I noticed that in some (many?) countries, screw sizes are listed as "M+number", e.g. M3, M4, M5 and so on.
What does this size designation mean?
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There are a few more numbers for a complete description. It's pretty much a world-wide system described by several ISO standards, but there are occasional oddities (slight differences between ISO and DIN standard sizes on a few items for instance)
You asked about M3, M4, etc - that describes (only) the outside diameter of the screw - or the size of an unthreaded hole the screw would slip through.
A full description is like M6-1X25 Which describes the outer diameter (6) , the distance from one thread to the next (1) (pitch) and the length (25) (all in millimeters.)
An M6 could also be an M6-0.75 (fine thread pitch) X whatever length. When not specified, the coarse thread is generally assumed, but sometimes the assumption is wrong...
Coarse and Fine pitch for the sizes you asked about:
Thread charts are easily found if you need to know about more sizes.
Mn for a number
n means that the diameter of the screw, projected onto the transverse plane, is
n milimeters - including the threading.
The "M" stands for "Metric" (as opposed to British-imperial units).
Mn designation does not tell you what the length of the screw (along its axis) is.