TITLE: ASCII prime plots and prime-rich quadratic polynomials
QUESTION [6 upvotes]: 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

REPLY [3 votes]: 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...
           


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.