## Sunday, September 03, 2023

## Thursday, August 10, 2023

### A million-digit Leyland prime (encore)

## Tuesday, August 08, 2023

### 833719/265381

Based on the simple continued fraction of π, its convergents (rational approximations) are: 3/1, 22/7, 333/106, 355/113, 103993/33102, 104348/33215, 208341/66317, 312689/99532, 833719/265381, 1146408/364913, ...

Prime numerators are at position 1, 5, 9, ... Prime denominators are at position 2, 4, 9, ... The ninth convergent therefore has *both* prime numerator *and* prime denominator, noted ~2003 in the OEIS. It seems unlikely that we will ever see another such.

I thought it might be useful to have here a listing of the *positions* of prime numerators (*p/*) and prime denominators (*/p*) so as to better assess the rarity of their confluence:

1/

/2

/4

5/**9/**

**/9**

11/

16/

/33

87/

230/

334/

594/

/595

840/

853/

/1127

1149/

/2003

2726/

/3611

3788/

/4356

/6926

7442/

8751/

/25333

/27652

/32395

/37722

42038/

/114199

143753/

...

## Thursday, July 20, 2023

### A million-digit Leyland prime (I got lucky)

Back in April, I wished myself "better luck" in my current million-digit Leyland prime search. I've won that lottery, so to speak:

191319^170462+1*170462^191319 is 3-PRP!

The number has 1000910 decimal digits. The current top-five Leyland prime leaderboard now looks like this (the first column is the number of digits):

1717671 (1343238,19) Ryan Propper May 2023

1433792 (300102,59935) Ryan Propper May 2023

1268947 (1139148,13) Ryan Propper Jul 2023

1000910 (191319,170462) Hans Havermann Jul 2023

1000175 (218767,37314) Gabor Levai Mar 2023

## Monday, July 10, 2023

### Trouble above

## Saturday, July 08, 2023

## Thursday, July 06, 2023

### Goodbye landline

My brand new iPhone 13 arrived yesterday. I wasted little time in trudging up to Freedom Mobile (1924 Weston Rd.) in the heat —

*twice*(I didn't bring any identification the first time)! So now I have a cellphone number. This meant that I could lose the "home" phone which I had been using only as a two-factor authentication device. When I called Bell, they dragged out the cancellation process and finally offered me the landline (I'm paying $60 per month) for something closer to $10 per month (if I heard correctly). "Just cancel the

*fucking*phone," I blurted out uncharacteristically.

## Wednesday, June 14, 2023

## Monday, May 29, 2023

### Little free library

Ross and Laurie (beside Denison Park at Lippincott) have set up this very nice "little free library" box in a corner of their place. I should be able to donate a few books from my own mostly-reference library. Not everyone thinks that these things are a good idea.

## Friday, May 19, 2023

### Agent orange

This late-evening marauder comes as close to being "orange" (especially in the tail) as one might deem possible. Close inspection suggests that the effect might just be a whitened mixture of "brown". A search finds that the descriptive word for it is "erythrism". This particular raccoon paced our back- and side-yards for a half-hour or so before finally climbing the fence into a neighbour's yard. This included ventures onto the back deck and peering into the kitchen door. When I went out in order to scare it away, it was somewhat unconcerned by my approach — perhaps even attracted to it — as though it was tame. This leads me to suppose that it might be suffering from distemper.

**Update:**Two days later it was on its way to be euthanized...

## Friday, May 12, 2023

### A million-digit Leyland prime (ryanp)

*another*(at a mere 582101 digits) that had a small

*y*[L(x,y) defines a Leyland integer as x^y+y^x, x≥y; here y=2]. The current top-five Leyland prime leaderboard now sports

*three*million-digit Leyland primes (the first column is the number of digits):

## Tuesday, May 09, 2023

### A million-digit Leyland prime (ramp-up)

This is an update to my previous "retry" post, wherein I announced a new million-digit Leyland prime search attempt. Today I finished ramping up the search from the initial 12 processes on 3 computers (covering 20% of the search space) to 57 processes on 18 computers (covering about 70% of the search space). My search last year had the same 18 machines doing 108 processes but I subsequently discovered that this overloading of processes was highly inefficient and detrimental to the effort. My current 57 processes should all be done (roughly) in early November and I can assign the remaining 30% of the search space — process by process, as they come due — at that time.

## Thursday, April 27, 2023

### A million-digit Leyland prime (retry)

After my last disappointing attempt, I have started again today on a new search. Initially, I will commit twelve processes (on my fastest three computers) to about 20% of the search space, with a completion date of late October. I'm hoping for better luck this time around.

## Thursday, April 06, 2023

### Bucket list #3

1300 Weston Road |

10-piece 'original' bucket; large fries, cole slaw, potato salad: $41.80, which included a $3charge to guarantee 7 'white-meat' pieces, but I got only 6 (plus a drumstick and 3 wings) |

## Tuesday, March 21, 2023

### All around my hat

It has been a long time since we were all excited by the aperiodic tilings of Penrose's kites and darts. There was even a version using images of chickens:

Penrose chickens |

Brad Klee: headliner |

## Monday, March 13, 2023

### A million-digit Leyland prime (lghu)

*not*capital i — discovered a 1000027-digit example that was in my purview.

