20190118, 13:57  #45 
Oct 2017
142_{8} Posts 
46solution
Having finally found a 46solution I would like to know, if anyone has found a 47solution.

20190118, 14:37  #46 
1976 Toyota Corona years forever!
"Wayne"
Nov 2006
Saskatchewan, Canada
1171_{16} Posts 
How about a 5x5?
Not me. 
20190118, 22:51  #47 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
3×5×383 Posts 
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

20190126, 17:40  #48 
Jan 2017
79 Posts 
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?

20190127, 06:06  #49 
Oct 2017
2×7^{2} Posts 
313
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.

20190127, 10:59  #50 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
13161_{8} Posts 
Given all the differences between squares need lots of factors I would expect solutions to get bigger and bigger as more factors are needed.

20190203, 13:43  #51 
"Mike"
Aug 2002
2^{7}·61 Posts 

20190203, 14:01  #52  
Jan 2017
79 Posts 
Quote:


20190203, 14:47  #53  
Jun 2003
17·281 Posts 
Quote:


20190204, 11:50  #54 
Mar 2018
3×43 Posts 
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 largest element of the set with the zero. (Assuming both sets contain nonnegative numbers, of course). I believe, though don't claim, that it is also the smallest possible largest element of both sets. My other 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. Last fiddled with by DukeBG on 20190204 at 11:50 
20190204, 15:09  #55 
Jan 2017
79 Posts 
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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