Monday, October 15

The nickel

In late August, a bag of potatoes kept in the basement showed obvious signs of having been attacked by a rodent. It had been many, many years since we last dealt with mice in the house and the idea that there was an unknown entry point somewhere was not a notion that I wanted to entertain. Perhaps it was a one-off, finding its way through the drainage system. At any rate, I resolved to live-trap the creature and release it outside.


My first attempt at a trap was ill-conceived. The container was too light. The twelve-sided Fiji 50-cents coin providing access to the container, too stable. And the paper underlay, too tearable. By early September I had corrected my deficiencies. A heavier plant pot propped up by a round Canadian nickel on a small glass plate:


The bait was peanut butter smeared on the inner wall. The next morning I found the nickel beside the fallen pot and, turning the whole thing over, a mouse inside. After releasing it I reset the trap and, looking every few days, found no more disturbances that month.

Last Wednesday, Catherine went into the basement to check on an unrelated matter and accidentally sprung the trap. A closer inspection showed the peanut butter gone and mouse droppings on the table. Argh! So I set it up again and the next morning snapped this photo of my second capture:


Yesterday morning I had a third one! So when in the pre-dawn hours today I heard a sound in the basement, I was fairly apprehensive. The pot had dropped but there was no mouse inside. The nickel was not on the table and it was not anywhere obvious on the floor underneath but I was in no mood to search for it. A few hours later I checked my wallet for another nickel but I did not have one. I must ask Catherine for one when she comes down for lunch later!

It was a rainy morning and I left Bodie's walk a little later than normal. Near my usual juncture on the walk I prepared to cross the street.


As I stepped off the curb I noticed something shiny on the road. I wondered: What are the odds?

Sunday, October 7

Consecutive primes summing to a conspicuous prime

After Friday's "What else?" I decided to tackle Q2 on Carlos Rivera's PrimePuzzles webpage.

Here are three large primes, each the sum of three consecutive primes:

Leyland(54,7) = 54^7+7^54 <46 digits>

1439371522465478854678431032569243142159152897
1439371522465478854678431032569243142159152957
1439371522465478854678431032569243142159152979
4318114567396436564035293097707729426477458833

Cullen(141) = 141*2^141+1 <45 digits>

131016878041367410956522344851809123847634759
131016878041367410956522344851809123847634887
131016878041367410956522344851809123847635187
393050634124102232869567034555427371542904833

Mersenne(127) = 2^127-1 <39 digits>

 56713727820156410577229101238628035201
 56713727820156410577229101238628035243
 56713727820156410577229101238628035283
170141183460469231731687303715884105727

And here's a smaller prime, the sum of 175 consecutive primes:

Mersenne(61) = 2^61-1 <19 digits>

  13176245766932173
  13176245766932207
  13176245766932219
  13176245766932231
  13176245766932279
  13176245766932321
  13176245766932363
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  13176245766938447
  13176245766938521
2305843009213693951

 I also had a look at A067377 and decided to update it a bit. I have an indexed 293768 terms of A067377 going to 10^7, listing for each prime the possible number of consecutive primes into which it may be decomposed. A 41 MB .txt file (2565345 terms, indexing not included) takes us to 10^8. Here is a 13 MB .zip compression of that.

Friday, October 5

What else?

Last month's seven consecutive primes summing to a repunit prime has made it to a PrimePuzzles website, where my "what else?" is transmuted into Q1. I'll note here my own negative results.

The seven-consecutive-primes is the only solution that I found for R19 (the repunit consisting of 19 ones) up to 10000 consecutive primes. I found no solution for R23 up to 10000 consecutive primes. I found no solution for R317 up to 1000 consecutive primes. I found no solution for R1031 up to 500 consecutive primes. Finally, I can show that R49081 is not the sum of three consecutive primes:

Let k equal (R49081 minus 1) divided by 3. This integer is not prime. The largest prime less than k is ((10^49081-1)/9-1)/3-71387 and the smallest prime greater than k is ((10^49081-1)/9-1)/3+84453. These two primes are therefore consecutive and both are required to be part of the three consecutive primes that might sum to R49081. It's an easy matter now to determine that the third number needs to be ((10^49081-1)/9-1)/3-13065 which (alas) is between our two consecutive primes and is therefore composite.

By the way, if you think that the 155839 composites between the two (above) consecutive primes constitutes a big prime gap, you'd be wrong. The gap has a "merit" of only 1.379. A website that charts large prime gaps won't even consider gaps with a merit less than 10.