We know that Archimedes learnt a lot from his father, Phidias (who was an astronomer) and most likely studied in Alexandria, alongside other great contemporaries such as Eratosthenes of Cyrene and Conon of Samos, with whom he would have collaborated and corresponded with about his work.
Do we however know at all whether Archimedes had any students that he would have taught personally?
A friend of his named Heracleides wrote a biography, but it was lost so the details of his biography is unknown.
But after reading its biography, (in my opinion ) I don't think he had any students. The quote below is from wikipedia
Unlike his inventions, the mathematical writings of Archimedes were little known in antiquity. Mathematicians from Alexandria read and quoted him, but the first comprehensive compilation was not made until c. 530 AD by Isidore of Miletus in Byzantine Constantinople, while commentaries on the works of Archimedes written by Eutocius in the sixth century AD opened them to wider readership for the first time.
I guess if he had any students, they would help more to spread his mathematical writings or
General Marcellus was reportedly angered by the death of Archimedes, as he considered him a valuable scientific asset and had ordered that he not be harmed.[11] Marcellus called Archimedes "a geometrical Briareus".
I guess If there are any students, I guess the general would ask for them or maybe there was a war the students had to participate in the war.
As a conclusion, because the there isn't a biography (and some assumptions I made above), it is hardly likely that he had any students.
It is, at this point in time, unknown as to whether Archimedes had any students or ran a school-(similar to Aristotle's Lyceum or Plato's Academy). If I was to provide an educated guess-(based upon the little that I know regarding Archimedes' biography), I would say......probably not. By today's standards, Archimedes would have been a professional Scientist and Inventor similar to someone, such as Thomas Edison. Due to his plethora of inventions and mathematically oriented writings, I am not sure Archimedes would have had the time to teach! He may have had his own version of Teaching or Research Assistants-(similar to Aristotle's Lyceum), though, based on our current evidence, it seems to have been unlikely.
However, I don't think it's an impossibility that Archimedes may have had a school or a group of students in Siracusa, Sicily or that perhaps Archimedes may have had the ancient equivalent of a Visiting Professorship or Teaching Fellowship at The Alexandrian University in Egypt. Many of the major Greek Scientific and Philosophical Thinkers dating back to Thales-(The Father of Western Science and Philosophy), were also Teachers and Educators. Democritus, Pythagoras, Zeno, Heraclitus, Socrates, Hippocrates, Plato, Aristotle, Epicurus and Hypatia, were all professional Teachers and Educators, while also pursuing scientific and/or philosophical writings and research. So it is not necessarily an impossibility for Archimedes to have been part of the centuries long Ancient Greek intellectual tradition.
However, at this point in time, there is just no primary evidence which supports such a theoretical or circumstantial claim and therefore, it continues to remain unknown.
