Find that I DO turn to numbers routinely. And the authority you feel when the numbers behave as mathematics requires is like no other to me.

The math does not care. But the math is never wrong.

And there is a comfort in that which I think turns into a sense of protection. So the math may not care but you can find shelter in truth.

So yes, will turn to numbers in comfort and love the conversation with the math even knowing the math does not care. But the math can talk to you. And you can talk to the math, and ask questions!

And if you ask the right questions, the math can give you the truth.

Without a doubt to me is one of the greatest phrases in the English language. And in mathematics truth can be found, without a doubt.

James Harris

# Some Math

For people who love numbers. Global resource of innovative mathematical ideas.

## Monday, June 12, 2017

### Summing seven squares to a square

Discovered a simple technique to build sums of as many squares as you want to a square. For example, here is a sum of seven squares to get a square:

_____________________________

My basic result used to get an arbitrary length of sums of squares is that in general there must always exist nonzero x and y, such that for an integer n equal to 1 or higher, and an integer m equal to 3 or higher:

Where n starts at 1. Often I like to start it at zero so it's a count of iterations but is prettier starting at 1.

For a sum of c+1 squares:

The BQD Iterator is:

Given nonzero integers u and v with

u

then it must also be true that

(u - (s

So for 7 squares, I'll need 6 s's and I'll use primes: 2, 3, 5, 7, 11 and 13

Then m = 4 + 9 + 25 + 49 + 121 + 169 + 1 = 378

1

First iteration: (-376)

**34967**^{2}+ 752^{2}+ 1128^{2 }+ 1880^{2 }+ 2632^{2}+ 4136^{2}+ 4888^{2}= 35721^{2}_____________________________

My basic result used to get an arbitrary length of sums of squares is that in general there must always exist nonzero x and y, such that for an integer n equal to 1 or higher, and an integer m equal to 3 or higher:

**x**^{2}+ (m-1)y^{2}= m^{n}Where n starts at 1. Often I like to start it at zero so it's a count of iterations but is prettier starting at 1.

For a sum of c+1 squares:

**m = s**_{1}^{2}+...+s_{c}^{2}+ 1

**x**^{2}+ (**s**_{1}^{2}+...+s_{c}^{2}**)y**^{2}= m^{n}The BQD Iterator is:

Given nonzero integers u and v with

u

^{2}+ (s_{1}^{2}+...+ s_{c}^{2})v^{2}= Fthen it must also be true that

(u - (s

_{1}^{2}+..+s_{c}^{2})v)^{2}+ (s_{1}^{2}+...+s_{c}^{2})(u + v)^{2}= (s_{1}^{2}+...+ s_{c}^{2}+ 1)*FSo for 7 squares, I'll need 6 s's and I'll use primes: 2, 3, 5, 7, 11 and 13

Then m = 4 + 9 + 25 + 49 + 121 + 169 + 1 = 378

1

^{2}+ 377*1^{2}= 378First iteration: (-376)

^{2}+ 377*(2)^{2}= 378^{2}
Second iteration: (-1130)

^{2}+ 377*(-374)^{2}= 378^{3}
Third iteration: (139868)

^{2}+ 377*(-1504)^{2}= 378^{4}
Which is: (139868)

^{2}+ 4*(-1504)^{2}+ 9*(-1504)^{2 }+ 25*(-1504)^{2 }+ 49*(-1504)^{2}+ 121*(-1504)^{2}+ 169*(-1504)^{2}= 378^{4}
Can divide both sides by 16, and get rid of negatives to get:

Which is: 34967

^{2}+ 752^{2}+ 9*(376)^{2 }+ 25*(376)^{2 }+ 49*(376)^{2}+ 121*(376)^{2}+ 169*(376)^{2}= 189^{4}
And now get final result where will show as all squares:

**34967**

^{2}+ 752^{2}+ 1128^{2 }+ 1880^{2 }+ 2632^{2}+ 4136^{2}+ 4888^{2}= 35721^{2}
Which is interesting to me, I think. Looks more impressive that way. Of course watch it get built maybe less impressive? But still is, a sum of seven squares to get a square.

Don't really see a practical use, so to me? Is just pure math.

