where is lightroom 6?

"i want to use lightroom6 on my mac ( caralina ). lightroom6 is good enough."

As you've stated, LR 6 can't be installed on Mac OS 10.15 (Catalina) because the installer is 32-bit and 10.15 doesn't run 32-bit applications. So your only option is to get a subscription to Lightroom Classic.

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i got lightroom6 ( mac ) but i don't use lightroom because it used 32bit setup app.

do I buy lightroom ( Payable monthly ) ? , It is not necessary for me

"the ProDesignTools web page does not have working links for LR 6 any more. When I try those links, I get "permission denied"."

You need to follow the "Note: Very Important Instructions" above the table of download links. There's a magic link at Adobe you must click on first before clicking the download link; that song and dance ensures you're logged in and seet some required browser cookie.

John, the ProDesignTools web page does not have working links for LR 6 any more. When I try those links, I get "permission denied".

As far as I know, the link for Lightroom 6 is available from https://helpx.adobe.com/download-install/kb/downloaded-older-app.html. If you didn't buy from Adobe, you can still register your LR 6 serial number, and then the method ought to work.

If you have a valid license key for LR 6, see here for where to download it:

https://community.adobe.com/t5/lightroom-classic/update-lightroom-6-0-to-6-14/m-p/10819370?page=1#M170107

But as David said, Adobe no longer sells new licenses to LR 6.

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Adobe no longer sells v6. If you have a v6 older than v6.14 (the last version) you might still be able to update. If you have v5 or even older, then, no, you cannot update/upgrade to v6.