sw00869math-3zs5zs7zs

(  3)   1+2     4+5      7+8     10+11    13+14    16+17    19+20    22+23

(  5)   2+3     7+8     12+13    17+18    22+23    27+28    32+33    37+38

(  7)   3+4    10+11    17+18    24+25    31+32    38+39    45+46    52+53

(  9)   4+5    13+14    22+23    31+32    40+41    49+50    58+59    67+68

( 11)   5+6    16+17    27+28    38+39    49+50    60+61    71+72    82+83

( 13)   6+7    19+20    32+33    45+46    58+59    71+72    84+85    97+98

( 15)   7+8    22+23    37+38    52+53    67+68    82+83    97+98   112+113

#
About 1+2, 2+3, 3+4, and:  The first # = the row #,
and the second # indicates the column # where the square of the ( #) is.
#
Next: If the two adjacent #s which sum to a ( #) exists in a prior row,
that ( #) is not a prime #.  Example: 4+5 is in row 1 column 2, and
4+5 is also in row 4 column 1.
#
Next:  Column 1 is the x1 column, C2 is the x3 column, C3 the x5,
C4 the x7, C5 the x9, C6 the x11, C7 the x13; C8 the x15.  So,
(column # x2) - 1 = the times # for that column.
#
Next:  Since 4+5 is in C2 of R1, which is 3s row, 4+5 = 3x3.
#
Next:  That # which is the second # in R1C1 is the first in R2C1.
This means that there are three distinct numbers C1 of any two
adjacent rows.  These three #s in C1 of R1 and R2 sum to 6.
These three #s in C1 of R2 and R3 sum to 9.  These three #s
in R5 and R6 sum to 18, or 3x6.  The ( #) in R5 is 11, and the
( #) in R6 is 13.  11+13=24, or 3x8.  24 - 18 = 6.  Go back.
3+5=8; 8 - 6 = 2.  5+7=12; 12 - 9 = 3.  So, the distinct #s in
adjacent rows rise by 3, and ( #)s in adjacent rows rise by 4 
when they are summed as terms in a set of adjacent rows.  I
am not yet certain what the value of knowing these facts is,
but I suspect there is an interpolation value.  
#
Next:  A known about primes is that there is a prime number
between n and 2n for every integer n > 1.
If n = 2, 2n = 4.  3 is between 2 and 4.  If n
= 3, 2n = 6.  5 is between 3 and 6.  If n = 4,
2n = 8.  Both 5 and 7 are between 4 and 8.  If n
= 5, 2n = 10.  Both 7 and 9 are between 5 and 10, but
we know that 9, or (10 - 1),  is not a prime.  Between 8 and 16
are 9, 11, 13, and 15.  Question:  Why do 5+6 and 6+7 equal
numbers which are prime numbers, while 4+5 and 7+8 do not? 



Brian A. J. Salchert