46solution
Having finally found a 46solution I would like to know, if anyone has found a 47solution.

How about a 5x5?
Not me. 
If ai+b is a square and ajai>0 is less than 2*sqrt(ai+b)+1 then aj+b can't be a square as it is less than the next square (sqrt(ai+b)+1)^2

I improved my search program a bit and have found 8 distinct 4+6 solutions. Looks like 4+7 or 5+5 solutions would have to be pretty huge. It's not obvious whether arbitrarily large solutions can be expected to exist at all. Has anyone tried to analyze that?

313
[QUOTE=uau;506908]I improved my search program a bit and have found 8 distinct 4+6 solutions. Looks like 4+7 or 5+5 solutions would have to be pretty huge. It's not obvious whether arbitrarily large solutions can be expected to exist at all. Has anyone tried to analyze that?[/QUOTE]
Using your approach I have found a 313solution  but I fear that this doesnât help very much. I continue to search for 4+7, but without analyzing. 
Given all the differences between squares need lots of factors I would expect solutions to get bigger and bigger as more factors are needed.

[url]http://www.research.ibm.com/haifa/ponderthis/solutions/January2019.html[/url]

[QUOTE=Xyzzy;507545][URL]http://www.research.ibm.com/haifa/ponderthis/solutions/January2019.html[/URL][/QUOTE]
This lists the same 4+6 solution many times as a "different" one. If you multiply all the numbers by a second power, all the sums obviously stay squares (square times square is a square). So to tell whether solutions are truly distinct, you should make sure to divide out any such common multiples. That obviously wasn't done when writing this solution page. 
[QUOTE=uau;507548]This lists the same 4+6 solution many times as a "different" one. If you multiply all the numbers by a second power, all the sums obviously stay squares (square times square is a square). So to tell whether solutions are truly distinct, you should make sure to divide out any such common multiples. That obviously wasn't done when writing this solution page.[/QUOTE]
Right. You should start out by normalizing such that smallest element is 0. Then divide out gcd (and list them in sorted order). 
My solution is not listed, as far as I can tell. Though I didn't try to normalize the listed ones.
[0, 36295, 233415, 717255] & [93^2, 267^2, 501^2, 1059^2] the second set expanded [8649, 71289, 251001, 1121481] I also claim that this pair of sets is the 44 solution with the smallest possible [I]largest element of the set with the zero[/I]. (Assuming both sets contain nonnegative numbers, of course). I believe, though don't claim, that it is also the smallest possible [I]largest element of both sets[/I]. My [I]other[/I] 44 solution is [0, 259875, 475875, 1313091] & [15^2, 447^2, 895^2, 1695^2]. In case anyone wants to make a registry of normalized solutions. I ended up not bothering to find 45 or larger solution. 
[QUOTE=DukeBG;507621]My [I]other[/I] 44 solution is [0, 259875, 475875, 1313091] & [15^2, 447^2, 895^2, 1695^2]. In case anyone wants to make a registry of normalized solutions.
I ended up not bothering to find 45 or larger solution.[/QUOTE] I doubt anyone would bother with a list of 4+4 solutions, or at least not one maintained by hand. I found over two thousand different 4+5 solutions, and 4+4 ones are more common (I didn't directly count those). Currently found 4+6 solutions could be listed by hand, but 4+5 and smaller are too common for that. 
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