this may be a silly question, but it's bothering me for a while.

I am going through the book "What is Mathematics?" by Richard Courant.

In the supplement to the first chapter, he speaks of the Euler function and the general way of finding the total number of relative primes of any number n.

For this, he gives a formula, and then says "The proof is quite elementary, but will be omitted here."

I tried to prove it without success, then looked up online to see various articles on multiplicative functions which don't seem elementary at all.

So do you think this is a joke in the spirit of Fermat? (his famous "elegant proof but too big for the margin?")

Or is it that multiplicative function is considered 'elementary'?

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    $\begingroup$ Yes, I expect the author finds multiplicative functions elementary. Remember, "elementary" isn't the same as "easy". It just means that no sophisticated techniques or heavy theorems are involved. $\endgroup$ – lulu Mar 7 '19 at 20:32
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    $\begingroup$ I don't see that it is humourous - google.com/url?sa=t&source=web&rct=j&url=https://… $\endgroup$ – Peter Foreman Mar 7 '19 at 20:32
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    $\begingroup$ Some mathematicians use "elementary," particularly in number theory, to mean "not requiring techniques outside a certain small subset of number theory." - basically, using techniques from before analytic and algebraic number theory. (Fermat's comment was likely not humor, but rather an error, but we'll never know.) $\endgroup$ – Thomas Andrews Mar 7 '19 at 20:32
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    $\begingroup$ Just to reinforce the point, Selberg famously gave us an "elementary proof of the prime number theorem". Here "elementary" really just means that subtle properties of analytic functions are never invoked. Needless to say, the proof is far from easy. $\endgroup$ – lulu Mar 7 '19 at 21:09
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    $\begingroup$ It means you will likely find the proof in any book titled "Elementary Number Theory". To prevent infinite regress, you aren't allowed to ask what book titles mean (by tradition). $\endgroup$ – Bill Dubuque Mar 7 '19 at 21:27

Usually in mathematics, the word "elementary" is reserved for things that can be proven without the need for some ingenious insight or complicated high-power theorem. Elementary number theory, for example, is basically number theory that doesn't require the tools of analysis and algebra (group, ring and field theory) to do. "Elementary" doesn't mean easy. For example, for obvious reasons, most high school level math competitions only have "elementary" problems, but that doesn't necessarily mean they are easy, especially not in the case of competitions like the International Math Olympiad.

I think that's what the author of your text meant by the sentence (though I don't have the book in question, so I can't say for sure). In fact, it's quite common for authors to omit proofs that detract from the general "narrative" they are developing, instead leaving them as exercises for the reader. I don't think there was any humour intended there.


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