Sunday, May 28

The road ahead

When I wrote my Movin' on up in February, I speculated that I might reach position #40 on the probable-prime (PRP) production-score leaderboard by August. I'm a couple of months ahead of the game:


I had back then anticipated generating in the time interval (to reach #40) some 27 new PRPs worth about .01 each. Instead, I have to date discovered only 15 new PRPs, but 11 of these are greater than 83000 decimal digits — worth about .02 each, thus the time discrepancy. The road ahead is pictured above. I have taken it as far as #28, Norbert Schneider. I know Norbert because he is searching for the same type of PRPs (Leyland primes) that have created the bulk of my production score, but he has been doing this for much longer. He searches for other types of primes as well and is quite active, so when I take another leaderboard snapshot in a year or two I somewhat expect his score to have kept pace with mine. The caveat being that the algorithm that creates the production score is not the simple additive procedure I have made it out to be and Norbert's score may be weighed down by the larger number of smaller primes that he has over time accumulated.

Sunday, May 21

4*4*4 Elevator


A companion purchase to my Mean Cube, these are the six pieces of Jos Bergmans' 2010 4*4*4 Elevator. This one does have rotations (two, of the bottom-left piece). The two top-right pieces are shifted a couple of times in the assembly/disassembly of the cube to allow those rotations, making this a satisfyingly comprehensible construction toy. Visible in the top-left piece is one of three brass pins that help to reinforce that particular piece's joints.

Mean cube


These are the six pieces of Tom Jolly's 2004 Mean Cube that I recently acquired from Brian Menold. The small metallic circle embedded in the top of the bottom-left piece is a magnet whose complement resides on the bottom-right piece. It's a cheat meant to prevent the first piece out of the finished 4 by 4 by 4 cube from coming out too easily. There are no rotations in the assembly/disassembly but still a very challenging puzzle to own and appreciate.

Wednesday, May 17

Countdown primes

The concatenation of the integers from 1 to n have been called Smarandache numbers, whereby the concatenation of the integers from n to 1 would be reverse Smarandache numbers. No Smarandache numbers are yet known to be prime but we have two for the reverse. I prefer to call them countdown primes.

The first is 82818079787776757473727170696867666564636261605958575655545352515049484746454443424140393837363534333231302928272625242322212019181716151413121110987654321, first noted by Ralf Stephan in 1998. The second countdown prime was found by Eric Weisstein in 2010. We can call them countdown(82) and countdown(37765) for short.

Surprisingly, a tabulation of countdown primes in bases other than ten appears not to have been tackled by anyone so I shall remedy that herewith:

 2 — 2, 3, 4, 7, 11, 13, 25, 97, 110, 1939, ...
 3 — 2, 5, 13, 57, 109, 638, 3069, ...
 4 — 4, 106, 118, 130, 1690, ...
 5 — 2, 313, 505, ...
 6 — 2, 6, 17, 28, 33, 37, 81, 5611, ...
 7 — 373, 1825, ...
 8 — 2, 9, 47, 50, 99, ...
 9 — 2, 5, 346, ...
10 — 82, 37765, ...
11 — 2, ...
12 — 3, 4, 5, 7, 17, 58, 106, 303, ...
13 — ?