proudly presents  Carnival of Mathematics Number 143

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Welcome to the 143rd edition of Carnival of Mathematics!
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Welcome to the 143rd edition of Carnival of Mathematics, meta-hosted by the remarkable aperiodical.com.

What about 143? Its main claim to fame seems to be that it solves Waring's Problem for $$n=7$$, meaning that any number can be written as a sum of at most 143 seventh powers of positive integers. In symbols, $$g(7)=143$$.

For example, 2175 requires the full 143 seventh powers: $$2175=1^7\times 127+2^7\times 16$$.

How do we know we never need more than 143 powers of 7? Because 7 obeys the inequality $$\left\{\left(\frac32\right)^n\right\}\leq 1-\left(\frac34\right)^n$$, where $$\{\,.\,\}$$ denotes the fractional part of a real number. And it was proved in the 1940s that if $$n$$ obeys this inequality then $$g(n)=2^n+\left[\left(\frac32\right)^n\right]-2$$, where $$[\,.\,]$$ denotes the integer part of a real number.

That was almost the complete solution to Waring's original 1770 problem. But the small print says that if the inequality fails then $$g(n)$$ is something a bit more complicated. And the small print to the small print says the inequality never fails for large enough integers. In fact it is conjectured that it never fails for any integers greater than 1. But this has only been checked up to $$n=471,600,000$$. So when you have finished enjoying this edition of Carnival you might like to think about looking for a counterexample to the inequality for some value of $$n$$ in the high end of the millions.

Apart from that small print, nowadays, the clever version of Waring's Problem asks about $$G(n)$$, the largest number of seventh powers required when $$n$$ gets big. It is known that, for large enough $$n$$, at most 33 seventh powers are needed: $$G(7)\leq 33$$. The conjectured value is 8(!!)

Anyway, enough about 143, it's time to get on with Carnival 143.

Wait! There's something else about 143: it's a product of twin primes: $$11\times 13$$. So its digital sum $$1+4+3$$ must be exactly $$8$$. How do I know? Because all twin prime products have digital sum 8. And this month saw a sweet blog from chalkdust magazine explaining exactly why. When you've read that, stay at chalkdust a little longer because issue 5 of their wonderful magazine has just now came out.

Another magazine-style blog is the 2nd of Evelyn J. Lamb's monthly round-ups "Stuff Evelyn wants you to read". EJL is so prolific she generates a veritable monthly mathematical carnival all on her own! This edition kicks off with a reminder from history of how an evil political group can destroy a noble mathematical one, a message of a particularly urgent resonance just now.

Evelyn has something about fake news too, also, as she says, feeling urgent in today's political mudslide. Will Davies at Things from Under my Hats finds fakeness permeating even the lower reaches of mathematics: 'fake maths' is little puzzles which pretend to be profound or routine algebra disguised using cutesy pictures or whatever. I sympathsise! The torrent of this stuff on social media is just burying the nature and art of mathematics ever deeper. It is well-meaning: it is trying to reach out and draw people in. But into what?

Manan Shah at Math Misery has a lovely example of reaching out in a much more productive manner in his account of introducing ideas of fairness into a 2nd grade mathematics class. Here mathematics touches "a much larger conversation about resource equity, access to opportunity, etc". A conversation which everyone needs to be engaging in just now!

A carnival is not the place to be political ... unless it's Germany's Rose Monday carnival. But events in the USA have surely had a nontrivial impact on the productivity of mathematicians. I've seen more than one distinguished mathematician tweeting that they wished an end to the ephemeral political nonsense so they could get back to the permanent value of mathematics. And echoing Evelyn Lamb, a certain Executive Order threatened more than just temporary distraction for the mathematical community who's universal reaction is eloquently expressed in Terry Tao's blog entry.

More in the carnival spirit is Ben Orlin's look at contemporary US politics at Maths With Bad Drawings: The Island of Democrats and Republicans is a delightfully topical homage to Raymond Smullyan, whose puzzles have always been just the opposite of Will Davies's fake maths target. But Orlin's post is topical partly for the sad reason that Raymond Smullyan died this February at the age of 97. There is a splendid tribute to him by Richard Lipton and Ken Regan at Gödel's Lost Letter.

A carnival, like an awards ceremony, is perhaps a fitting place to celebrate the dear departed. I don't have blog submissions for the following who like Smullyan, left us in February; I will just link to obituary notices or whatever (I'm sure there are some omissions—post in the comments below): Kenneth Arrow (1921–2017), Bertram Kostant (1928–2017), Hans Rosling (1948–2017), Igor Shaferevich (1923–2017). The mathematical world is poorer without them but they leave mathematics so much richer.

And that brings me to my last citation and, to match my opening citation from Chalkdust, let it be a completely unpolitical bit of beautifully written purely mathematical explanation: from Brent Yorgey at The Maths Less Travelled we have The MacLaurin series for sin(x). There: un-fake, apolitical, non-ephemeral; simply beautiful.

Post a comment below! Submit here to Carnival 144, to be hosted by Frederick at White Group Mathematics!