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Forum: Miscellaneous Math 2017-06-29, 18:11
Replies: 192
Views: 12,949
Posted By mahbel
You may be right. It could be just an artifact....

You may be right. It could be just an artifact. But as long as we don't have a proof, we can't rule it out.
Forum: Miscellaneous Math 2017-06-29, 13:31
Replies: 192
Views: 12,949
Posted By mahbel
my post showed that the expansion of squares...

my post showed that the expansion of squares (a,b,c,d) into sum of squares helps in response to your post in which you said it doesn't. And I gave an example which clearly shows that the expansion...
Forum: Miscellaneous Math 2017-06-29, 12:21
Replies: 192
Views: 12,949
Posted By mahbel
It does. Expanding the squares of a given 4-sq...

It does. Expanding the squares of a given 4-sq rep helps as a general rule. Look at the simple example of N=7*13=91, the first 4-sq rep is (5,5,5,4). There is not a single combination of squares that...
Forum: Miscellaneous Math 2017-06-28, 19:46
Replies: 192
Views: 12,949
Posted By mahbel
not necessarily a lower pair of Pythagorean...

not necessarily a lower pair of Pythagorean triple. 8^2=4^2+4^2+4^2+4^2 but 4^2=2^2+2^2+2^2+2^2. We may get some Pythagorean triplets but not as a general rule.
Forum: Miscellaneous Math 2017-06-28, 13:25
Replies: 192
Views: 12,949
Posted By mahbel
I cannot do the calculations by hand. Because...

I cannot do the calculations by hand.
Because we are dealing with a big number, the squares (a,b,c,d) themselves will have many 4-sq reps. The idea of my comment is to limit the search for factors...
Forum: Miscellaneous Math 2017-06-27, 23:34
Replies: 192
Views: 12,949
Posted By mahbel
It simply means that we only need to find the...

It simply means that we only need to find the first 4-sq rep, then expand its squares into sum of 4 squares and then use these expansions to form combinations of square and non-squares as done before...
Forum: Miscellaneous Math 2017-06-27, 18:50
Replies: 192
Views: 12,949
Posted By mahbel
[/LIST] It turned out that one 4-sq...

[/LIST]
It turned out that one 4-sq representation can be transformed into another simply by expanding the squares in the first one and re-arranging them to create new 4-sq representations. It can...
Forum: Miscellaneous Math 2017-06-24, 20:46
Replies: 192
Views: 12,949
Posted By mahbel
Yes.

Yes.
Forum: Miscellaneous Math 2017-06-24, 20:34
Replies: 192
Views: 12,949
Posted By mahbel
It is simply a confirmation that using...

It is simply a confirmation that using combinations of non-squares (a,b,c,d) to get a factor is justified.

In the original 4-sq rep used, (10,4,2,1), by just adding 10+1=11, we get a factor. But...
Forum: Miscellaneous Math 2017-06-24, 19:30
Replies: 192
Views: 12,949
Posted By mahbel
tiling an 11x11 square with squares from the 4-sq rep (10,4,2,1)

I made the tiling of an 11x11 square. I started with the 4-sq rep (10,4,2,1). Now using combinations of non-squares, we can find a factor by adding 2 terms 10+1=11. In this picture, it will be shown...
Forum: Homework Help 2017-06-24, 19:20
Replies: 1
Views: 2,118
Posted By mahbel
Forum: Homework Help 2017-06-24, 19:12
Replies: 1
Views: 2,118
Posted By mahbel
just a test of uploading a picture - please ignore

here it is, tiling an 11x11 square with smaller squares.

wrong dimensions, need to resize.
Forum: Miscellaneous Math 2017-06-24, 13:59
Replies: 192
Views: 12,949
Posted By mahbel
The answer Will Jagy gave in the thread on the...

The answer Will Jagy gave in the thread on the statckexchange site proved that writing 2N as a sum of two squares is equivalent to writing N as a sum of two squares. And he is one of the most...
Forum: Miscellaneous Math 2017-06-24, 12:55
Replies: 192
Views: 12,949
Posted By mahbel
I don't know if the algorithm is faster than...

