That's an interesting thought, Morsul, but it doesn't survive comparison to the book. The birthday customs of the Hobbits are described as existent long before any of the Shire Hobbits knew of Gollum, let alone of his background as Sméagol. Even after the War of the Ring, it doesn't appear that his story was common knowledge among the common people. And if I remember rightly, not even Sméagol's own people were aware that he possessed the Ring, and no one there knew what had happened to Déagol, much less that Sméagol had murdered him.
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'Mercy!' cried Gandalf. 'If the giving of information is to be the cure of your inquisitiveness, I shall spend all the rest of my days in answering you. What more do you want to know?' 'The whole history of Middle-earth.. .'
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