The equation F(n^k - 1) = 1 for k = 2, 3 suggests F is identity, indicating key properties.
We prove that if F is a completely multiplicative function and k ∈ \2, 3\ such that the equation F(nᵏ - 1) = 1 holds for every n ∈ N, $n > 1,$ then F is the identity function. A similar result is proved for the equation F(n⁴ - 1) = 1 assuming a suitable conjecture concerning prime numbers. The equation F(n³ + 1) = 1 is also studied.
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Kátai et al. (2025) studied this question.
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