Did the Japanese word for "thank you" come from Portuguese?

The Japanese word for "thank you" is "arigatou" (ありがとう). It is often suggested that this word derives from the Portuguese word "obrigado", both of which mean "Thank you." However, evidence indicates that "arigatou" has a purely Japanese origin, so these two words are false cognates.

"Arigato" is an "u"-sound change of "arigataku". In turn, "arigataku" is the adverbial form of an adjective "arigatai", from older "arigatashi", itself a compound of "ari" + "kashi".

"Ari" is a conjugation of the verb "aru" meaning "to be", and "kashi" is an adjective meaning "difficult". So, "arigatashi" literally means "difficult to exist", hence "rare" and thus "precious", with usage shifting to indicate gratitude for receiving an outstanding kindness. The phrase to express such gratitude is "arigatou gozaimasu", or "arigatou" for short.

The Portuguese arrived in Japan in the 16th century, and there was some cultural exchange between the two countries. However, there is no evidence that the Portuguese word "obrigado" had any influence on the Japanese word "arigatou". The two words are simply coincidental in form.