sw00882math--descension-by-squares-groups

28


 pewn = positive even whole number      pown = positive odd whole number
 sqrt = square root      resultant = remainder      G = Group      sq = square

 In a descension-by-squares/ pown squares are subtracted from a pewn square
in a lesser to greater order, or pewn squares are subtracted from a pown sq in
a lesser to greater order.  Each subtraction produces a resultant integer which
is always a pown.  That pown is also the product of two integers which can be
discovered by inspecting the sqrt values in a subtraction.  If the values are "18"
and "11", simply subtract the 11 from the 18 to get the multiplier and add the
11 to the 18 to get the multiplicand.  As I will show, the construction of a d-b-s
can quickly become cumbersome, tending to make such constructions less and
less useful.  The main use, however--finding the final resultant, is easily done.
Here are some relevant formulas:
(2 x G#) - 2 = pewn sqrt for that group      G# = (pewn sqrt + 2)/2
(2 x G#) - 1 = pown sqrt for that group      G# = (pown sqrt + 1)/2
(final resultant # + 1)/2 = sqrt      (sqrt x 2) - 1 = final resultant #
Example using the above:
G28      pewn sqrt is 54      final resultant is 107
            pown sqrt is 55      final resutant is 109
[ Note:  Because they are at level one (the x 1 level), some final resultants are
   prime numbers. ]

First ten groups:
-
G1 has one object: 1² - 0² = 1² = 1
-
G2, wherein 2 + 1 = 3 and 3 + 2 = 5:
2 x 2 = 4 _ _ _ _ _ _ _ _ _ _ 3 x 3 = 9
2² - 1² = 3 _ _ _ _ _ _ _ _ _ 3² - 2² = 5
-
G3
4 x 4 = 16 _ _ _ _ _ _ _ _ _ 5 x 5 = 25 
4² - 1² = 15 _ _ _ _ _ _ _ _ 5² - 2² = 21
4² - 3² = 7 _ _ _ _ _ _ _ _ _5² - 4² = 9
-
G4
6 x 6 = 36 _ _ _ _ _ _ _ _ _ 7 x 7 = 49
6² - 1² = 35 _ _ _ _ _ _ _ _7² - 2² = 45
6² - 3² = 27 _ _ _ _ _ _ _ _7² - 4² = 33
6² - 5² = 11 _ _ _ _ _ _ _ _7² - 6² = 13
-
G5
8 x 8 = 64 _ _ _ _ _ _ _ _  9 x 9 = 81
8² - 1² = 63 _ _ _ _ _ _ _ 9² - 2² = 77
8² - 3² = 55 _ _ _ _ _ _ _ 9² - 4² = 65
8² - 5² = 39 _ _ _ _ _ _ _ 9² - 6² = 45
8² - 7² = 15 _ _ _ _ _ _ _ 9² - 8² = 17
-
G6
10 x 10 = 100 _ _ _ _ _ _ 11 x 11 = 121
10² - 1² = 99 _ _ _ _ _ _ 11² - 2² = 117
10² - 3² = 91 _ _ _ _ _ _ 11² - 4² = 105
10² - 5² = 75 _ _ _ _ _ _ 11² - 6² = 85 
10² - 7² = 51 _ _ _ _ _ _ 11² - 8² =57
10² - 9² = 19 _ _ _ _ _ _ 11² - 10² = 21
-
G7
12 x 12 = 144 _ _ _ _ _ _13 x 13 = 169
12² - 1² = 143 _ _ _ _ _ 13² - 2² = 165
12² - 3² = 135 _ _ _ _ _ 13² - 4² = 149
12² - 5² = 119 _ _ _ _ _ 13² - 6² = 129
12² - 7² = 95 _ _ _ _ __13² - 8² = 105
12² - 9² = 63 _ _ _ _ _ _13² - 10² = 69
12² - 11² = 23 _ _ _ _ _ 13² - 12² = 25
-
G8
14 x 14 = 196 _ _ _ _ _ _ 15 x 15 = 225 
14² - 1² = 195 _ _ _ _ _ _15² - 2² = 221
14² - 3² = 187 _ _ _ _ _ _15² - 4² = 209
14² - 5² = 171 _ _ _ _ _ _15² - 6² = 189
14² - 7² = 147 _ _ _ _ _ _15² - 8² = 161
14² - 9² = 115 _ _ _ _ _ _15² - 10² = 125
14² - 11² = 75 _ _ _ _ _ _15² - 12² = 81
14² - 13² = 27 _ _ _ _ _ _15² - 14² = 29
-
G9
16 x 16 = 256 _ _ _ _ _ _ 17 x 17 = 289
16² - 1² = 255 _ _ _ _ _ _17² - 2² = 285
16² - 3² = 247 _ _ _ _ _ _17² - 4² = 273
16² - 5² = 231 _ _ _ _ _ _17² - 6² = 253
16² - 7² = 207 _ _ _ _ _ _17² - 8² = 225
16² - 9² = 175 _ _ _ _ _ _17² - 10² = 189
16² - 11² = 135 _ _ _ _ _ 17² - 12² = 145
16² - 13² = 87 _ _ _ _ _ _17² - 14² = 93
16² - 15² = 31 _ _ _ _ _ _17² - 16² = 33
-
G10
18 x 18 = 324 _ _ _ _ _ _ 19 x 19 = 361
18² - 1² = 323 _ _ _ _ _ _19² - 2² = 357
18² - 3² = 315 _ _ _ _ _ _19² - 4² = 345
18² - 5² = 299 _ _ _ _ _ _19² - 6² = 325
18² - 7² = 275 _ _ _ _ _ _19² - 8² = 297
18² - 9² = 243 _ _ _ _ _ _19² - 10² = 261
18² -11² = 203 _ _ _ _ _ _19² - 12² = 217
18² - 13² = 155 _ _ _ _ _ 19² - 14² = 165 
18² - 15² = 99 _ _ _ _ _ _19² - 16² = 105
18² - 17² = 35 _ _ _ _ _ _19² - 18² = 37

[ Note: I was surprised when 4 and 28 had prime pairs which do not match
   when digit addition is applied to them, but have since learned that there 
   are congruities in two other ways.  28 - 4 = 24, which is divisible by both
   3 and 8; 107 - 11 = 96, which is divisible by both 3 and 8. ]

Guess what.  Unless I've made an error someplace, I've just been thrown
another dropping curve ball I cannot hit.  191, 193; 197, 199 are primes.  
G49      pewn sqrt is 96      final resultant is 191
            pown sqrt is 97      final resultant is 193

Plainly put, if the even integer between two primes is not
divisible by "4", those primes will not be in the same group.

[ Note:  What I have been calling prime pairs are usually
   called twin primes; so I guess one could call 11 13 and
   17 19 paired twin primes.  In every such pairing, only
   the even integer between one of the twins is divisible by
   "4".  Not sure, but the integer between the twin primes
   which is not divisible by "4" may always be divisible by
   "6". ]

9:27 PM - With help from a table on page 211 of the paperback
copy of The Mathematical Experience I own, just discovered
that 9431 9433 and 9437 9439 are paired twin primes, and that
9432 is divisible by "4" and 9438 is divisible by "6".

Conjecture:  Regarding paired twin primes, the even integer 
between one of the twin primes is always divisible by "4"
and the even integer between the other of the twin primes   
is always divisible by "6". 




#
Brian A. J. Salchert