**p^a+q^b=r^c **

How many distinct solutions of the equation above are there, subject to the following constraints:

p, q, & r distinct primes

a, b, & c distinct positive integers,

each
more than one

None of the powers exceeds 1111.

[2,4, 3, 2, 5, 2]

[2, 5, 7, 2, 3, 4]

[2, 7, 17, 3, 71, 2]

[7, 3, 13, 2, 2, 9]

2^4+3^2 =5^2 -->a, b, and c are NOT distinct. So, this is ignored.

2^5+7^2 = 3^4

2^7+17^3 =71^2 --> 5041 exceeds 1111. So, this is ignored.

7^3+13^2 = 2^9

Therefore, there are only 2 valid solutions, and these are:

(p, a, q,b,r,c) = (2,5, 7,2,3,4), (7,3,13,2,2,9)

*Edited on ***September 23, 2023, 9:03 pm**