deletions | additions       diff --git a/Ones.tex b/Ones.tex  index 58f6c57..94a45d8 100644  --- a/Ones.tex  +++ b/Ones.tex 
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This section notes some numbers of the family $1|a;n|1|b;n|1$.  \begin{equation}  \begin{array}  a & b & n & p? & smallest \;div \\  \hline  a & b & n & pr? & smallest \;div & a+b\\  \hline  0 & 0 & 0 & 0 & 37 \\ & 0\\  0 & 0 & 1 & 0 & 3 \\ & 0\\  0 & 0 & 2 & 0 & 3 \\ & 0\\  0 & 0 & 3 & 0 & 3 \\ & 0\\  0 & 0 & 4 & 0 & 3 \\ & 0\\  0 & 1 & 0 & 0 & 37 \\ & 1\\  0 & 1 & 1 & 1 & s \\ & 1\\  0 & 1 & 2 & 0 & 181\\ 181& 1\\  0 & 1 & 3 & 0 & 3 \\ & 1\\  0 & 1 & 4 & 0 & 89 \\ & 1\\  0 & 1 & 5 & 0 & 10789 \\ & 1\\  0 & 2 & 0 & 0 & 37 \\ & 2\\  0 & 2 & 1 & 0 & 29 \\ & 2\\  0 & 2 & 2 & 0 & 13 \\ & 2\\  0 & 2 & 3 & 0 & 3 \\ & 2\\  0 & 2 & 4 & 1 & s \\ & 2\\  0 & 2 & 5 & 0 & 89 \\ & 2\\  0 & 3 & 0 & 0 & 37 \\ & 2\\  0 & 3 & 1 & 0 & 3 \\ & 2\\  0 & 3 & 2 & 0 & 3 \\  0 & 3 & 3 & 0 & 3 \\  0 & 3 & 4 & 0 & 3 \\ 
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1 & 7 & 3 & 0 & 3 \\  1 & 7 & 4 & 0 & 389\\  1 & 7 & 5 & 0 & 677\\  1 & 8 & 1 & 0 & 3 \\  1 & 8 & 2 & 0 & 3 \\  1 & 8 & 3 & 0 & 3 \\  1 & 8 & 4 & 0 & 3 \\  1 & 9 & 1 & 0 & 19 \\  1 & 9 & 2 & 1 & s \\  1 & 9 & 3 & 0 & 3 \\  1 & 9 & 4 & 1 & s \\  1 & 9 & 5 & 0 & 7 \\  2 & 0 & 1 & 1 & s \\  2 & 0 & 2 & 0 & 23 \\  2 & 0 & 3 & 0 & 3 \\  2 & 0 & 4 & 0 & 67 \\  2 & 0 & 5 & 0 & 31 \\  2 & 1 & 1 & 0 & 3 \\  2 & 2 & 1 & 0 & 17 \\  2 & 2 & 2 & 1 & s \\  2 & 2 & 3 & 0 & 3 \\  2 & 2 & 4 & 1 & s \\  2 & 2 & 5 & 0 & 17 \\  2 & 3 & 1 & 0 & 7 \\  \hline  \end{array}  \end{equation}  It would seem that if $N(a+b) \; mod \; 3 =0$ then the particular number $1|a;N|1|b;N|1$ will be divisible by $3$ due to digit summation, this has the benefit of having $1+1+1=3$, other families will have an offset rule. Primes of this form can never be divisible by $5$ as they will never end in a $5$ [Check this], with a similar argument for $2$ [Check this also]. Thus we may expect a set of primes which are able to divide a family with a given 'topology' in this case the $1|-|1|-|1$ part. This set appears to include $3,7,13,17,19,23,29,31,37,41,43,47,53,89,389,677,827,1201,...$ and others. We only list the smallest divisor above and there must be many more.   Denote these topology triplet as $t_1,t_2,t_3$, such that any given family $F(a,b)$ under any topology $T(t_1,t_2,t_3)$, to any order $N$ may be expressed as the number \begin{equation}  Z(F,T)=t_1|a;N|t_2|b;N|t_3  \end{equation}  We may add the rule that if $(t_1|a;N|t_2|b;N -2t_3) \; mod 7 = 0$ then the particular number $1|a;N|1|b;N|1$ will be divisible by $7$ due to digit summation.  Prime pairs \begin{equation}  \begin{array}  a&b&N&a+b\\ 
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1&6&2&7\\  1&7&1&8\\  1&7&2&8\\  1&9&2&10\\  1&9&4&10\\  2&0&1&2\\  2&2&2&4\\  \hline  \end{array} 
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Always Non-prime family pairs \begin{equation}  \begin{array}  \hline  a & b & div\\ div & a+b\\  0 & 0 & 3 & 0  \\ 0 & 3 & 3 & 3  \\ 0 & 6 & 3 & 6  \\ 0 & 9 & 3 & 9  \\ 1 & 2 & 3 & 3  \\ 1 & 5 & 3 & 6 \\  1 & 8 & 3 & 9 \\  2 & 1 & 3 & 3  \\ \hline  \end{array}  \end{equation}