Prove or disprove: there do not exist positive integers $a, b, c$ such that $a^7 − b^5 = c^4$ 
Prove or disprove: there do not exist positive integers $a, b, c$ such that $a^7 − b^5 = c^4$.

Normally for exponential functions, I would find a way to rewrite them in terms of their factors and then prove or disprove them. I'm not sure how to go about this
 A: We can prove the statement. In fact, we can prove that infinite many such triplets of positive integers $(a, b, c)$ exist.
Pick two arbitrary numbers. Let's say $3$ and $4$.
$$3^7 - 4^5 = 1163$$
Multiply both sides by $1163^{\operatorname{lcm}(7, 5)} = 1163^{35}$ to get
$$(3*1163^5)^7 - (4*1163^7)^5 = (1163^9)^4$$
For the arbitrary equation having $n$ variables similar in form to this one, you would pick $n-1$ numbers, compute the sum on the LHS, and then multiply both sides by that sum raised to the power of the lcm of the exponents on the left. If the desired exponent on the RHS variable does not divide the real exponent on the RHS then we can multiply by a constant to allow it to do so.
A: Above equation shown below:
$a^7-b^5=c^4$ ----$(1)$
Equation $(1)$ has numerical solution:
$(a,b,c)=(8,16,32)$
A: There do exist such positive integers. Developing the answer by Sam to show how his solution can be found other than by a random search ...
We look for values of $a,b,c$ which are powers of $2$ and such that $a^7 = 2b^5 = 2c^4$ (the latter ensuring that $a^7 - b^5 = c^4$).  Since $lcm(5,4)=20$, we must find $x,y$ such that:
$$(2^x)^7 = 2(2^{20y})$$
To find suitable values of $x,y$ we simply have to solve the linear diophantine equation:
$$7x = 20y+1$$
and then put $a=2^x$, $b=2^{4y}$ and $c=2^{5y}$.
The smallest solution is $(x,y) = (3,1)$ yielding $(a,b,c)=(8, 16, 32)$, ie Sam's solution.  The next smallest is $(x,y) = (23,8)$ yielding $(a,b,c) = (2^{23}, 2^{32}, 2^{40})$.
