My home server is connected to the router. This is the output of the /etc/resolv.conf file
How do I determine the IP address of my ISP's nameservers?
Go to Network Tools
Select the "DNS Records" option. By default, your external IP is currently in the search box. Click the "Go!" button. This will show you all the DNS records including the official IP address of your ISP's name servers.
Here's an example:
However, you may also find it better to browse the help pages for your ISP. They usually publish what numbers should be used for DNS. What you find using Network Tools is actually the IP addresses of the DNS servers that server the hostnames for your ISP's domain. They may have other caching DNS servers with different IP's that you should use for resolving DNS queries on the internet.
I think I better point out the difference between a straight nameserver, and a caching recursive nameserver or DNS Cache
A nameserver simply responds to name queries for the records that it is responsible for. In the case of an ISP, it's only for the ISP's domain and those that have been delegated to it.
A caching recursive nameserver. Is a server that'll actually resolve addresses on behalf of the one querying it that can be outside of the those delegated to the server.
BIND, the most common nameserver on the internet combines both functions. It's both a nameserver and a recursive nameserver.
There is a lot of DNS terminology going around and many names of the same thing. But basically, one just answers for it's domains, while another will answer for other domains too. That's why you can often get away with pointing your DNS queries at your IP's authoritive DNS servers. Don't do that though, use the IP's they give you or allow DHCP to do it.
So, my first answer to your question "How do I determine the IP address of you ISP's nameserver" is correct. BUT, I think you want "How do I determine the IP address of my ISP's DNS Cache?" is probably what you want.
Hopefully, the answer is found published on your ISP's website in the help or support pages. This should also be set through DHCP in your modem/router and can be viewed in it's built in web page or through the devices command line.
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You can get this information from the connecting device (your router). Usually, you can find this information along with the IP address and default gateway (ISP) in the connection status. I can not tell exactly because there are many different routers models and vendors.
(Yes, I'm assuming that you meant proxy DNS servers. M. Schwartz is right, but it's a fair assumption given that you mentioned
Ask your ISP, or look at where this information already resides on your machines.
This is one of those situations where the right answer is open your mouth and ask the relevant person. Sometimes the world of computers involves actually talking to human beings. ☺
Matt H's answer is — alas! — dead wrong for all but one paragraph buried right at the end. Don't expect your ISP to provide proxy DNS service from its content DNS servers. You don't expect proxy and content servers to be in the same place for HTTP. You don't expect submission and relay servers to be in the same place for SMTP. You shouldn't expect proxy and content servers to be in the same place for DNS, either. A good ISP that is security aware won't be providing DNS proxy service to the outside world.
Ask your ISP where its proxy DNS servers are. Almost always, the ISP provides this information without your asking, or without your needing to contact technical support. There'll be a welcome pack, a FAQ page on the ISP's WWW page, or (for poor quality ISPs) a we-don't-understand-you-non-Windows-users-so-here's-a-dump-of-technical-gibberish WWW page with this information in, somewhere.
Of course, your router almost certainly has a forwarding proxy DNS server in it, configured to use the DNS servers that your ISP gave to it via DHCP. If you can access the information about the DHCP lease that is on the WAN side of your router from your router's configuration utility, you can just read the IP addresses from it.