Saturday, January 23

Indexing the Leyland primes, reprise

When I wrote Indexing the Leyland primes last May, I suggested that (of the then-known Leyland primes) "the smallest 954 (or slighly more) are indexable". Letting a Leyland number L(x,y) = x^y+y^x, x>=y, and [excepting L(2,1)] y>1. Two and a half weeks ago, I found that L(12876,2447) was prime and that this wasn't a known Leyland prime. As a result of having now examined all 75545875 smaller Leyland numbers, I can say that 955 of the then-known Leyland primes were in fact indexable!

I had made a bit of a fuss over the three Leyland primes I discovered last October. So far this month, chronologically after L(12876,2447), I found L(13307,3442), L(13227,2200), L(13371,3068), and L(13051,2448).* It seems that I'm picking low-hanging, perhaps mistakenly ignored fruit. Norbert Schneider is also looking for Leyland primes not far from where I'm currently searching, so I have competition! Regardless, I will continue my effort to close the gaps between known Leyland primes — extending their indices as far as I am able.

* January 25: L(13343,3150)

Wednesday, December 23

Double Noeud

The photo shows the seven pieces and cage (crafted by Brian Menold) of Gregory Benedetti's Double Noeud. First the two lower-left pieces are together (with some shifting) fitted into the cage. Then the two lower-right pieces are together (again, with some shifting) inserted and rotated into place. The three remaining pieces will now complete the cube. I like this much better than Triagonal Agony.

Thursday, December 10

The underpass

The photo (taken in 2012) shows Catherine aiming her camera at a billboard showing an artist's conception of what the local Denison Road underpass would look like once completed. Note that the completion date was predicted to be spring 2013. It was finally opened last Saturday! Some other misrepresentations on the billboard are the sloping grassy areas which have been replaced with high retaining walls and the dedicated bicycle lanes which are nonexistent. The treed landscape backdrop was always pure fiction. Here's my photo — taken yesterday — of what the thing actually looks like:

Friday, November 27

Triagonal Agony

This is another puzzle just purchased from Brian Menold. Designed by Laszlo Molnar, the six pieces form a cube. But that's not all. You have to construct that cube in a box that has only two triangular openings on adjacent faces. Just creating the cube is not easy. After some time I managed to luck into a solution but got confused trying to place a couple of its pieces into the box. Removing them from the box I found that I could no longer recreate my cube. Argh! Once I did finally construct the cube again I smartly drew myself a placement picture. I held the cube a number of ways and decided which might be the final two pieces into the box. Setting those aside I tried to position the other four pieces inside the box. I managed three only and then had a difficult time getting them out again. Extremely frustrating!


The six pieces of my just-purchased Pirouette. This is another Jos Bergmans creation crafted by Brian Menold. The two pieces in the bottom corners of the photo are together pirouetted into the structure at top-left before the other three pieces can be inserted to complete the cube. Very nice.

Friday, October 16

A senior moment

My first waking hours as an official senior citizen did not go well. Having taken Bodie for his morning walk, I was crossing the street intent on depositing his poop bag into the Denison Park garbage container across from the cemetery. I kept a wary eye on the cemetery because a dog that does not like Bodie was in there. That was a mistake. Somehow I failed to negotiate the curb and ended up face-down on the sidewalk. My camera took a hit but there was no significant damage: a small crack on the lens housing is all. My right knee seems to have borne the brunt of it.

My third Leyland prime find

13350^9739 + 9739^13350 is prime! This one falls into the same smallish gap (between two previously-known primes) as my second find.

Saturday, October 3

My first Leyland prime find

Back in May when I wrote my Indexing the Leyland primes I saw an opportunity to discover some of my own Leyland primes simply by looking where no one else was. My plan was to first check the gaps between the smallest 954 known Leyland primes in order to verify that these gaps contained no additional primes. That process is ongoing. Somewhat bored with the tedium, about a month ago I also started checking some small gaps beyond Leyland prime #954. That search has now borne fruit: 13739^4600 + 4600^13739 is prime!