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I have have scanned an A4R paper sheet to a 2048x1443 jpg file.

EDIT1 : It's already rotated to landscape (2048x1443).
END OF EDIT1

Now if I try convert it with this command :

$ convert -page A4R A4R_Scanned.jpg A4R_Scanned.pdf

$ echo $?

0

EDIT2 : There was no error message regarding the A4R argument.
END OF EDIT2

I get a blank US Letter, Portrait page :

$ identify A4R_Scanned.pdf

   **** Warning:  File has an empty MediaBox. Using the current page size instead.

                  Output may be incorrect.

   **** Warning:  File has an empty MediaBox. Using the current page size instead.

                  Output may be incorrect.

A4R_Scanned.pdf PDF 595x842 595x842+0+0 16-bit Bilevel DirectClass 63.2K 

EDIT3 : I tried another solution :

$ convert -page A4 -rotate 90 A4R_Scanned.jpg A4R_Scanned2A4.pdf

This gives a satisfying A4 document with no visible quality loss but I need to rotate it back to have an A4R (landscape).

When I do that, more quality is lost and that's visible :

$ convert -rotate -90 A4R_Scanned2A4.pdf A4R_Scanned2A4R.pdf

Now, my question is : How do I rotate the document back without loosing too much quality ?

END OF EDIT3

EDIT4 : Thanks to dirkt, the conversion can be done in one step and the quality is good :-) :

$ convert -page 842x595 A4R_Scanned.jpg A4R_Scanned2A4R.pdf

END OF EDIT4

A simple look at the documentation of the -page command line options shows that A4R doesn't exist (and I've never heard of a format with this name). If you mean A4, use -page A4 instead. If you mean something like "A4 with an extra feature that's called R", you possibly have to do some extra things for "R", whatever that is.

Edit: If "R" means "rotated", have a look at the -rotate option.

I wouldn't be surprised if you also got an error message while calling convert which complained about this.

Edit

If I understand the edit's correctly, the problem is that you want landscape without rotating. The problem here is that the PDF format has various ways to flag "landscape": You can either have actual landscape (bounding box width greater than height), or you can store a portrait image and add a Rotation entry. (Also see this question). And what various tools do in this respect is always a bit hit or miss.

One of the simplest things you can try is to use -page 842x595 (A4 is just a shorthand for 595x842) and see what happens.

Anything else will probably require to take a closer look at what happens by inspecting the PDFs that are generated by your imagemagick/graphicsmagick convert and you current version.