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Pogorzelski published his purported proof in Crelle's Journal (292, 1977, 1-12) and has a Ph.D. in mathematics (his advisor was Raymond Smullyan). If his claimed proof has not been disproven after 33 years, I am curious why this would be the case, given that Shanks considers it important enough to mention in his book.
This is not a disagreement with what Carl Mummert says but it is worth remembering that when Zermelo first proposed his axioms for set theory, there was considerable scepticism that they really would avoid contradictions. People like Bertrand Russell, Philip Jourdain and Henri Poincaré criticised his axioms. Russell wrote that "I suspect that his axioms will not really avoid contradictions, i.e., I suspect new contradictions could be manufactured specially designed to be consistent with his axioms." [quoted on p. 91 of Ebbinghaus's biography of Zermelo tinyurl.com/2fskff7 ]
(continued): which suggests that prehistoric cultures may have had some familiarity with the golden ratio. In any case, the question was just about which came first, the golden ratio or the Fibonacci numbers. The golden ratio was definitely understood in the ancient world. The first unequivocal mention of it appears to be by Euclid in The Elements. There is evidence that the Fibonacci numbers were understood in ancient India, with Wikipedia citing a date as early as 200 B.C. That is 100 years after Euclid, but close enough that one could claim the question is not settled.
The section I quoted is at the bottom of page 53; the link I gave should direct you to that page. You have to scroll down the page to see it. You are right that the mere use of a pentagram does not prove that the ancient Egyptians were aware of the golden ratio and its significance. But if you read further in that section, entitled "Examples Before Pythagoras (before c. -550)," the author lists many examples from various times and geographical locations, that show at least some understanding of the golden ratio. I think the earliest evidence he cites is from 4,500 B.C. in Palestine (page 57),
