20170621, 21:04  #100 
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 

20170621, 23:36  #101  
"mahfoud belhadj"
Feb 2017
Kitchener, Ontario
2^{2}·3·5 Posts 
Quote:
But I can see that we can get a factor by adding 10+1=11. Finding a factor can be done even with 4sq reps that cannot tile a square. And that's the whole point of what I said previously. And in fact, there are two other 4sq reps that would give a factor (9,6,2,0) and (7,8,2,2). we can see that 9+2=11 and 7+2+2=11 although both of these 2 resp won't tile the 11x11 because 9+6>11 and 7+8>11. To get a tiling, I suppose we would have to expand 10^2 of the (10,4,2,1) rep as a sum of smaller squares. Last fiddled with by mahbel on 20170621 at 23:44 Reason: added one example 

20170622, 01:35  #102  
Aug 2006
175B_{16} Posts 
Quote:
I don't have good geometric intuition so I'm not sure I have much to contribute here. But aren't there parameterized dissections of rectangles into (fixed numbers of) squares? It might be interesting to see what factorizations they correspond to. 

20170622, 12:45  #103  
Feb 2017
Nowhere
2^{2}×3^{2}×139 Posts 
Quote:
None of this tiling business helps with the actual problem of factoring N, and this thread began with a claim of having a new method for doing that. I have produced examples in which your originallydescribed method simply cannot work. Your response was, "change the method." But you have not given any reason to justify the notion that combining the squares with their roots has any more chance of finding a factor of N than does trying random numbers. Or even that this approach always works. But I imagine that if someone went to the trouble of finding a counterexample to your "strong extended" method, you would just come up with another ad hoc variation that works on that particular example. And, since generating random numbers is so much easier than finding representations of N as a sum of four squares, experiment has shown that trying random numbers can find a factor faster than your "method," by many orders of magnitude. Your response to that was an insulting troll. Sorry, but merely repeating "it works!" no matter how many times, even in the face of contrary facts and evidence, and insulting those who provide such facts and evidence, doesn't go anywhere towards proving it does. The subject of tiling rectangles with squares has been around for a while. Back in the day, Martin Gardner devoted one of his "Mathematical Games" column to "Squaring the Square." There is even a doctoral thesis on the subject of tiling rectangles with squares here. Last fiddled with by Dr Sardonicus on 20170622 at 12:53 

20170622, 13:07  #104  
"mahfoud belhadj"
Feb 2017
Kitchener, Ontario
60_{10} Posts 
Quote:
As to using combinations of nonsquares (a,b,c,d) to find a factor, I have provided at least 3 examples to show that it works. Instead of commenting on those examples and telling us why they need to be rejected, you continue to ignore those results because it's difficult to argue with numbers when they are correct. The last example is 11^2= 10^2+4^2+2^2+1^2. Anyone can see that 10+1=11 is a factor. You are the one who is saying 10+1 sometimes makes 11 and sometimes doesn't. Please give us half a reason why we should reject 10+1=11 as a factor. Then we can all move on to something else. 

20170622, 14:44  #105  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
Quote:


20170622, 16:01  #106 
Aug 2006
3×1,993 Posts 

20170622, 18:30  #107  
"mahfoud belhadj"
Feb 2017
Kitchener, Ontario
2^{2}×3×5 Posts 
Quote:
We first try to use combinations of squares (a^2,b^2,c^2,d^2), then combinations of nonsquares (a,b,c,d) and if we still haven't find a factor, we use a mixture of square and nonsquares combinations. I have not experimented with the order, that is which combinations should be first used. I can't tell which one will lead faster to a factor. If by now, we haven't found a factor, we use the next 4sq representation and do the same thing. Last fiddled with by mahbel on 20170622 at 18:46 

20170622, 18:44  #108  
"mahfoud belhadj"
Feb 2017
Kitchener, Ontario
2^{2}×3×5 Posts 
Quote:
But in more than 18 months of testing, I have never encountered a number, among the small ones that were considered, that cannot be factored by a combination of the 3 combinations (squares, nonsquares, and squares + nonsquares). But of course, I realize that it's not enough and the numbers tried are small. Last fiddled with by mahbel on 20170622 at 18:47 

20170622, 19:21  #109  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}×131 Posts 
Quote:
Last fiddled with by science_man_88 on 20170622 at 19:59 

20170622, 20:31  #110  
"Forget I exist"
Jul 2009
Dumbassville
2^{6}·131 Posts 
Quote:
Last fiddled with by science_man_88 on 20170622 at 20:37 

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