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I am a beginner and I found that there are some links between the divisions of a large number which I don't understand. I couldn't find any way to explain them mathematically so I am hoping someone can explain this to me.

I am using python. We have the following division operations:

20011548789665855544566//6 = 3335258131610975924094

20011548789665855544566//7 = 2858792684237979363509

If we subtract the two results we get:

3335258131610975924094 - 2858792684237979363509 = 476465447372996560585

If we continue further along with dividing the same number we get:

20011548789665855544566//20 = 1000577439483292777228
20011548789665855544566//21 = 952930894745993121169

And if we subtract these two results we get:

1000577439483292777228 - 952930894745993121169 = 47646544737299656059

The result of the two subtractions is very close although the divisions are quite further apart:

476465447372996560585 vs. 47646544737299656059

How can this be explained? And is there any way to know the next 'pair' of divisions of the initial number which will result in this 'pattern' when subtracted?

4764654473729...

Thanks!

Robert
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1 Answers1

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Note that $\frac 16-\frac 17=\frac 1{42}$, so the result of your first subtraction is $\frac 1{42}$ of the original number. If the first two divisions happen with no remainder (not the case here) the division by $42$ will happen with no remainder because $6$ and $7$ are coprime. Similarly $\frac 1{20}-\frac 1{21}=\frac 1{420}$ The results of your two subtractions are not close, one is ten times the other because it has one more digit. The pattern of digits is the same because $420=42 \cdot 10$. The next pair would be $n$ such that $n(n+1)=4200$ but there is none.

Thanks to the Alpern Diophantine equation solver (needs Javascript, so doesn't work in Chrome for me), we can find more cases. We are looking for integers $x,y$ where $10x(x+1)=y(y+1)$. Your example is $y=20,x=6$ Another is $y=4,x=1$. There is a recurrence $x_{n+1}=19x_n+6y_n+12,y_{n+1}=60x_n+19y_n+39$ which gives the next solution as $x=55,y=175$ and they keep going.

Ross Millikan
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  • Thanks so much. This is really interesting. So this means that similar but different of course patterns exist for all divisions of the same number? I just found links between divisions with 6,7,20,21 but are there any such links for all divisions until it gets to 0? – Robert Aug 13 '19 at 02:40
  • "Similar but different patterns" is a vague phrase. It depends on what you mean by that whether they exist. What makes this work is $7 \cdot 3=21, 6 \cdot \frac {10}3=20, 21-20=1$. You can look to express $\frac 1{4200}$ as a difference of fractions, but the numerators may not be $1$ and the denominators will not be $1$ apart. – Ross Millikan Aug 13 '19 at 03:17
  • I see. By "Similar but different patterns" I mean this: Let's say we divide x/2 and x/3 and subtract the results of these divisions. How can we know which are the next divisions that will follow the pattern except by continuing to divide manually and find out of course. Thanks! – Robert Aug 13 '19 at 03:29
  • As $\frac 12-\frac 13=\frac 16$ you want to find two fractions that differ by $\frac 1{60}, \frac 1{600}$ or some such. We have $\frac 1{24}-\frac 1{25}=\frac 1{600}$ so that is wonderful. We have $\frac 1{10}-\frac 1{12}=\frac 1{60}$ but $10$ and $12$ differ by $2$. Does that count? – Ross Millikan Aug 13 '19 at 03:43
  • Yes, great. There is also a comment above from J. W. Tanner and his formula allows for a 'prediction' of any of these patterns. Thanks again! – Robert Aug 13 '19 at 03:48