I don't know if the algorithm is faster than factorization. I only know that such an algorithm exists. It is based on solving quadratic equations and finding values of a parameter k that makes the...
Forum: Miscellaneous Math 2017-06-23, 23:42
Replies: 192
Views: 12,949
Posted By mahbel
There are algorithms that can provide the...

There are algorithms that can provide the decomposition of an integer N into a sum two squares which do not require factoring N.
Forum: Miscellaneous Math 2017-06-23, 23:38
Replies: 192
Views: 12,949
Posted By mahbel
How I found them was described before in one of...

How I found them was described before in one of my post. You take a 4-sq rep (a,b,c,d) and you combine the elements a,b,c,d, in this case the non-squares.
a+b, a+c, a+d, b+c, b+d, a+b+c, a+b+d,...
Forum: Miscellaneous Math 2017-06-23, 19:33
Replies: 192
Views: 12,949
Posted By mahbel
1457 = 31*47 = 36^2 + 11^2 + 6^2 + 2^2 ...

1457 = 31*47 = 36^2 + 11^2 + 6^2 + 2^2

factor: 36+11=47

1457 = 31*47 = 27^2 + 20^2 +18^2 +2^2

factor: 27+20=47

1457 = 31*47 = 30^2 + 21^2 + 10^2 + 4^2
Forum: Miscellaneous Math 2017-06-23, 13:26
Replies: 192
Views: 12,949
Posted By mahbel
I share the conclusions you drew in the last...

I share the conclusions you drew in the last paragraph.

I don't know if a faster way to generate sums of 4 squares can be found. The online calculator I used is very slow when fed large numbers....
Forum: Miscellaneous Math 2017-06-22, 18:44
Replies: 192
Views: 12,949
Posted By mahbel
I can only agree with the above. The problem will...

I can only agree with the above. The problem will persist even if I were to provide 10^6 examples because the next example, the 10^6 +1, may not work. I realize that the method is not based on a...
Forum: Miscellaneous Math 2017-06-22, 18:30
Replies: 192
Views: 12,949
Posted By mahbel
yes, you are right. The second part of the quote...

yes, you are right. The second part of the quote above is exactly what we are trying to do. And we can't find a factor if we tie our hands behind our back.

We first try to use combinations of...
Forum: Miscellaneous Math 2017-06-22, 13:07
Replies: 192
Views: 12,949
Posted By mahbel
I think you are the one doing all the insulting....

I think you are the one doing all the insulting. Just re-read your comments in case you forgot.

As to using combinations of non-squares (a,b,c,d) to find a factor, I have provided at least 3...
Forum: Miscellaneous Math 2017-06-21, 23:36
Replies: 192
Views: 12,949
Posted By mahbel
the 11x11 square cannot be tiled by (10,4,2,1)...

the 11x11 square cannot be tiled by (10,4,2,1) squares, simply because if you start with 10x10 square you won't be left with enough space for the other squares.

But I can see that we can get a...
Forum: Miscellaneous Math 2017-06-21, 18:42
Replies: 192
Views: 12,949
Posted By mahbel
No, you don't need to "have already factored N"....

No, you don't need to "have already factored N". It was done to justify using combinations of non-squares (a,b,c,d) to get a factor. And just like combinations of squares (a^2,b^2,c^2,d^2), some...
Forum: Miscellaneous Math 2017-06-21, 13:23
Replies: 192
Views: 12,949
Posted By mahbel
Not really. Fitting squares into squares looks...

Not really. Fitting squares into squares looks more complicated to me. Fitting squares into rectangles is what I am trying to do to justify using non-squares (a,b,c,d) to find factors. If one (or...
Forum: Miscellaneous Math 2017-06-21, 12:33
Replies: 192
Views: 12,949
Posted By mahbel
On a possible justification of using combinations...

On a possible justification of using combinations of non-square (a,b,c,d) to find factors.

There are two ways to look at using the 4-sq reps, the arithmetic way and the geometric way.

The...
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