .o.Oo .oO o
++ + ++ +++   + +   O    O    O    O    O    O    O
         + +         .oO o .O. o Oo. oO.o. O .o.Oo .o
 ++ +   +  + +     +    o  o  o  o  o  o  o  o  o  o
 +   +   +     + + +   . .O. . O . .O. . O . .O. . O
+ ++    +     ++ ++ +   Oo  oO o  O  o Oo  oO o  O  o
 +   +     +       +     O o .O. o Oo. oO.o. O .o.Oo
  ++   + +  ++  +++    +    O    O    O    O    O
   + +   +     +       +   .o. o .o. o .o. o .o. o .o
 ++ ++     +    +  ++ +  +    oO o  O  o Oo  oO o  O
   + +   +   + + + +         O . .O. . O . .O. . O .
+  +  +  + +  ++     + ++ +   Oo  oO o  O  o Oo  oO o
 +     +   +     +             . O .o.Oo .oO o .O. o
++  +++++ +  +   ++++++ + +
         + +         +           .o.Oo .oO o .O. o Oo
  +  + +  +  +   + ++  +    +     O o  O  o Oo  oO o
 +   +   + + +     + +   +   +     O . .O. . O . .O.
  ++  + +  ++    ++       ++ +  +    oO o  O  o Oo  o
 +   + +   + +   + +     +   +       . o .o. o .o. o
+++ +     ++  +++   + +  + +  ++   +     O    O    O
   + +         +                 +     Oo. oO.o. O .o
    ++    ++  + +  +        +  +        O o  O  o Oo
 + +   +     +       + +   + + + +   +   . O . .O. .
+    ++  +  + ++  +  +       +  ++  +  +   o  o  o  o
 +               + +         +       +     . oO.o. O
+ + +  ++++ + ++ +  + ++++    +  ++  +  +   O    O
   +     +           + +     +     +   +     Oo. oO.o
 +     ++    ++    +  +     ++ ++               O  o
   + +     + +   +   +   +       + +   +       . . .
+ +  +   +  + +  +  ++  +  + ++    +     ++      o Oo
     +       +     +   + +         + +     +     O o
 + ++ + +  + + +    +  ++++   +     + +  + ++     O
   +     +       +         + +   +                 .o
 ++ +  +   + ++     + ++ +  +  +   + ++     + +

Based on this plot, I suspect my conjecture about the prime "nearest" to and greater than $n*m$ being at most $4nm$ not only holds (as a sort of 2-dimensional Bertrand's conjecture), but that this prime differs in taxicab distance by $O(\log(nm)^2)$. In other words, there is an absolute constant $C$ such that there is a prime $p$ with $p-1 = n'm'$, and with $n \leq n' \leq n + C\log(nm)^2$ and also $m \leq m' \leq m + C\log(nm)^2$. I am interested in information supporting or refuting my suspicion (and I suspect Aaron Sterling shares this interest), but that is incidental to what follows.

The primary question is a reference request: has anyone seen a plot like this before in the literature? I know of Ulam Spirals ( http://en.wikipedia.org/wiki/Ulam_spiral ) and it seems that artefacts in the plot might be related to a conjecture of Hardy and Littlewood regarding primes of the form $ax^2 +bx +c$. What I find striking are the diagonals that occur in the plot, especially those starting at $(a,a)$ and continuing in the direction $(2,-1)$. In particular, the sequence 101,109,113,113,109,101,89,73,53,29 appears as such a diagonal. Is it possible that primes of the form $a^2 + 1$ lead to prime rich polynomials of the form $(a+2t)(a-t)+1$? The secondary question series is: what is known about prime rich quadratic polynomials, and does such knowledge follow naturally from studying plots like those above?

Gerhard "Ask Me About System Design" Paseman, 2011.10.11

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2 Answers

joro's plots are much more detailed and informative, but here is a plot closer to your original ASCII art: red is prime, green is composite with no prime factor in $\lbrace 2,3,5,7,11 \rbrace$, and the other colors indicate composites with smallest factor among those primes (blue, purple, magenta, orange, yellow). The range here is $n,m \in [1,300]$, with $(1,1)$ in the lowerleft corner. I find this display relatively unrevealing, only repeating the grid-like pattern evident in joro's plots. Edit: I saved the image through a different process and the grid-like structure disappeared, pretty much confirming that the grid-pattern was an imaging artifact. This less-deceiving image is, alas, not very interesting...
           Gerhard Primes


To respond to joro's question, here is just primes vs. not-primes (red and white), $n,m \in [1,1000]$, which should be directly comparable to his last plot.
           Gerhard Primes

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btw, my prime plot appears counterintuitive to me (it may be either a bug or some strangeness in visual perception). Do you get similar results if you plot prime $xy+1$ as white and nonprime as black? –  joro Oct 11 '11 at 13:14
    
@joro: Perhaps we are seeing some effect of the interpolation performed by JPEG conversion? My plot of just the primes does not show the bands. Incidentally, for me (1,1) is the lowerleft corner. In any case, these plots seem not as interesting as prime spirals... –  Joseph O'Rourke Oct 11 '11 at 13:30
    
When I look at my first plot at different scales, the bands sometimes disappear. So perhaps they are an artifact of the interpolation algorithms that underlay the displays? –  Joseph O'Rourke Oct 11 '11 at 13:34
    
Hm, I don't see a bug in my program and I am using PNG which to my knowledge is lossless. I "zoomed" the plot in sage and this might have artificially caused the grid-like effect - I don't see a grid for the primes at original size. Can you try black and white and zooming before interpolating? –  joro Oct 11 '11 at 13:41
    
Oh, your image is less than 1000 pixels and your range is 1000? The shrinking might cause loss - jpeg compresses and loses information... –  joro Oct 11 '11 at 13:45
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Here is a sage plot of the number of divisors of $xy$ and $xy+1$ and primality of $xy+1$ (darker is smaller):

EDIT

As Joseph O'Rourke and Gerhard Paseman pointed out, the original plots were incorrect.

The grid like structure visible in some of the plots was caused by image processing, most likely by interpolation in the shrinking.

Updated the images (and the constants in the code) to hopefully something saner.

number of divisors of $xy$: number of divisors of $xy+1$

number of distinct prime factors of $xy+1$:

$xy+1$ is prime:

$xy+1$ is prime zoomed:

Just for recreational purposes plot of $x^2+y^2 \mod 2^9$ in HSV colormap.

Code:

sage: lim=750
sage: D=[[ len(divisors(i*j)) for i in range(1,lim)] for j in range(1,lim)]
sage: M=matrix_plot(D,vmin=2,vmax=max(flatten(D)),cmap='gray')
sage: M.show(figsize=[10,10]) #might be dependent on dpi

for the second

sage: D=[[ len(divisors(i*j+1)) for i in range(1,lim)] for j in range(1,lim)]
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It's interesting that your last plot does not seem to be symmetric across the main diagonal. Then again, it may be my monitor. Gerhard "Ask Me About System Design" Paseman, 2011.10.11 –  Gerhard Paseman Oct 12 '11 at 0:41
    
@Gerhard on second thought you may be right. Looking at zoom x2 x4 the last image doesn't appear symmetric indeed. Well may be buggy... –  joro Oct 12 '11 at 6:37
    
@Gerhard thanks for spotting this. Corrected the plots, hopefully they are saner now. –  joro Oct 12 '11 at 7:36
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