We know that Tom Riddle was obsessed with immortality and was planning about it ever since he was a little boy. If he had the time to meticulously plan for those Horcruxes, why did he not go in search for the three things that could guarantee his immortality and provide him with the power he had always desired?

The Dark Lord likely didn’t know about the other two Hallows.

Dumbledore explains to Harry when they meet in King’s Cross that he doesn’t think the Dark Lord knew about the Hallows, because he turned the ring containing the Resurrection Stone into a Horcrux without ever having recognized what the Stone actually was.

“And Voldemort never knew about the Hallows?’

‘I do not think so, because he did not recognise the Resurrection Stone he turned into a Horcrux.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 35 (King’s Cross)}}

The Dark Lord likely never knew about the Hallows, so therefore couldn’t choose to seek them.

Even if he did, he likely wouldn’t have been interested in them.

Dumbledore also says that had the Dark Lord known about the Hallows, it was unlikely he’d have any interest in the Cloak or the Stone, he’d still only want the Elder Wand.

“But even if he had known about them, Harry, I doubt that he would have been interested in any except the first. He would not think that he needed the Cloak, and, as for the stone, whom would he want to bring back from the dead? He fears the dead. He does not love.”
_{^{- Harry Potter and the Deathly Hallows, Chapter 35 (King’s Cross)}}

So, if he had known about the Hallows, he still likely wouldn’t go searching for all three of them, when the only one he wanted was the unbeatable Wand to use against Harry’s.

Harry himself thinks about this in Chapter 22 when he realizes that Voldemort is after the Elder Wand:

Voldemort had been raised in a Muggle orphanage. Nobody could have told him The Tales of Beedle the Bard when he was a child, any more than Harry had heard them. Hardly any wizards believed in the Deathly Hallows. Was it likely that Voldemort knew about them?

Harry gazed into the darkness.... If Voldemort had known about the Deathly Hallows, surely he would have sought them, done anything to possess them: three objects that made the possessor master of Death? If he had known about the Deathly Hallows, he might not have needed Horcruxes in the first place. Didn’t the simple fact that he had taken a Hallow, and turned it into a Horcrux, demonstrate that he did not know this last great Wizarding secret?

Which meant that Voldemort sought the Elder Wand without realizing its full power, without understanding that it was one of three... for the wand was the Hallow that could not be hidden, whose existence was best known... The bloody trail of the Elder Wand is splattered across the pages of Wizarding history...

Essentially Harry assumes that Voldemort did not know about the Deathly Hallows for three reasons:

1. Voldemort would likely never have heard of them, growing up as a Muggle.


2. The fact that he didn't seek them indicates that he was unaware of them.


3. The fact that he made a Hallow into a Horcrux indicates that he did not realize what it was.

Voldemort only knew about the Elder Wand because the Elder Wand had a history of it's own, independent from the Hallows. Voldemort knew of the Elder Wand qua Elder wand, not qua Deathly Hallow.

Now we can speculate as to whether Voldemort would have sought the Hallows if he had known about them. First, there's always the possibility that, like Hermione, he wouldn't believe that the legend was true. But what if he would believe that it was true?

Dumbledore, for one, seemed to think that Voldmeort wouldn't care for the Hallows. In Chapter 35 he tells Harry:

"But even if he had known about them, Harry, I doubt that he would have been interested in any except the first. He would not think that he needed the Cloak, and, as for the stone, whom would he want to bring back from the dead? He fears the dead. He does not love."

But is this a valid argument? Even if Voldemort would have no use for the other two Hallows, might he still not seek them in order to unite the three Hallows and become master of death?

In fact, Dumbledore himself admitted that when he and Grindelwald were seeking the Hallows they didn't really need the Invisibilty Cloak, but they sought it anyway in order to complete the set and become invincible (my emphasis):

"And the Cloak... somehow, we never discussed the Cloak much, Harry. Both of us could conceal ourselves well enough without the Cloak, the true magic of which, of course, is that it can be used to protect and shield others as well as its owner. I thought that, if we ever found it, it might be useful in hiding Ariana, but our interest in the Cloak was mainly that it completed the trio, for the legend said that the man who united all three objects would then be truly master of death, which we took to mean 'invincible.'

Thus, it is certainly possible that Voldemort would have felt the same way, and would have sought all the Hallows for the purpose of uniting them to conquer death, even though he had no use for two of them.