1. Generalization: We need to show that there are infinitely many tuples $(a_1, a_2, \ldots, a_n)$ and $(b_1, b_2, \ldots, b_m)$ of natural numbers such that
$$\sum_{i=1}^n a_i^{p_i} = \sum_{j=1}^m b_j^{q_j}$$
where $p_i, q_j, n, m$ are natural numbers and fixed, and $\gcd(\text{lcm}(p_1, p_2, \ldots, p_n), \text{gcd}(\text{lcm}(q_1, q_2, \ldots, q_m))) = 1$.

2. Lemma: We first show that there are infinitely many tuples $(x_1, x_2, \ldots, x_n)$ of natural numbers such that
$$x_1^{t_1} + x_2^{t_2} + \cdots + x_{n-1}^{t_{n-1}} = x_n^m$$
where $m, n, t$ are natural numbers and fixed, and $\gcd(m, \text{lcm}(t_1, \ldots, t_{n-1})) = 1$.

3. Proof of Lemma:
- Consider the sequence of natural numbers $\{b_n\}$.
- Set
$$a_i = b_i \left(\sum_{i=1}^n b_i^{p_i}\right)^{\frac{Mk}{p_i}}$$
where $M = \text{lcm}(p_1, p_2, \ldots, p_n)$ and $k$ is an integer such that $m \mid Mk + 1$ (according to Bezout’s theorem).

4. Back to the main problem:
- Case 1: $\min(m, n) = 1$. This is equivalent to the lemma.
- Case 2: $\min(m, n) > 1$. According to the lemma, the following equations have infinitely many solutions in natural numbers:
$$\sum_{i=1}^{n-1} a_i^{p_i} = \sum_{j=1}^1 b_j^{q_j}$$
$$\sum_{i=n}^{n} a_i^{p_i} = \sum_{j=2}^m b_j^{q_j}$$
Combining these, we conclude that the equation
$$\sum_{i=1}^n a_i^{p_i} = \sum_{j=1}^m b_j^{q_j}$$
has infinitely many solutions in natural numbers.

5. Conclusion: By the lemma and the cases considered, we have shown that there are infinitely many tuples $(a, b, c, d)$ of natural numbers such that $a^3 + b^4 + c^5 = d^7$.

$\blacksquare$