Two Rules of Successive Odd Numbers Apply to Square Numbers

Obviously, 1, 4, 9, 16, 25, 36, 49 ... is as we have been taught in school

1 + 3 = 4, 4 + 5 = 9, 9 + 7 = 16, 16 + 9 = 25, 25 + 11 = 36, 36 + 13 = 49 ...

Now take a look at 49, it is 7 * 7. Imagine we are dealing with a square of pebbles or where dimensions are required to be one precise set of length units. Now, take the same parameter and make the rectangle oblong, by shuffling the units from width to length.

7 * 7 = 49
	49 - 1 = 48
6 * 8 = 48
	48 - 3 = 45
5 * 9 = 45
	45 - 5 = 40
4 * 10 = 40
	40 - 7 = 33
3 * 11 = 33
	33 - 9 = 24
2 * 12 = 24
	24 - 11 = 13
1 * 13 = 13

Is this a law? Check 64.

8 * 8 = 64
	64 - 1 = 63
7 * 9 = 63
	63 - 3 = 60
6 * 10 = 60
	60 - 5 = 55
5 * 11 = 55
	55 - 7 = 48
4 * 12 = 48
	48 - 9 = 39
3 * 13 = 39
	39 - 11 = 28
2 * 14 = 28
	28 - 13 = 15
1 * 15 = 15

I think it is. How about cubes, now?

First of all, what are the successive differences between cube numbers?

1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.

1 + 1 + 1*6 = 8; 8 + 1 + 3*6 = 27; 27 + 1 + 6*6 = 64; 64 + 1 + 10*6 = 125; 125 + 1 + 15*6 = 216; 216 + 1 + 21*6 = 343 ...

If you don't recognise the multiples of six involved, 1, 3, 6, 10, 15, 21 are triangular numbers. 1 + 2 = 3; 3 + 3 = 6; 6 + 4 = 10; 10 + 5 = 15; 15 + 6 = 21. They augment by successive numbers.

If we scale down by shuffling units from factor to factor, as before, we will however get another kind of rule.

8 * 8 * 8 = 512
	512 - 1*8 = 504
7 * 8 * 9 = 504
	504 - 3*8 = 480
6 * 8 * 10 = 480
	480 - 5*8 = 440
5 * 8 * 11 = 440
	440 - 7*8 = 384
4 * 8 * 12 = 384
	384 - 9*8 = 312
3 * 8 * 13 = 312
	312 - 11*8 = 224
2 * 8 * 14 = 224
	224 - 13*8 = 120
1 * 8 * 15 = 120

Yes, we are scaling down with exactly same steps as from a square, except that each subtrahend is now multiplied by the cube root of the original cube number. Let's doublecheck:

7 * 7 * 7 = 343
	343 - 1*7 = 336
6 * 7 * 8 = 336
	336 - 3*7 = 315
5 * 7 * 9 = 315
	315 - 5*7 = 280
4 * 7 * 10 = 280
	280 - 7*7 = 231
3 * 7 * 11 = 231
	231 - 9*7 = 168
2 * 7 * 12 = 168
	168 - 11*7 = 91
1 * 7 * 13 = 91

This is the kind of mathematics that Boethius enjoyed - unless one should say he put in his textbook of arithmetic no such lopsided things as going from 7*7 to 6*8 or from 7*7*7 to 6*7*8. I don't recall those examples from his textbook, but that could just be my sloppy and tired reading back then or bad memory since then.

Check ye it out!

Anicii Manlii Torquati Severini Boetii
De Institutione Arithmetica
Libri duo
E Libris Manu scriptis Edidit Godofredus Friedlein MDCCCLXVII
https://la.wikisource.org/wiki/De_Arithmetica

Oh, if you know Latin, of course. And if someone is prepared to fill in the remaining 50 % from the text of Friedlein (since it is 1867, copyright has expired, there is no copy right violation for this edition by numerising it).

Hans Georg Lundahl
Nanterre UL
Vigil of St Matthew
Apostle and Gospeller
20.IX.2016