Tuesday, May 26

Indexing the Leyland primes

A Leyland number is x^y+y^x, technically x >= y > 1 except that (x,y) = (2,1) has lately been admitted. Ordered by size, every Leyland number has associated with it an index number. Leyland primes are Leyland numbers that are prime, proven or probable. Most of the known Leyland primes can also be indexed — but not all of them! The difficulty lies in the manner in which Leyland primes have historically been discovered: restricting y to small values while allowing x to be very large. This method divorces finds from their Leyland number indices (i.e., their size), with the possibility of unchecked primes existing between two known examples. Of the currently 1091 known Leyland primes, I believe that only the smallest 954 are indexable, give or take.

There's a Numberphile video on Leyland numbers and Leyland primes. In it is mentioned the currently largest known Leyland prime: (x,y) = (328574,15). What might be this 386434-digit prime's index? Step one is to figure out the number's Leyland index. Using a Mathematica program to count, I believe it to be Leyland #3808683611. Step two is to fit the Leyland number indices of the 954 indexable primes to a curve:

The suggestion here is 17*index^2.23 as a decent fit. This equation is not meant to be infallible: a database of further-along primes might necessitate adjusting the multiplier and exponent somewhat, but for our purposes it is good enough. What Leyland prime index will generate a Leyland number index of ~3808000000? The number 5553 comes close. So, I expect the currently largest known Leyland prime to be roughly #5550 of all Leyland primes. That leaves thousands of smaller Leyland primes still to be discovered!

Saturday, May 16


Year after year, I see here a lot of the same bird species (and, I rather suspect, the same individuals where those species are poorly represented). I first saw this trumpeter swan in the river on May 7, and again the following day. I thought it had moved on, but there it was yesterday — flying a couple of circles before heading upriver. I know there are trumpeters on the lake (Ontario) but this is the first one I've seen in this locale. Ontario trumpeters sport yellow wing-tags for easy identification.

Thursday, April 16

O Canada

The image is a section of Canada's northwest, part of a new political map of Canada, a copy of which I have placed here. It's a big picture, so be patient. For me, it loads and handles ok in Chrome and Firefox — not so much in Safari. For those who didn't know, I'll point out that all of the islands in Hudson Bay — including its southerly James Bay extension — are part of Nunavut.

Wednesday, April 1

My hood

The picture is an Apple 3D representation of where in Toronto I live and take my walks. I'll try to describe the map in words without recourse to symbols planted onto the image. The bird's eye view looks west toward Raymore Park on the farther side of the Humber river, scene of significant devastation when hurricane Hazel hit in 1954. A built-in-1995 footbridge across the river is visible at the top. The main street is Weston Road, seen curving at the right toward the highrises. Just beyond is downtown Weston — once a village (the orange lot near the bottom right of this 1878 map is my reckoned property), then town, outside of Toronto. The railroad corridor shown under reconstruction at bottom right will hopefully be finished this year.

If you look for an h — let's call it a chair — with legs abutting Weston Road, my street — Sykes Avenue — is the seat and front leg of that chair. My residence is on the south side of the leg part, sixth house in. Sykes runs into Denison Road West, the back of the chair, which then curves and continues until it is stopped by St. John's Cemetery on the Humber, a private cemetery that — strictly speaking — is in Mount Dennis, the community south of Weston. To the right (north) of the cemetery is Denison Park. It is behind this park — looking down — that I get my photos of Raymore island (as I call it) sitting in the deeper water held back by the curved whitewater of Raymore weir, downstream to its left (south).

The cemetery and most of the streetscape is on high ground, contrasted with the low ground of Raymore Park, adjacent bits on the nearer side of the river including the autumn-hued treescape above (west of) the cemetery, and the large school building and mega-housing structures at bottom left, below (east of) the cemetery.

A photo I took this morning of a beaver in the river returning to its den under the island:

Wednesday, March 11

Visitors from infinity, reprise

When I posted my Visitors from infinity on New Year's day, I had been working with a database of two-and-a-half billion (10^9) numerical correspondences. On February 6, I completed a computation taking the Yellowstone permutation to five billion — but only as another two-and-a-half-billion-term file, because I cannot (with only 64 GB RAM) store all five billion terms into Mathematica at the same time.

The limitation of that shortcoming is that when I map n into A098550(n) or A098550(n) into n, any time the mapping crosses over into the other-file regime I have to reboot Mathematica with that other file. Just the reading of it takes about nine hours and working the 302 currently unknown-outcome trajectories (with minima < 10^4) backwards (towards the left in the graph) might take another day or two on each reading. But I finally completed the task yesterday! The picture shows all 302 orbits with their minima synced (although many of the more-or-less random-hued paths are obscured by others).

One can see in the graph that my backward reach is limited to five billion, while moving forward (to the right) always ends in a point beyond (sometimes well beyond) that. If you are interested in the raw data, it is still here — and I have placed individual graphs for all 302 chains here.

Friday, February 6

(101*10^120072-87)/7 is prime

A good part of my dabbling in recreational arithmetic is searching for large primes. My previous best was 2*(10^39960+10^63+10^36+10^12)+2223 which has 39961 digits and dates to October 2011. Today I bettered that with (101*10^120072-87)/7 which is made up of 120074 digits — large enough to be on page three of Henri & Renaud Lifchitz's top probable prime records. The find arises out of an observation I made for A254005.

