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It's always amusing to read proofs that use massive sledgehammers to crack nuts. Christian Elsholtz has recently posted a paper of this kind to arXiv. It is entitled Fermat's Last Theorem Implies Euclid's Infinitude of Primes. 1/

arxiv.org/pdf/2009.06722…

arxiv.org/pdf/2009.06722…

You might think that there was a danger of circularity here: can one really prove Fermat's Last Theorem without using the fact that there are infinitely many primes? However, the title turns out to be slightly misleading, and in fact Elsholtz only needs to use ... 2/

Fermat's theorem for a single exponent, and he has checked that the proof that a^3+b^3=c^3 has no solutions does not require Euclid's result.

The paper goes on to do the same with other results that seemingly have nothing to do with primes. 3/

The paper goes on to do the same with other results that seemingly have nothing to do with primes. 3/

Here's an attempt at an answer to the question "How would anyone think of algebra?" (a genre of question that I very much like).

Suppose I'm a certain number of miles away, and I come back at a certain number of miles per hour. 1/

tiktok.com/@gracie.ham/vi…

Suppose I'm a certain number of miles away, and I come back at a certain number of miles per hour. 1/

tiktok.com/@gracie.ham/vi…

How many minutes will it take me? To work it out, I take the number of miles I was away, divide it by the number of miles per hour, and multiply by 60.

That was a bit of a mouthful, but there's a nice way to say it more succinctly. 2/

That was a bit of a mouthful, but there's a nice way to say it more succinctly. 2/

Let's call the first number D (for "distance") and the second number S (for "speed"). Then the number of minutes I'll take is 60 x D / S (or, as we usually write it, 60D/S). That's a lot quicker to write than "60 times the number of miles away divided by the number of mph". 3/

This poll seems to have amused some people and annoyed others: my guess is that the former mainly voted "No" or "It's complicated" and the latter mainly voted "Yes". Interestingly, "Yes" was in a clear minority for the first few thousand votes, but ended at just over 50%. 1/

I think that's because at first most voters came from my followers, who are largely mathy and techy, but then @ConceptualJames sent his troops over to save the day.

What follows is a brief (if I can manage it) statement of my philosophical position about mathematics. 2/

What follows is a brief (if I can manage it) statement of my philosophical position about mathematics. 2/

Incidentally, for those worried that my views will dangerously infect my teaching, I can provide some assurance: it's basically impossible to deduce from the way a university-level mathematician teaches what his/her philosophical views about mathematics are. 3/

A bizarre discussion is going on on Twitter at the moment, concerning whether 2+2=5. Apparently, it's "woke" to try to undermine the truth we all know, that 2+2=4. As a mathematician who fondly imagines that he is towards the woke end of the spectrum ... 1/

(but not right at the extreme) I feel I ought to comment.

So first, as many have pointed out, the truth of a mathematical sentence depends on the definitions involved and on the system to which that sentence belongs. For example, there is a very important context, ... 2/

So first, as many have pointed out, the truth of a mathematical sentence depends on the definitions involved and on the system to which that sentence belongs. For example, there is a very important context, ... 2/

arithmetic base 2, in which the sentence '1+1=10' is true.

Secondly, which definitions and rules we adopt depends a lot on what we happen to find useful, either for mathematical purposes or for the purposes of modelling the world. I can imagine a (slightly contrived) ... 3/

Secondly, which definitions and rules we adopt depends a lot on what we happen to find useful, either for mathematical purposes or for the purposes of modelling the world. I can imagine a (slightly contrived) ... 3/

Sarah Peluse has just proved a result with a very appealing statement. It concerns the character tables of the symmetric groups S_n. It can be shown that the entries of these tables are integers. (To get a feel for why, see e.g.

www-users.math.umn.edu/~tlawson/old/1….) 1/

www-users.math.umn.edu/~tlawson/old/1….) 1/

Experimental evidence appeared to suggest that as n gets large, almost all the entries of the character table are not just integers, but even integers. Peluse shows that this is indeed the case: the proportion of even entries tends to 1. 2/

arxiv.org/abs/2007.06652

arxiv.org/abs/2007.06652

At least from a quick glance, the proof (actually, she gives two proofs) looks very clever, but I don't know enough about the area, so some of the techniques that look surprising to me may be more standard to experts. But given that Peluse has surprised me in areas ... 3/

An exciting result on arXiv this morning. Thomas Bloom and Olof Sisask have proved the first non-trivial case of a very famous conjecture of Erdös. 1/

Their result shows that if A is a set of integers such that the sum of the reciprocals of its elements is infinite, then it contains an arithmetic progression of length 3. 2/

Another consequence is a new proof of a result of Ben Green: a subset of the primes that is relatively dense must contain a 3AP as well. 3/