My lightroom mobile says it has 500 pictures but doesn't sync?

Could you try to re-sign-in to Lr Mobile and let me know if that helps.

If not please try this:

First remove the sync.lrdata file and restart LrD. This would restart/refresh the sync (metadata only).

Removing the Sync.lrdata might fix whatever the issue is.  Or, at least, might trigger it to occur again. So remove the Sync.lrdata, start Lr, and then wait until it either syncs or gets stuck again and create at this point a Lr Desktop diagnostic log which can hopefully provide some clues.

You can find the  sync.lrdata file here:

On Mac it's at /Users/<user>/Library/Caches/Adobe/Lightroom/Sync Data/Sync.lrdata

On Win it would be at C:\Users\<user>\AppData\Local\Adobe\Lightroom\Caches\Sync Data\Sync.lrdata

Please send me the LR Desktop diagnostig log  - best as a private message with a downloadable dropbox link.

You can trigger the log via LR Desktop preferences -> Lightroom Mobile and when you hold down the alt key you will notice a generate diagnostic log button

In addtition send me a Lr Mobile app log as well. This can be triggered when you open up the settings and long press the to LR Icon a diagnostic log will be generated and attached to your local mail client.

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Guido