sw00881math--pown-and-pewn-squares-notes

27


 pown = positive odd whole number         da = digit addition
 sq = square         tpo = term position         DF = descension family
 pewn = positive even whole number        sqrt = square root

There are no integer squares in the 2, 3, 5, 6, 8 da families.
[ The even squares run: 4 7 9 1 1 9 7 4 9 ]
The sq of any pown/ divisible by "3" is in the 9da family.

tpo          pown          sq          da
-
 1              1               1           1
 2              3               9           9
 3              5              25          7
 4              7              49          4
 5              9              81          9
 6             11            121          4
 7             13            169          7
 8             15            225          9
 9             17            289          1

The da family rotation above is Group 1.  19 through 35 comprises Group 2.
Here is a different view:

pown          1          3          5          7          9          11          13         15         17
-
sq da          1          9          7          4          9           4           7           9           1
--
pown         19        21        23         25         27         29          31         33         35
-
pown         37        39        41         43         45         47          49         51         53

The goal I have may have already been reached, and it may be I cannot reach it;
but that is neither here nor there, and at this moment I am reluctant to reveal it.
-
In the 1da family for pown squares, the powns are separated from each other
in a 16 2 pattern.  In the 4da family for pown squares, the pattern is 4 14; and
in the 7da family for pown squares, the pattern is 8 10.  In each case, 18 apart
rules when going from a given column in one group to the same column in the
group above or below it.
-
Also, there is a times-10 rule helper.  Example:  49 + (7 x 10) = 119 (or 7 x 17).
This is leading to final-digit math regarding both pown and pewn squares.  I may
need to change the title of this entry.  Anyway, here's the info:
Am skipping "1".  Am skipping "5".  9 + (3 x 10) = 39 (or 3 x 13),
121 + (11 x 10) = 231 (or 11 x 21); 361 + (19 x 10) = 551 (or 19 x 29).
So, where the final digit is "9" in the above, a 120 - 1 and a 40 -1 occur.
Where, likewise, the final digit is "1", a 230 + 1 and a 550 + 1 occur.
Similar outcomes occur with pewn squares.  4 + (2 x 10) = 24 (or 2 x 12)
and 64 + (8 x 10) = 144(or 8 x 18).  16 + (4 x 10) = 56 (or 4 x 14) and
36 + (6 x 10) = 96 (or 6 x 16).  So, where the final digit is "4", a 25 - 1 and
a 65 - 1 occur.  Where the final digit is "6", a 55 + 1 and a 95 + 1 occur.
Obviously, pown squares revolve around "0" and pewn squares revolve
around "5". 

May 08, 2008
Late last night before I lost my broadband connection, I subtracted
119 from 144.  The 25 reminded me of descension-by-squares and
Descension Families.  Visit this entry.  Moments ago I was there and
I think it is better than what I had planned to write here.  Still, I never
know when a rethinking will move me to change terminology and/or how 
an idea is presented.  For instance: there I let 3² - 2² be a DF of one
level; but it could be a DF of two levels.  The former is as above/
where the resultant integer is "5".  Were I to show it as two levels,
the second level would be 3 x 3 = 9, and would be there mainly for
informational reasons.  "5" would still be the resultant integer, the
final resultant integer, that is.

Do want to add some further thoughts.
-
It could be said that "3" and "5" are the only two-level (or one-level) DF
primes.  . . .  After the 8DF (4 x 4 = 16) comes the 10DF (5 x 5 = 25).
[ Haven't done an extensive check, but it appears the DF's rise in pairs.
If so, they could be grouped so, and this might aid in understanding the
secrets of the final resultants.  Will probably be heading that direction. ] 
Proceeding with my changed thinking, these are the only three-level DF's,
and the final resultant for the 10DF is not a prime.  Here is the new look:
5 x 5 = 25
5² - 2² = 21
5² - 4² = 9 = 3²
Points of interest: 
5² descends to 3², 13² to 5², 25² to 7², 41² to 9²,
61² to 11², 85² to 13², 113² to 15²; 145² to 17².
This generates the following:
8        12        16        20        24        28        32
For going beyond 113² there is this, in which multiples of "36" reign:
5²     41²     113²     221²     365²     545²     761²
These are for descensions to squares which are multiples of "3".
The resulting squares are:  9, 81, 225, 441, 729, 1089; 1521.
Here:  761² - 760² = 1521 = 39 x 39.    



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Brian A. J. Salchert