Carnival of Mathematics #43

43 has the distinction of being connected to one of the most (in)famous open questions in mathematics: the twin prime conjecture.

The twin prime conjecture states that there are an infinite number of primes p such that there also exists a prime p+2. The first five twin primes are (3, 5), (5, 7), (11, 13), (17, 19), and (29, 31). Here’s a list of the first 100,000 twin primes: Link.

A Chen prime is a prime p such that p+2 is either a prime or the product of two (possibly equal) primes. They are, in other words, the twin primes plus some almost-but-not-quite-twin-primes. For example, 47 is a Chen prime because 49 = 7 * 7.

In 1966, Chen Jingrun proved that there are infinitely many such primes. Chen primes also make an appearance in a recent paper by Ben Green and Terence Tao where

As a second application we deduce from Chen’s theorem, Roth’s theorem, and a transference principle that there are infinitely many arithmetic progressions of primes, such that is either a prime or a product of two primes for each .

We’ve got a good number of puzzles to solve in this Carnival, so keeping with the theme I’ll leave this as a puzzle: what does all this have to do with 43? Answers in the comments are welcome. Now, on to the entries!

Speaking of Terence Tao, I recommend his recent post on when eigenvalues are stable. If you’re intimidated by some of his other posts (say, his 19-part series on Ricci flow and the Poincaré conjecture) you may still find this post approachable given you’re familiar with linear algebra and a spectrum.

David Haley writes about a neat probability problem from the Putnam exam which looks harder than it really is.

Jonathan asks if you can change one winning percentage to another without having a particular one in between. (He reports this is another question from the Putnam.)

Todd Trimble posts about a number theory problem which looks easier than it really is. Hint: You’ll need more base cases than you’re likely used to before you start induction. The solution is in this post.

And there’s still more puzzles! Head to MathNotations for an Algebra 2 Math Contest problem. There’s also an investigation involving inscribing squares into right triangles.

Edmund Harriss argues the case for using computers in mathematics.

Michael Croucher argues that numerical computer bugs are good for us and remind us that computers are limited by the humans behind their programs.

Alexandre Muniz takes the Look and Say Sequence studied by John Conway and visually translates it into two dimensions. He also gives the source code he used in Python. I’m very curious if this method can yield results for other sequences.

David Speyer at the Secret Blogging Seminar gives a discussion of quivers and roots. (A quiver is a directed graph which allows both loops and multiple arrows between vertices.) This is likely the most difficult-to-read post in the Carnival, but it’s worth the effort.

Michael O’Connor at XOR’s Hammer discusses something called Constraint Logic which can be used to determine the computational complexity of a particular game. (Game here meaning both one-player like Rubik’s Cube and two-player like Tic-Tac-Toe.)

Yoo at Stochastic Scribbles discusses the mind-bending notion of a probability of a probability. Try the puzzle first before reading the rest of the post. Keep in mind it is assuming the coin may not be fair.

Sameer Shah investigates three-dimensional curves using parametric equations, complete with pretty animations.

Mark Dominus has an addenda to two prior articles of his, in particular a discussion of the “cofinality” of an ordinal number (which I hadn’t heard of either).

Test review getting too tiring? Maria Andersen comes to the rescue with a test review game and accompanying Powerpoint file.

John Cook looks at the cost of breaking things apart and putting them together in computer science terms and the impressively named “master theorem”.

For something a little different, learn just what the difference is in finance between APR and APY.

The perennial Carnival favorite 360 never sent an entry, so I’m going to pick for them! Check out how Fourier analysis proved a piano was used in A Hard Day’s Night.

Finally I’m going to take editor’s license and include my own post at Invisible Math on ten ways to write the equation of a line.

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And that’s a wrap! The next Carnival does not have a host yet The next Carnival will be hosted at Maxwell’s Demon.