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Background

In chapter 5 of the book "Number Theory in Science and Communication" by Manfred E. Schroeder, the author goes into continued, Egyptian and Farey fractions. On p. 65, he writes:

"We also mention the (almost) useless Egyptian fractions (good for designing puzzles, though, including unsolved puzzles in number theory) ..."

By the 'unsolved puzzles in number theory', Schroeder probably refers to open problems like the Erdős–Straus conjecture and the odd greedy expansion problem for fractions with an odd denominator.

I'd like to zoom in on the '(almost) useless' part of the quote, and I wonder to what extent it's true. I am aware that there are real-world applications of Egyptian fractions, for instance in the study of the design of electrical circuits. Although I would consider this an interesting application, I am currently most interested in purely mathematical applications of Egyptian fractions.

Question

What applications does the study of Egyptian fractions have to other branches of mathematics?

Klaus
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Max Muller
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  • Why vote to close? – Max Muller Sep 03 '22 at 14:36
  • Egyptian fractions have interesting applications in modern number theory. Keywords are Erdős–Graham problem, Znám's problem, and Engel expansion, see here. Moreover it is related to the Erdős–Straus conjecture. – Dietrich Burde Sep 03 '22 at 14:44
  • @DietrichBurde Yes, I've read the wiki page. I also mention the Erdős–Straus conjecture in the question. I'm looking for applications that haven't been listed on this page, preferably in areas outside number theory. – Max Muller Sep 03 '22 at 15:21
  • I think that the wiki page would list any interesting application "outside" number theory. So I think your question does not have an interesting answer. Egyptian fractions are, more or less, of number-theoretical interest (and have applications induced by number theory). – Dietrich Burde Sep 03 '22 at 15:26
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    I think the longevity of the idea - Egyptian fractions have been found interesting for thousands of years - suggests there is more to it. There are questions, for example, about the design of currency systems (how do you divide your basic unit into subunits - a note represented y coins) and in various sharing problems, which raise the questions. However, quite often simpler "non-Egyptian" systems have been found to be more practical than the theoretical systems suggested by deeper analysis. FYI I solve a Sudoku puzzle recently where my knowledge of Egyptian fractions helped significantly ... ! – Mark Bennet Sep 03 '22 at 20:05
  • @MarkBennet Thank you for your comment - your point about longevity makes sense to me. Interesting you mention Sudoku puzzles as an application of Egyptian fraction analysis - could you elaborate on that? – Max Muller Sep 04 '22 at 15:30
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    @MaxMuller Here is a link to the puzzle on Logic Masters Germany https://logic-masters.de/Raetselportal/Raetsel/zeigen.php?id=000ALV (there is an option for English rather than German). There are some fascinating puzzles on there with rules which don't make it into the newspapers. Dangerously addictive. – Mark Bennet Sep 04 '22 at 17:00

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