Field with one element

So, what’s up with that, right…?

In case you haven’t heard of the field with one element, it’s a totally stupid non-existent thing that doesn’t exist. Nevertheless, its blatant non-existence did not deter several people from extensively investigating it, with the intention of proving the Riemann hypothesis. There’s quite an extensive Wikipedia entry about it, but the same applies to the Loch Ness Monster.

More interesting are elusive things which may or may not exist (not unlike many rumours involving me!). Let’s have a quick run-though of the various objects in this category:

  • Wall-Sun-Sun primes: These are primes p for which is a divisor of the Fibonacci number F(p−l), where l is defined to be 1 if p = 5k ± 1 or −1 if p = 5k ± 2. If Fermat’s last theorem were false, there would exist examples of Wall-Sun-Sun primes. However, the converse is not true: we cannot conclude that there are no such primes, and it has been conjectured that infinitely many exist. A related notion is that of a Wieferich prime, where p² divides 2^(p-1)-1. Unlike Wall-Sun-Sun primes, we do have a couple of explicit examples, namely 1093 and 3511. If any larger Wieferich primes exist, they exceed 17 quadrillion.
  • A 57-regular graph with diameter 2 and girth 5: A k-regular graph is a graph where every vertex is connected to k other vertices. If we insist that the diameter (maximum distance between any two vertices) is 2 and the girth (length of shortest cycle) is 5, then a theorem states that k must be either 2, 3, 7 or 57. We have examples for the first three cases (the pentagon, the Peterson graph and the Hoffman-Singleton graph, respectively), but no 57-regular graph with this property is known.
  • Perfect cuboids: A 3-by-4 rectangle has a diagonal length of 5, which is also an integer. If all faces of a cuboid have this property, it is called an Euler brick (the smallest example is 240 by 117 by 44). A hypothetical Euler brick whose diameter is also an integer is called a perfect cuboid. No such examples are known.
  • Connected aperiodic monotile: The two Penrose tiles together can tile the plane, but not in a periodic repeated pattern. Can a single tile have this property? Socolar and Taylor found a disconnected hexagonal example, but no connected example is known. A rough measure of ‘how close a tile is to being aperiodic’ is its isohedral number. The connected tile with the greatest isohedral number (10) was discovered by Joseph Myers and shown below.

The unique (up to similarity) known example of a 10-isohedral tile.

My calendar informs me that it will be Cipher Tuesday in a few days. I’ll make this one entirely textual, since I’ve had numerous complaints from a particular person about the impossibility of inputting images into his favourite cryptanalysis software.

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2 Responses to Field with one element

  1. Edward says:

    I think you may find Hunter’s comments on the field of one element reasonably illuminating.

    (I believe you here!)

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