Patterns in A129783?

[Andrew Plewe]

"So m^2 is almost certainly the smallest square >= pq for all prime p"

I believe that this is true for all odd (and even) integers; as long as 
p and q are either both odd or both even and q is a bounded distance 
away from p. For p = 7:

7*9 = 63
8*8 = 64
diff = 1

7*11 = 77
9*9 = 81
diff = 4

7*13 = 91
10*10 = 100
diff = 9

7*15 = 105
11*11 = 121
diff = 16

7*17 = 119
12*12 = 144
diff = 25 <-- first square >= 2p

7*19 = 143
13*13 = 169
diff = 36 (but 144 is actually the next square, so the pattern ends here)

So, as long as m^2 - pq <= 2p+r (where 2p+r is the smallest square 
greater than 2p), then m^2 will always be the next largest square.

    -Andrew Plewe-