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In this article on Hilbert's 10th problem in the March 2008 Notices of the AMS, Bjorn Poonen uses the equation

$$x^{1729}y^{1093}z^{196884} - 163xyzt^{262537412640768000} = 561$$

to suggest how deep the rabbit hole (of integer solutions to polynomial equations) goes. Clearly this is a cheeky equation. I recognize 1729 as the Hardy-Ramanujan number, 163 as the 9th (and greatest) Heegner number, and 196884 as the linear coefficient in the q-expansion of the j-invariant (and so, by Monstrous Moonshine, also 1 more than the degree of the smallest irreducible representation of the Monster group). Of course this made me curious about the others. Internet searches told me that 1093 is the smallest Wieferich prime and 262537412640768000 appears in the Chudnovsky algorithm for fast approximation of $\pi$. But all I could find out about 561 was that it's the area code of Palm Beach, FL.

What's the number-theoretic significance of 561?

Addendum: This question has been marked as a duplicate of a question asking "Why is 561 the smallest Carmichael number?" This is not a duplicate of that question, since a person who has the present question (as I did when I asked it) does not know that 561 is the smallest Carmichael number.

Ben Blum-Smith
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    Please search first, e.g. if you google "561 number theory" the first match is the Wikipedia page on Carmichael numbers. – Bill Dubuque Nov 25 '19 at 16:31
  • In fact typing your exact question in the title into google also yields an entire first page of results that immediately mention Carmichael numbers. So I'm puzzled what search you did that failed. Do you remember the exact query you used? – Bill Dubuque Nov 25 '19 at 19:04
  • The exact search query was "561". The analogous search was successful for "1093" and "262537412640768000" and I confess it did not occur to me to add "number theory" to the search string after those two successes. (Tbh, I also assumed that it would be easy and fun for someone who wanted to answer it here.) – Ben Blum-Smith Nov 26 '19 at 01:46
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    Re: your addendum.. You accepted an answer that says "561 is xyz", yet you seem to think a question that explains why "561 is xyz" doesn't answer your question. Duplicates don't signify that the question is identical, rather, as the text says above the dupe link "this question already has an answer here". In fact the answer is already in the title of the dupe, along with much more of interest in the answer. – Bill Dubuque Nov 26 '19 at 03:08
  • @BillDubuque - Fair enough. The language of the dupe link is "already has an answer here..." That's satisfied here. I was responding to the language in the "marked as duplicate" box, which is "This question has been asked before and already has an answer." – Ben Blum-Smith Nov 26 '19 at 05:15
  • Incidentally, I tried the title of this post as a search query in DuckDuckGo (my preferred search engine). The top hit is this very post. After that, the first hit to mention Carmichael numbers is the 15th. https://duckduckgo.com/?q=what%27s+the+number-theoretic+significance+of+561%3F&t=ffab&ia=web (Not a surprise that Google is better at this sort of thing, but still.) – Ben Blum-Smith Nov 26 '19 at 14:18
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    Wow, the DDG search results are horrible, turning up mostly numerological matches, whereas the google results are spot on. But DDG does ok with "561 number theory", probably because the "theory" helps to exclude the nonmathematical matches. – Bill Dubuque Nov 26 '19 at 17:37

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$561$ is the smallest Carmichael number.

JP McCarthy
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