Jigsaw Puzzles and the Riemann Hypothesis

[This is a back-issue of one of this site’s newsletters]

Here’s a simple-sounding puzzle: I give you a bunch of triangular tiles. What shapes can you make with them? The tiles are all the same size, all equilateral triangles. You have to use all the tiles I give you.

For example, can you make a triangle with four tiles? (Hint: yes, you can.) What if I give you five tiles instead? What if I ask you to make a quadrilateral instead? How many tiles might you need?

Some German mathematicians recently wondered how many tiles are needed to make a pentagon. (Not a regular pentagon – you couldn’t fit equilateral triangles neatly into the corners of a regular pentagon.) They added the rule that the pentagon must be convex – no inward-pointing corners. They discovered that any number of tiles will do, with the exception of a small handful of numbers, all of which are “Idoneal numbers”. It turns out that there’s only a small list of Idoneal numbers, they’re related to finding whole number solutions to x2+ny2=m. One through ten are all idoneal, but 11 is not. This means you can make a convex pentagon using 11 equilateral triangles. Try it!

The next non-idoneal number is 14. After that, the idoneal numbers get sparser. The last few known are 840, 1320, 1365 and 1848, and that might be all.

Mathematicians don’t know yet if there are any more, but the answer depends on what’s called the Generalized Riemann Hypothesis. If the Generalized Riemann Hypothesis is true, there are no more idoneal numbers. Otherwise, there’s one more, but it’s bigger than a hundred million.

It would be nice to say that any number of equilateral triangles bigger than 1848 can be arranged into a pentagon. However, to be sure that’s true, someone will have to prove the Generalized Riemann Hypothesis. The Generalized Riemann Hypothesis is a statement linking prime numbers and functions of complex numbers. The goal is to either prove the statement true or false. This turns out to be such a difficult problem mathematicians have not been able to crack even a simpler version of it, despite over a century of trying. Since 2000, there’s been a million-dollar prize on the head of a simpler version of the problem. Nobody yet has stepped up to claim the prize. The German mathematicians’ paper is behind a paywall here

4 thoughts on “Jigsaw Puzzles and the Riemann Hypothesis”

  1. Your proof is lacking in detail. That is why it is not accepted by others you’ve shown it to. For example, for your Theorem 2, you show the formula holds for a few particular values of m, and say “The general case is similar”.

    For theorem 3, you observe a pattern. However, that doesn’t prove the pattern always holds. A classic, easy-to-understand example is “n2 + n + 41 is always prime”: if you check it for n=1, n=2, n=3, n=4 and so on, it does seem to always be prime…. until you hit n=40, and the pattern breaks down.

    For a proof, you need a series of statements, each following with watertight logic from the earlier ones, that leave not even the faintest logical possibility that your statement could be wrong. This is why things like the Riemann hypothesis, the Goldbach conjecture, P vs NP are still unproven. Not because nobody saw a pattern, or nobody thinks they know the answer, but because nobody has found a sufficiently watertight step-by-step proof.

  2. nnn~



    If you think deep,then it is clear.

    Your example do not fit my theorem3,I think.

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