Friday, January 9


It's no secret that there are coyotes in Toronto. Over the years, neighbourhood acquaintances have told me they had seen one — generally down by the river, but just a few days ago in the local cemetery wherein a lot of folk walk/run their dogs. I don't always take my camera with me anymore — the way I used to when going for a walk — because having Bodie with me constrains my picture-framing ability. But this morning I did have my camera and certainly glad of it.

Initially, I spotted one coyote walking up the east side of Raymore Island:

A minute or so later, I noticed another one on the west side:

They both looked to be in good health, in spite of the bitter cold of the last few days. I like this more-distant capture of that second coyote rounding the upstream tip of the island:

Thursday, January 8


Twenty-five years ago I was still buying books, magazines, journals, and newsletters, as though I didn't know that the internet was just around the corner. Of course I didn't know. One of those newsletters was Cubism For Fun and there must have been a contest therein for me to have submitted to them an offbeat musical puzzle, complete with a mixtape. I never heard any more about it — until two days ago, when a Christian Halberstadt emailed me and asked if I was the author of it. I wasn't sure; it had been so long!

Unbeknownst to me, my "mystery" puzzle was published in 1997. Christian supplied me with a copy. He and his friend, Rolf Braun, had subsequently made an effort to solve it, or at least deconstruct some of the verbiage. Here is a reprint (with a few minor editorial accruements) of my 1990 submission to CFF:

Chet Ritter was a Mohawk from Kanesatake. He was young, handsome – but had a somewhat brutish attitude. He was well-educated; certainly well-read. He had trouble holding onto money, friends, or a place to live. However, unlike most people with that misfortune, he never succumbed to the despair and depravities of society’s outcasts. Instead, you could find him at the library – reading some mathematical journal and, headphones on, listening to some recent compact disc. (Remember when all you could find at a library was books?)

It was thus that I first met him some two years ago, listening to Bruce Cockburn’s Inner City Front and studying Raymond Smullyan’s Chess Mysteries of Sherlock Holmes. I placed in front of him a note on which I had written: CAN YOU CHAT? After turning around to look at me, but before removing his headphones, he crossed out my CAN, wrote underneath it the word KEN, placed quotation marks around the word CHAT, and printed his answer on the handed-back note: OUI. C’EST MOI! I must have looked a little puzzled, for he explained, “All my friends call me Cat.” I smiled as the joke sank in.

I think it was his witticism, this perversity, that attracted me to him… A child-like honesty in the face of come-what-may.

We had many delightful discussions on our haphazard encounters, detailing everything from the failings of democracy to the meaning of life. Much of what we dealt with were problems and puzzles in the realm of recreational mathematics. He was most interested in perusing a few of the less-well-known periodicals and newsletters to which I subscribed. I think he even wrote a couple of letters-to-the-editor to some of these.

Several months ago, he disappeared. He had lost his most recent part-time job, availing himself to the charities of one of his women-friends and spending the nights (near as I could tell) under a bridge by the river. Some weeks prior, he had handed me a slip of paper titled “DUODENE” on which were listed the following twelve items:

Who Cares?
School Days
Black Diamond Bay
10:15 Saturday Night
Me And The Boys
The Walk
Norwegian Wood
Get Off Of My Cloud
Planet Claire
The Magnificent Seven

I recognized at least a few of these as songs. In addition, Cat gave me a good-quality, blank 60-minute tape, saying simply, “You’ll need this.” I reacted to his generosity by asking, “Did you pay for this?” Cat shot back: “You gonna pay anybody for recording this music?” Embarrassed, I said no more.

It took a little doing, but I finally got that tape together. I had to have help however from a few of my more musically-inclined acquaintances; at least half of the songs (and three of the artists) were unknown to me. The first eight selections fit neatly on ‘side one’ of the tape. By this time I had a pretty good idea what the organizing principle was. ‘Side two’ confirmed my hunch although (I have to admit) I listened to Bob Dylan’s Hurricane three times before I heard it. When next I saw Cat, I gloated, “Some of the song-placings are context-sensitive. In fact Joey is downright ambiguous… You could have put it somewhere else.” Cat, grinning from ear to ear, jabbed a finger gently into my ribs and said, “But then, the songs wouldn’t have fit on the tape.”

That was a week before Cat’s disappearance and the second-last time I saw him. He never said good-bye but did mention a certain “home-sickness”. Life in the big city had (I think) gotten to him. “You know,” he once said, “this world is run by jerks for jerks.”

Thursday, January 1

Visitors from infinity

Sometimes a sequence in the OEIS doesn't do justice to the bigger picture. The blue trajectory, forward from the zero point to where it ends on the right is A251412. This integer sequence also has a backward history. Combined with an infinite number of like-minded sequences (four of which are shown; the identifying numbers at the forward end are the trajectories' minima at point zero) coming in from infinity at the left and going back out to it at the right, these trajectories meander for position in number space. Any outgoing trajectory running into an integer-point of an incoming one would of course merge with it. (Well, it was a single trajectory all along. We just didn't know it.) What if an outgoing trajectory were to run into its own tail? In that case, the trajectory is seen to be — not infinite — but finite. There are currently 34 known finite orbits in this mapping (which includes 7 fixed points, orbits of length 1). The currently longest orbit is one of length 91.