James Harris

## Friday, June 09, 2017

### Talking my modular inverse discovery

The modular inverse is a rather simple thing from modular arithmetic, like consider:

2(3) ≡ 1 mod 5

Here 2 is the modular inverse of 3 and vice versa because they multiply to have a residue of 1 modulo 5, which is the modulus. That can be written as:

2 ≡ 3

Importantly in my research as I use SO much modular long ago tired of copying and pasting the modular congruence symbol so just use equals, so I have:

2(3) = 1 mod 5 and then: 2 = 3

May seem small but in my experience you can face purists who will be dismissive on such small matters! When am someone who is NOT going to waste time copying and pasting something all over a vast amount of research just to appease such people as is human convention. The math does not care.

The modular inverse as a concept has been around for some time, but only a few basic approaches for finding it were previously known. One relies on something called the extended Euclidean algorithm, which is very simple but will just link. The other approach depends on something called Euler's theorem, where learned the above from article on Wikipedia on what they call the modular multiplicative inverse.

And now can add another basic approach which relies on a system of equations I discovered about a month ago. For some residue r modulo N, its modular inverse is:

Where y

and

---------------------------------------------------

Copied from my post: Modular Inverse Innovation

It is also derived there. And my system, is kind of more direct like that which follows from Euler's theorem, but is also iterative like what follows from extended Euclidean algorithm. But allows you to fiddle with things in a way that neither does. So you can pick two key variables:

And they are so named because of how I discovered the system, as was just kind of puzzling over some things.

It is one of my most direct discoveries which surprised me a bit, as it just

Research about the modular inverse on the web I've done since, when yeah got REALLY interested, has indicated is also a practical result as calculations of the modular inverse are part of modern techniques, according to that research. So it has applied and pure math aspects.

The result is definitive though in terms of evaluating social aspects of how discovery is actually treated versus how one might imagine. I've had lots of experience in this area with prior results.

Mathematics discovered belongs to the human race.

James Harris

2(3) ≡ 1 mod 5

Here 2 is the modular inverse of 3 and vice versa because they multiply to have a residue of 1 modulo 5, which is the modulus. That can be written as:

2 ≡ 3

^{-1}mod 5Importantly in my research as I use SO much modular long ago tired of copying and pasting the modular congruence symbol so just use equals, so I have:

2(3) = 1 mod 5 and then: 2 = 3

^{-1}mod 5May seem small but in my experience you can face purists who will be dismissive on such small matters! When am someone who is NOT going to waste time copying and pasting something all over a vast amount of research just to appease such people as is human convention. The math does not care.

The modular inverse as a concept has been around for some time, but only a few basic approaches for finding it were previously known. One relies on something called the extended Euclidean algorithm, which is very simple but will just link. The other approach depends on something called Euler's theorem, where learned the above from article on Wikipedia on what they call the modular multiplicative inverse.

And now can add another basic approach which relies on a system of equations I discovered about a month ago. For some residue r modulo N, its modular inverse is:

rr

^{-1}= (n-1)(r + 2my_{0}) - 2md mod NWhere y

_{0}is chosen as is m, with m not equal to r, and n and d are to be determined. They are found from:**2mdF**_{0}= [F_{0}(n-1) - 1](r + 2my_{0}) mod Nand

**F**_{0}= r(r+2my_{0}) mod N---------------------------------------------------

Copied from my post: Modular Inverse Innovation

It is also derived there. And my system, is kind of more direct like that which follows from Euler's theorem, but is also iterative like what follows from extended Euclidean algorithm. But allows you to fiddle with things in a way that neither does. So you can pick two key variables:

**m**and**y**_{0}And they are so named because of how I discovered the system, as was just kind of puzzling over some things.

It is one of my most direct discoveries which surprised me a bit, as it just

*flowed*. And I'm beginning to accept that I have years of experience which has given me a certain level of expertise.Research about the modular inverse on the web I've done since, when yeah got REALLY interested, has indicated is also a practical result as calculations of the modular inverse are part of modern techniques, according to that research. So it has applied and pure math aspects.

The result is definitive though in terms of evaluating social aspects of how discovery is actually treated versus how one might imagine. I've had lots of experience in this area with prior results.

Mathematics discovered belongs to the human race.

James Harris

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