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Some may find the question trite, but it really is intriguing. Primes were not invented, but exist as part of number theory. When we became curious about numbers, things like the nature of primes were uncovered. Have prime numbers always existed? It is hard to say 'no' to that, although before anyone existed who even had a clue about them, what was the nature of their 'existence'? Like the many things that 'are' but are not yet known, they were floating in the.. I don't know what... waiting to be caught. Maybe YOU will catch something out of the mathematical air that no one has ever known about before.

You may be asking about the first time they were identified. The first documented discussions of primes come to us from Euclid in ancient Greece (circa 300 BCE). There are hints that the ancient Egyptians may have known about them; in some materials they used a representation for primes different from that used for composites.

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Q: Where were prime numbers invented?
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