The Golden State Warriors guard made the statement during a taping of the "Winging It" podcast.
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Curry didn't explain why he doesn't believe it happened.
Since the podcast was published, people on social media have been tweeting at him, trying to get him to explain more.
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NASA has reportedly invited Curry for a personal visit to see the evidence.
'One giant leap for mankind': Apollo 11 moon landing
Nasa workers Apollo 11