all of the sudden I get a message that there is a software problem with my HP printer regarding 32- bit app compatibility with Mac OS system. Any suggestions so that I can continue and get this older printer to work as it has been great all these ye

Have you recently upgrade the OS on your mac to macOS Catalina or macOS Big Sur.

These OS's do not run 32bit software and that includes 32bit drivers for printers etc.

If you go to the HP support site for this printer you will see that there are no updates available for it,

HP stopped supporting this printer in Mac OS X 10.9 Mavericks which was released in 2013.

https://support.hp.com/gb-en/drivers/selfservice/hp-photosmart-c4500-all-in-one-printer-series/3575173 If you have been using this printer in newer OS systems then you have been luck it has continued to work.

Maybe it is time you treated yourself to a new printer.

Hi,

If necessary, you could download the (Post Date: Dec 11, 2020) HP Printer Drivers v5.1 for macOS (https://support.apple.com/kb/DL1888?viewlocale=en_US&locale=en_US) from Apple. An HP Photosmart C4500 series driver version 5.0.5 is mentioned in the referenced article Printer and scanner drivers for Mac - Apple Support.

Reset the Mac printing system to solve a problem - Apple Support

Add a printer on Mac - Apple Support

As long as you don't upgrade your Mac to run macOS to Catalina or Big Sur, your printer should continue to function as normal.

Did you receive a message that your printer would no longer work in the current version of macOS? That's a reference to Big Sur as being Apple's current OS.

Catalina and Big Sur do not run 32-bit apps and utilities. If you wish to run either OS, you will have to update your apps.

Use GO64 to check yours.