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Without a Disquisitiones Arithmeticae at hand, I may ask... When the unique factorization theorem was first called the Fundamental Theorem of Arithmetic?

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This is difficult to answer since the first time it was called fundamental theorem it might have been in German or Russian. In any case Eric Temple Bell used it in 1915, on p.~2 of An Arithmetical Theory of Certain Numerical Functions.

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According to "Earliest Known Uses of Some of the Words of Mathematics" it's:G. H. Hardy and E. M. Wright 'An Introduction to the Theory of Numbers' first published in 1938 that first used the terminology Fundamental Theorem of Arithmetic to refer to this result.

In more detail it says:

FUNDAMENTAL THEOREM OF ARITHMETIC is another name for the unique factorization theorem, that any positive integer can be represented in exactly one way as a product of primes. The name was used by G. H. Hardy and E. M. Wright An Introduction to the Theory of Numbers (1938, p. 3) who remark that the theorem "does not seem to have been stated explicitly before Gauss (Disquisitiones arithmeticae (1801, §16)). It was, of course, familiar to earlier mathematicians; but Gauss was the first to develop arithmetic as a systematic science." See also A. G. Ağargün & E. M. Özkan "A Historical Survey of the Fundamental Theorem of Arithmetic," Historia Mathematica, 28, (2001), 207-214.

The term appears with another meaning in 1895 in the Century Dictionary: "the proposition that any lot of things the count of which in any order can be terminated is such that the count in every order can be terminated, and ends with the same number."

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It is true that Gauss (1801) did not call his theorem no. 16 “theorema fundamentale arithmeticae”, but he does include it under the heading “theoremata praeliminaria de numeris primis, factoribus etc.” (page 14), which is not very much different.

http://gdz.sub.uni-goettingen.de/pdfcache/PPN235993352/PPN235993352___LOG_0007.pdf

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