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In https://www.math.dartmouth.edu/~carlp/ordertalkunder.pdf Carl Pomerance writes: "... Over two centuries ago, Gauss asked if this deal with the decimal for $1/p$ occurred for infinitely many primes $p$. I.e., do we have $l_{10}(p) = p − 1$ for infinitely many primes $p$?"

Meanwhile in http://guests.mpim-bonn.mpg.de/moree/surva.pdf Pieter Moree writes: "... Hence we expect that there are infinitely many primes $p$ having 10 as a primitive root mod $p$. This conjecture is commonly attributed to Gauss, however, to the author’s knowledge there is no written evidence for it."

What evidence is there that Gauss raised the question, and what evidence is there that he hazarded a guess?

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(Not really an answer, but too long for a comment.)

I found many references writing that Gauss conjectured that there exst infinitely many primes, with $10$ a primitive root, some referring to Disquitiones, others without any reference. Gauss studied the period of $1/p$ from his early days. A link to a manuscript (in German) by Siebeneicher, including handwritten tables from the Göttingen archive. https://www.math.uni-bielefeld.de/~sieben/fermat.pdf

In Disquisitiones, section 316, Gauss lists the relevant primes $7, 17, 19, 23, 29, 47, 59, 61, 97$ below 100 and writes he has the decimal periods of prime powers up to 1000 and might publish that list on some other occasion. It seems to me that he did not explictly write in Disquitiones (and possibly elsewhere) that there are infinitely many such primes. This makes this question by James an interesting one, and it should clearly be clarified why it eventually became a "conjecture" of Gauss in the modern literature!

However, it seems very very likely to me that Gauss observed from these small primes that this type of primes is not rare: Gauss famously counted primes in very young age, and conjectured a good approximation formula to $\pi(x)$, and the list above "looks like" a positive proportion of all primes below 100.

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