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When I do query site:www.mydomain.com with "some string on each page", several thousand pages are returned.
What this contradiction could mean? What is your experience with these queries? What is right way to check number of pages in index?
Another issue is the following. On first page of results it is printed that there are, say, 10000 results. But if I go down and click last page in row of links "1 2 3 4 ... Next", then at this last page different, say 500, less number of results is printed. I wonder, why it happens, and what number is real?
The results you see with the common phrase included are probably right - but I'm surprised that the plain old site: query doesn't line up with site: plus a common phrase. Even more, different Google data centers will often have a different number of urls for the same site: query, so there's no "true" answer these days to yuour question.
When I'm trying to get a handle on a larger site, I'll often make several site: operator queries with different directories included, like this:
... and so on.
But as I said, the "exact" number of pages indexed by Google - especially for a larger site - can be a slippery thing, and it's not work worrying about except in a general, "in the area of" kind of way.