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