Sorted Multiplicative Inverse Pairs (Plotted) [closed]

Does symmetry for (M2-M1) hold for all $P$, where $P$ is prime? Please read the info-graphic for clarification. Right click on the image and "open in a new tab". I cannot find a proof showing that for all $P$, their sorted multiplicative inverse pairs will have symmetrical plots. ----- I'm not asking for a proof, I'm asking if there is a proof... so an answer of yes or no is fine.

$P$ = 47; (multiplicative inverse pairs sorted and in correct column)

$P$ = 47; (multiplicative inverse pairs sorted and NOT in correct column)

Solution 1:

Notice that if $ab\equiv1$ then $(-a)(-b)\equiv1$. This means that for each pair of inverses $(a,b)$ with $a\le b$, there is a corresponding pair of inverses $(P-a,P-b)$ with $P-a\ge P-b$. So the set of numbers in your second column just consists of the negatives (mod P) of the numbers in your first column.

As a result, after you sort the columns separately, the $n$th element from the top in the second column is just $P-a$ where $a$ is the $n$th element from the bottom in the first column. If the difference on the $n$th row from the top is $(P-a)-b$, then the difference on the $n$th row from the bottom is $(P-b)-a$. These expressions are equal to each other, so your plots are symmetrical.