A Greek business has two numbers you will meet on invoices and in contracts. They get confused constantly, though the difference is simple.
Αριθμός ΓΕΜΗ
The General Commercial Registry entry number. Nine to twelve digits, no letters, assigned on registration. Every registered entity has one — public limited companies, private companies, partnerships, cooperatives and sole traders alike.
It never changes — not on a change of name, address or ownership — and it is not reissued after a company is struck off. That is what makes it a reliable way to identify a company years later.
ΑΦΜ
The tax identification number, nine digits. It is issued by ΑΑΔΕ on tax registration, not by ΓΕΜΗ. For intra-EU trade it is written with an EL prefix — not GR. Greece is the one member state whose VAT prefix is not its country code, and it is a mistake made often.
An important detail: an entity can have an ΑΦΜ without being registered for VAT. Tax registration and VAT registration are different things.
How they relate
The Αριθμός ΓΕΜΗ identifies the registration; the ΑΦΜ identifies the taxpayer. Neither is derivable from the other. Nor is it one-to-one: a branch can carry its own Αριθμός ΓΕΜΗ while sharing its parent's ΑΦΜ — which is why the Αριθμός ΓΕΜΗ is the safe identity key and the ΑΦΜ is the join key to tax data and public contracts.
How to check them
Check an Αριθμός ΓΕΜΗ in the General Commercial Registry: enter it and see whether the company that comes back is the one you expected. Check an ΑΦΜ in VIES, which answers for the whole EU and tells you whether the number is valid right now. Both are free and take seconds.