*exhaustive*. Having already looked at all Leyland number pairs of 1000000 digits up to 1000100 digits, finding a new PRP in the range of 1000100 digits up to 1000200 digits would allow me to (eventually) say that that the new PRP and the previous 1000027-digit one are

*consecutive*Leyland primes.

**Update (March 15):**I have ceased my new run on the understanding that Gabor will himself check all of the intervening Leyland candidate pairs for primality. Indications are that he can do this faster than I. Coincidentally, I found a new 386805-digit prime early this morning, the first such since 2 May 2022.

## Tuesday, February 28, 2023

### A million-digit Leyland prime (start of a new run)

Five days ago I started a new run of testing candidates for the property of being a million-digit Leyland prime. The million-digit part is relatively easy; the primality testing, not so much. My last run required nine months, not counting the month it took to sieve. I'm not pushing it for now. Of the 59364 candidates in this run, I'm only doing 9000 on my three iMacs. This should be done in July. My Mac minis are testing much smaller Leyland numbers (currently ~386750 digits) for primality and I'll keep that going until I exhaust my current crop of sieved numbers in that range. Only then will I divert them to help in the million-digit hunt.

## Monday, February 13, 2023

### Small-string final non-appearance coincidences in base-ten powers of two

If we look at OEIS A094776, one sees the beginnings of sequences that apply the inherent concept to strings of more than one digit. Keith Lynch suggested the idea (tongue-in-cheek, I thought) on MathFun a few days ago and Maximilian Hasler actually worked out the numbers for strings 10 to 18. I decided to chart a more comprehensive listing...

My results: sorted by strings and sorted by powers of two. There are "coincidences" where two or more strings share the same power of two as their final non-appearance exponent (for example, the two 71s in A094776 for digits 5 and 7). I'll list those here after brief summaries of each n-digit result:

**1-digit strings**

{71,5}

{71,7}

...

range: 119.5 ± 48.5

average: 1026/10

...

{153,3}

{168,2}

string coincidence

71: {5,7}

**2-digit strings**

{1300,91}

{1416,07}

...

range: 2399.5 ± 1099.5

average: 215386/100

...

{3493,28}

{3499,95}

string coincidence

2146: {33,48}

**3-digit strings**

{20589,141}

{20729,713}

...

range: 37290.5 ± 16701.5

average: 28860154/1000

...

{51375,552}

{53992,661}

string coincidences

22044: {024,275}

24486: {404,675}

25305: {410,947}

25440: {317,604}

25668: {442,815}

25704: {123,766}

25980: {096,868}

26046: {378,588}

26136: {422,677}

26316: {227,929}

26477: {152,690}

26695: {085,256}

26792: {048,732}

27003: {737,974}

27121: {545,932}

27479: {183,687}

28196: {300,554}

28252: {116,641}

28270: {099,575}

28317: {578,656}

28425: {287,392}

28532: {171,910}

28609: {017,919}

28784: {033,719}

28850: {164,647}

28891: {346,505}

29173: {648,787}

29705: {668,997}

29711: {335,799}

29976: {665,995}

30977: {131,395}

32637: {076,426}

33550: {555,796}

33607: {582,598}

33631: {117,735}

39571: {021,622}

**4-digit strings**

## Tuesday, February 07, 2023

### Fenestron

The loud sound of a helicopter just before 10:30 p.m. last night got me to look out the window and, because of the unusual appearance of the tail, run to get my camera and take a photo from the front porch. The aircraft disappeared behind the apartment buildings on Weston Rd. — seemingly landing (in the vicinity of Weston's UP Express train station; I'm going to guess at the Toronto Paramedic Services lot). Coincidentally, a UP Express service alert appeared one-and-a-half hours later.

## Friday, February 03, 2023

### Lake effect

click to enlarge |

*ocean*effect.

click to enlarge |

## Sunday, January 29, 2023

## Wednesday, January 25, 2023

### A million-digit Leyland prime (end of run)

manual distribution worksheet for 59536 primality tests |

What have I discovered? Of Leyland numbers with at least one million decimal digits, but fewer than one million one hundred decimal digits, there is only one prime. That prime was discovered by Gabor Levai long before I got to it. I saved all of my primality-test output where the 59536 entries are listed smallest to largest. If you want to see the one prime, search for "PRP".

The execution times vary wildly (due to processor circumstances) with an average of 9.45 hours per test. That would work out to 64 years if I hadn't been able to multi-process. I know now how to keep the execution times to 6 hours or less per test but that means running fewer processes per machine. Still, I might be able to shave a couple of months off the total time required for the next run.

## Thursday, January 12, 2023

### A digit-spine sequence

Éric Angelini presented a proposal for three "digit-spine" sequences on his blog here, as well as to the MathFun community. I decided to take on the first one:

s = 1, 10, 2, 0, 3, 26, 9, 119, 532, 4, 6, 896, 118, 34, 15, ...

p = 2, 11, 2, 2, 3, 23, 7, 113, 523, 3, 5, 887, 113, 31, 13, ...

d = 1, 1, 0, 2, 0, 3, 2, 6, 9, 1, 1, 9, 5, 3, 2, ...