What is the name of this solution algorithm for $x^2-61y^2=1$?

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I'm writing a paper on Pell's equations and found a solution technique for $x^2-61y^2=1$.
At first glance I thought it was the Chakravala method (I'm very new to this topic), but upon further inspection it seems to be something else entirely. Could someone please tell me what method this is and/or give me resources for learning more? screencap of forum posting