Perform For-loop and show output which meets a certain criterion

Question

I am new to Mathematica and I have a For loop like so:

For[i = 0, i < 15, i++, PowerMod[i, -1, 15]]

Currently it doesn't output anything. The calculation I am performing is i^-1 mod n. I want t his loop to output the values of i ONLY where i = i^-1 mod n. How can i accomplish this?

Edit

I can print the output by wrapping PowerMod in Print[]. How can I accomplish printing i and then the result from PowerMod? Here is what I tried, but it is incorrect since I don't know the syntax to use here:

For[i = 0, i < 15, i++, Print[i] Print[PowerMod[i, -1, 15]]]

This prints i and then the result of PowerMod below it. How can I get them on the same line?

Answer 1

Avoid the For-loop. Here is way to do this with functional programming tools, which are more idiomatic to Mathematica.

Quiet[Table[{i, PowerMod[i, -1, 15]}, {i, 1, 14}]] // Cases[{x_, x_} :> x]

{1, 4, 11, 14}

Answer 2

Let's rephrase your question a little:

I want this loop to, IFF $i\equiv i^{-1}\mod n$, output the value of $i$.

Translating this into Mathematica:

If[i == PowerMod[i, -1, n], Print[i]]

We can make a few improvements. First, I'll use Do instead of For (Try to avoid constructs like For and While in Mathematica, they're less idiomatic). I'll also make your loop a function, so you can easily call it for different moduli:

selfInverses[n_] := Do[If[i == PowerMod[i, -1, n], Print[i]], {i, 0, n - 1}]

We're still getting those "not invertible" errors, so let's try a slightly different tack. By multiplying both sides of the condition by $i$, we get:

$$ \begin{align} i(i)&\equiv i^{-1}(i)\mod n \\ i^2&\equiv 1\mod n \end{align} $$

Since you can always multiply two numbers [citation needed], you don't get those errors. Putting in these changes:

selfInverses[n_] := Do[If[1 == PowerMod[i, 2, n], Print[i]], {i, n}]

(If for some reason you needed to keep PowerMod[i, -1, n], I suggest checking that i is invertible first with Not[CoprimeQ[i, n]] && PowerMod[i, -1, n] == i)

Note that the iterator for this Do, {i, n}, loops from 1 to n, not 0 to n - 1 as your code did. However, since $0^2\equiv 0\not\equiv 1$ and $n^2\equiv n\equiv 0\not\equiv 1\mod n$, neither end case will change our results.

Printing the results is less helpful if you want to use the results later, so we'll have the function generate a list instead:

selfInverses[n_] := 
 Reap[Do[If[1 == PowerMod[i, 2, n], Sow[i]], {i, n}]][[2, 1]]

An example of usage:

selfInverses[15] (* returns {1, 4, 11, 14} *)

If you're having trouble understanding what this function is doing, look at the documentation of the functions used.

Answer 3

Here is a simple function, which lets you calculate your result for any modulus n:

pm[n_Integer/;n>1]:=Quiet@Select[Range[0,n-1],PowerMod[#,-1,n]==#&]

e.g.:

pm[15]   (* {1,4,11,14} *)
pm[30]   (* {1,11,19,29} *)
pm[3129] (* {1,148,895,1042,2087,2234,2981,3128} *)