Primitive Pythagorean Triples + Exponent

Question

So, a while ago, I watched a YouTube video about the positive integer solution of $$3^x + 4^y = 5^z$$ and the result was $ x = y = z = 2 $.

My question, now, is:

For any primitive Pythagorean triples $x, y, z$, does $x^a + y^b = z^c$ have any positive integer solution, other than $a = b = c = 2$?

And, does the following conjecture holds?

For any primitive Pythagorean triples $x_n, y_n, z_n$, make a set $P_n$, consisting of $x, y,$ and $z$, such that $\min(P_1)<\min(P_2)<\min(P_3)<\min(P_4)$ and so on. Then, the number of positive integer solutions of $x_k^a+y_k^b=z_k^c$, with $x_k, y_k, z_k$ a member of the set $P_k$, is less than $k.$

I tried attempting the first question, with $(5, 12, 13)$, but I haven't tried further triples yet. (The answer was, still, $x = y = z = 2$).

Answer 1

Andrew Wiles proved that $A^x+B^x=C^x$ is true only for $x=2$ but, if you allow different exponents, I can think of $$(1^1+2^3=3^2),\quad (9^1+3^3=6^2),\quad (17^1+4^3=9^2),\\ (19^1+5^3=12^2),\quad (40^1,6^3,16^2),\quad (18^1,7^3,19^2)$$

There are an infinite number of these triples and you can find them by subtracting any cube $(B)$ from the next higher square $(C)$ to get $(A)$

There are other easy ones like $(2^1 +5^2 +3^3).\space $ A $6$-deep For-loop finds many more like $(3^5 +10^2 =7^3),\space (7^3 +13^2 =2^9)$.

I'm just an amateur so I do not understand your conjecture.

Here is a simple program that may help you explore these latter triples.

  110 print "Enter HI num";:input h1
  120 print "a^x”,”    b^y ","c^z"
  130 for a1 = 2 to h1
  140  for b1 = a1+1 to h1
  150   for c1 = 2 to h1
  160    for x1 = 2 to h1
  170     for y1 = 2 to h1
  180      for z1 = 2 to h1
  181         If a1 <> c1
  190         if y1 <> a1
  200         if x1 <> y1
  210         if x1 <> z1
  220         if y1 <> z1
  230         if a1^x1+b1^y1 = c1^z1
  240            print a1 x1,b1 y1,c1 z1
  241         endif
  250         endif
  260         endif
  270         endif
  280         endif
  290         endif
  300      next z1
  310      next y1
  320      next x1
  330      next c1
  340      next b1
  350      next a1