Very little is known about Euclid's life--much less than about other famous ancient Greek mathematicians, which is puzzling. It is also strange to me that Euclid didn't write about the Eratosthenes sieve.
Thus I'd like to ask mathematical historians and everybody else: is there any clear proof that Euclid and Eratosthenes are two different people?
I wonder if Euclid's Elements were considered by the top geometers of Archimedes times (before and afterward for awhile) as teaching materials (or even philosophical too) rather than mathematical research. Especially that Elements were not mentioned much in those years, they became popular quite a bit later.
For a longer time, even after Archimedes, nobody talked about Euclid of Alexandria. Instead, they talked only about "the author of Elements". Possibly, the geometric part of Elements was a cumulation of the work by several mathematicians, of which Eudoxes was a big part. But the number theoretical part of Elements and the Eratosthenes Sieve were perhaps by the same author, namely Erathostenes.
The Elements had no name nor any pseudonym assigned to it. That's why nobody was talking explicitly about the author.
Bourbaki used to have meetings together in a single room (perhaps a different room at the different occasions). They have established common notation and conventions, etc. Thus their textbooks had a fairly consistent style. They used to write in that style also monographs by single Bourbakist authors. In the case of the ancient authores such consistency was impossible due to the time and space span. This is why I conjecture that Eratosthenes combined the past results by several authors (especially by Eudoxes) and Eratosthenes rewritten these results in his own hand. This would explain why there was no author mentioned but the style of Elements was consistent.