How to explain the connection between divisions of a large number?

Question

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!

Answer 1

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.