All-Russian Regional Preparation NT Exercise 5

Does there exist an arithmetic progression of $3$ distinct positive integers such that the first is a perfect square, the second is a perfect cube, and the third is a perfect $5$-th power?

Replace sorry in the template below with your solution. See Answer Bank for acceptible answer declarations. Mathlib version used by the checker is v4.29.0.

import Mathlib.Algebra.Group.Nat.Defs

def answer : Prop := sorry

theorem solution : (∃ a b c : ℕ, 0 < a ^ 2 ∧ a ^ 2 < b ^ 3 ∧ b ^ 3 < c ^ 5 ∧
    ∃ d, a ^ 2 + d = b ^ 3 ∧ b ^ 3 + d = c ^ 5) ↔ answer := sorry

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