TITLE: Prove $13^{17} \ne x^2 + y^5$
QUESTION [10 upvotes]: I've been going in circles trying to prove $$13^{17} \ne x^2 + y^5$$ 
A friend of mine hinted to use modular arithmetic, but I'm not familiar with that field of study. 
Any suggestions are appreciated. 
Edit: $x,y \in\mathbb {N} $

REPLY [2 votes]: Check your equation mod 11:
$$13^{17} = 7 \pmod{11}$$
$$\begin{array} {r|r} X & X^2 \pmod{11} \\ \hline
0 & 0\\
1 & 1\\
2 & 4\\
3 & 9\\
4 & 5\\
5 & 3\\
6 & 3\\
7 & 5\\
8 & 9\\
9 & 4\\
10 & 1\\
\end{array}$$
$$\begin{array} {r|r} Y & Y^5 \pmod{11} \\ \hline
0 & 0\\
1 & 1\\
2 & 10\\
3 & 1\\
4 & 1\\
5 & 1\\
6 & 10\\
7 & 10\\
8 & 10\\
9 & 1\\
10 & 10\\
\end{array}$$
The java program I used to find $11$:
public class ThirteenSeventeen {

  public static void main(String[] args) {
    // does exist X,Y : 13^17 = x^2 + y^5

    for (int ModBase = 1; ModBase < 100; ModBase++) {
      boolean ExistFound = false;
      for (int X = 0; X < ModBase; X++) {
        for (int Y = 0; Y < ModBase; Y++) {
          if (ModExpt(13, 17, ModBase) == (ModExpt(X, 2, ModBase) + ModExpt(Y, 5, ModBase)) % ModBase) {
            ExistFound = true;
            break;
          }
        }
        if (ExistFound) break;
      }

      if (!ExistFound) {
        System.out.println("No Existance found for modular base: " + ModBase);
        System.out.println("13^17 = " + ModExpt(13, 17, 11));

        System.out.println("\\begin{array} {r|r} X & X^2 \\pmod{" + ModBase + "} \\\\ \\hline");
        for (int X = 0; X < ModBase; X++) {
          System.out.println("" + X + " & " + ModExpt(X, 2, ModBase) + "\\\\");
        }
        System.out.println("\\end{array}");

        System.out.println("\\begin{array} {r|r} Y & Y^5 \\pmod{" + ModBase + "} \\\\ \\hline");
        for (int Y = 0; Y < ModBase; Y++) {
          System.out.println("" + Y + " & " + ModExpt(Y, 5, ModBase) + "\\\\");
        }
        System.out.println("\\end{array}");

        break;
      }

    }

    System.out.println("Done");
  }

  // compute X^Y % P
  public static int ModExpt(int X, int Y, int P) {
    int R = 1;
    int S = X % P;
    while (Y > 0) {
      if (Y % 2 != 0) {
        R = (R * S) % P;
      }
      S = S * S % P;
      Y = Y / 2;
    }
    return R;
  }

}