Something I’ve been going over with baker/Jan (and briefly VA) on Discord:

So (x+n)(x+n) = 2d(n-1) + nn + f – 1. We’ve known this for months now. You can create the smaller square that you’d add to c to make the bigger square with 2d(n-1), nn, f and -1. I was just thinking about why VQC brought up sqrt(2d) and sqrt(e). 2d(n-1)+nn+f-1 is always going to be a square with a valid cell. So if we take the square root of 2d(n-1)+nn+f-1, we’ll get x+n, right?

x+n = sqrt(2d(n-1)+nn+f-1)

This is true, and can be verified with test cases. e.g. c559, (30:5:6) = {30:5:23:10:13:43} f=17

x+n = sqrt(2d(n-1)+nn+f-1)

10+5 = sqrt(46(4)+25+17-1)

15=sqrt(225) (this is true)

What’s weird is when you use algebra to try to isolate sqrt(2d) or any of the other elements.

x = sqrt(2d(n-1)) + sqrt(f) - 1

This can also be represented as

x = sqrt(2d(n-1)) + sqrt(2d) – sqrt(e)

which isolates sqrt(2d) and sqrt(e), two newer elements that VQC brought up probably about a month ago now. This seems like it makes algebraic sense, right? I mean, as an example, 16=28, and sqrt(16)=sqrt(2)sqrt(8). So if you plugged in the numbers, surely you’d calculate the correct x, right? Well, this x should be 10. Let’s plug in some numbers. First with the sqrt(f) equation:

x = sqrt(2d(n-1)) + sqrt(f) – 1

x = sqrt(46(4)) + sqrt(17) – 1

x = 16.687765592

Clearly that isn’t correct. Let’s also do it with the 2d and e equation.

x = sqrt(2d(n-1)) + sqrt(2d) – sqrt(e)

x = sqrt(46(4)) + sqrt(46) – sqrt(30)

x = 14.869764374

This not only also isn’t equal to 10 but it also isn’t equal to the sqrt(f) version of the equation.

So does anyone know why this would happen? Is it us making a mistake with the algebra, or is there something weird going on with these squares?

>>7212

See that's what I thought, but it doesn't seem like any of us know how to isolate sqrt(2d) and sqrt(e). Do you?

I've been looking into this crumb a bit:

"The values of a in the first cell where e=1 contains ALL factors for odd numbers that are the sum of two squares"

I haven't found anything groundbreaking (that I'm aware of) but I did find some interesting patterns. I'll dump numbers in this post and then explain some things in some replies. The numbers on the left are the sequence of a values in (1,1). The numbers on the right are the factors of these numbers.

1

5

13

25 5

41

61

85 5 17

113

145 5 29

181

221 13 17

265 5 53

313

365 5 73

421

481 13 37

545 5 109

613

685 5 137

761

841 29

925 5 25 37 185

1013

1105 5 13 17 65 85 221

1201

1301

1405 5 281

1513 17 89

1625 5 13 25 65 125 325

1741

1861

1985 5 397

2113

2245 5 449

2381

2521

2665 5 13 41 65 205 533

2813 29 97

2965 5 593

3121

3281 17 193

3445 5 13 53 65 265 689

3613

3785 5 757

3961 17 233

4141 41 101

4325 5 25 173 865

>>7524

The first most obvious thing is that when a number is the sum of two squares and it's odd, it will only be divisible by other numbers that are also the sum of two squares and odd. So if you have a c value and it's in an e column where e is a square (because this means it's the sum of two squares, based on another crumb), its factors will be in columns where e is a square too. I'll post more in a while.

>>7524

The second useful thing to mention, that I probably should have mentioned first, is that every value of a in (1,1) is the sum of two squares and is odd. VQC's crumb said that these values "contain ALL factors for odd numbers that are the sum of two squares", but they are also themselves odd numbers that are the sum of two squares.

Third thing: while these a values are each odd sums of two squares, sum odd sums of two squares are missing. Here's the first ten odd sums of squares: 1, 5, 13, 17, 25, 29, 37, 41, 45, 53. Here are the first ten a values in (1,1): 1, 5, 13, 25, 41, 61, 85, 113, 145, 181. So, as you can see, while a in (1,1) are all odd square sums, some of the odd square sums are missing, such as 17, 29, 37, 45 and 53.

Fourth thing: the sequence of a values starts at 1 and increases by 4(x-1) where x is the place in the sequence. In other words, 1, +4=5, +8=13, +12=25, +16=41, etc. This follows the pattern of sums of squares that are odd in that they are all equal to a multiple of 4 plus 1.

Fifth thing: these odd numbers that are the sum of two squares appear as factors of others in the series in a pattern. Where a number appears as a factor, it will appear as a factor again the number of numbers away that is equal to itself. If that was worded confusingly, take this example: where 5 appears as a factor, it will appear again 5 numbers in the sequence away, then 5 away again, and so on. It will also always appear once between these times. This seems to hold for all of them. For 5, it appears two away, then three away (this second one being 5 away from the origin), and this pattern repeats. When 13 appears as a factor, it appears 8 away, and then appears again 5 away, and so on. Where 17 appears as a factor, it appears 4 away, and then appears again 13 away, and so on. For 25, it's 18 and 7. For 29, it's 12 and 17. This seems to hold for all numbers that are odd and the sum of two squares.

>>7524

>>7525

>>7526

In summary, the crumb "the values of a in the first cell where e=1 contains ALL factors for odd numbers that are the sum of two squares" is true because all of these a values are themselves odd numbers that are the sum of two squares, and because every number that is odd and the sum of two squares can only be divided by other numbers that are odd and the sum of two squares. I'm not sure if every c value in the columns where e is a square are the sum of two squares, but every c value that is the sum of two squares does appear in these columns. If you have a c value that is the sum of two squares, it will appear as an a value in (1,1) and its factors will appear in (1,1).