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Table image link http://postimg.org/image/v57ixyddr/

The above table tells about periodicity of occurrence of products of two numbers that can be represented by general formula $6k +1$ or $6k-1$

I don't understand the meaning of last column. What do numbers $30, 42, 66, 78, 102$ and $114$ represent in last column ? How the values of last column is being calculated ?

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1 Answer 1

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If you look at one line of a table it looks something like this: $$\begin{bmatrix} x_{1} & empty & x_{2} & empty & x_{3} & empty & x_{4} & empty & x_{5} & empty \\ empty & y_{1} & empty & y_{2} & empty & y_{3} & empty & y_{4} & empty & y_{5} \end{bmatrix}$$ And in the last column you have $$\begin{bmatrix} A \\ B-C \end{bmatrix}$$ The formula for this number is as follows:
$\forall i$ $A = x_{i+1}-x_{i} = y_{i+1} - y_{i}$, $B = y_{i}-x_{i}$ and $C = x_{i+1} - y_{i}$.
Hope this helps.

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Is this some kind of standard for writing about periodicity in such form ? –  divanshu Apr 20 '13 at 13:34
    
Okay so assyme this is the line for $a_{1}$. Then $x_{i}=a_{1}a_{i}$ so $x_{i+1}-x_{i} = a_{1}a_{i+1} - a_{1}a_{i} = a_{1}(a_{i+1}-a_{i}$. Now since $a_{i+1} = a_{i} + 6$, so it follows $x_{i+1}-x_{i} = 6a_{1}$. So for $a_{1} = 5$ you have $A=30$ and for $a_{1} = 7$ $A=42$ and so on. Then using this you can get same type of formulas for each one. –  Belov Apr 20 '13 at 16:12

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