sw00892math--tpo-double-view-paired-6nET

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  ET = Elimination Table 

Explanation of entry title:  tpo means term position - this is the first time
I am using two tpo streams together - the reason is/ I will be showing both
the usual integer results of a 6n Elimination Table (all integers "3" does
not divide into evenly) and the integers required using the formula 1 + 6n
to get the squares of the usual integer results; BUT this will be done in 
both cases through the use of magic numbers.  Therefore, instead of 
dealing with each usual integer result separately, the minor result and its
related major result will be dealt with as a pair, and so also the integers
which when multiplied by "6" will get those respective products needed
to get the respective squares (once + 1 has been added) for the pair's
members.  The intial minor member is "5" and the initial major member is
"7".  Their respective integers to be multiplied by "6" are "4" and "8".  A
single tpo (in this case tpo1) will suffice for both magic number streams.
Showing this, however, might be a bit tricky.

Was working on this last night, and went through several changes, but
this is how it began (without showing "1"):
The times 6 squares row:
4   8      20   28      48   60      88   104      140   160
The 6n minor and major square root row:
5   7      11   13      17   19      23    25        29    31
Multiples of 6 apply throughout, but only in the first set is that multiple
the same integer: "1".  So, each tpo serves two functions: one for the
square roots stream and another for the squares stream.

Have concluded that envisioning 1 x (1 x 6) = 6, 2 x (2 x 6) = 24,
3 x (3 x 6) = 54, 4 x (4 x 6) = 96, 5 x (5 x 6) = 150, and so on, is
the best way to see each tpo.  These are my reasons:
6 -1 = 5 and 6 + 1 = 7, and 1 + 1 = 2 so that 6 - 2 = 4 and 6 + 2 =
8.  Just don't forget that the tpo# represents the n for "6" in the square
roots stream.  This then is how it works:

tpo4 nets 24-1 (23) and 24 + 1 (25) for the square roots, wherein 24 
is the magic number in that stream;
and 24 x 4 (96) from which 4 + 4 must be subtracted (88) and to which 
4 + 4 must be added (104) for the squares, wherein 96 is the magic number
in that stream, and once + 1 is added to 6 x 88 the resulting integer will
be the square of 23, and once + 1 is added to 6 x 104 the result will be
the square of 25.

The magic numbers for the first five pairs are:
6 and 6, 12 and 24, 18 and 54, 24 and 96, and 30 and 150.
Knowing the tpo# isall that is necessary as long as the handling of it
uses the approaches shown above.

Each term position represents a set of four integers.







#
Brian A. J. Salchert