How to generate GPG-compatible and PEM-encoded keys using Elixir?

I’m trying to create an Elixir code that will generate GnuPG-compatible key pairs and also to be able to encrypt, decrypt and sign messages. But it’s also important that it can export PEM-formatted keys and messages, so users can use it with the GnuPG implementation.

Here’s how I began my attempt:

defmodule GPG do
  def test do
    private_key = generate_private_key()
    public_pem = pem_encoded_public_key(private_key)

    IO.puts("Public Key PEM:")
    IO.puts(public_pem)
  end

  defp pem_encoded_public_key(private_key) do
    private_key
    |> public_key_from_private_key()
    |> pem_entry_encode(:RSAPublicKey)
  end

  defp generate_private_key do
    :public_key.generate_key({:rsa, 2048, 65537})
  end

  defp public_key_from_private_key(private_key) do
    {:RSAPublicKey, elem(private_key, 2), elem(private_key, 2)}
  end

  defp pem_entry_encode(key, type) do
    :public_key.pem_encode([:public_key.pem_entry_encode(type, key)])
  end
end

If I run GNU.test() on iEx, it will output:

iex(3)> GPG.test()
Public Key PEM:
-----BEGIN RSA PUBLIC KEY-----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==
-----END RSA PUBLIC KEY-----

But if I create a test_public_key.txt file with this given key as content and run GnuPG to evaluate it, this happens:

$ gpg --show-keys test_public_key.txt 
gpg: no valid OpenPGP data found.

I’m wondering what I’m missing.