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?

## Monday, October 15

## 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

4318114567396436564035293097707729426477458833

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

131016878041367410956522344851809123847634759

131016878041367410956522344851809123847634887

393050634124102232869567034555427371542904833

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

56713727820156410577229101238628035201

56713727820156410577229101238628035243

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

13176245766932411

13176245766932441

13176245766932473

13176245766932477

13176245766932497

13176245766932587

13176245766932639

13176245766932803

13176245766932819

13176245766932837

13176245766932867

13176245766932873

13176245766932887

13176245766932917

13176245766932951

13176245766932977

13176245766933017

13176245766933049

13176245766933079

13176245766933113

13176245766933121

13176245766933133

13176245766933269

13176245766933343

13176245766933407

13176245766933437

13176245766933481

13176245766933521

13176245766933523

13176245766933551

13176245766933583

13176245766933593

13176245766933749

13176245766933797

13176245766933803

13176245766933829

13176245766933883

13176245766933979

13176245766933983

13176245766934007

13176245766934027

13176245766934069

13176245766934111

13176245766934153

13176245766934177

13176245766934219

13176245766934297

13176245766934307

13176245766934361

13176245766934373

13176245766934379

13176245766934391

13176245766934457

13176245766934477

13176245766934489

13176245766934511

13176245766934543

13176245766934561

13176245766934661

13176245766934721

13176245766934769

13176245766934781

13176245766934787

13176245766934823

13176245766934841

13176245766934883

13176245766934931

13176245766934951

13176245766934961

13176245766934979

13176245766934981

13176245766935021

13176245766935039

13176245766935081

13176245766935123

13176245766935183

13176245766935323

13176245766935327

13176245766935347

13176245766935351

13176245766935357

13176245766935443

13176245766935477

13176245766935479

13176245766935501

13176245766935527

13176245766935569

13176245766935639

13176245766935663

13176245766935687

13176245766935731

13176245766935773

13176245766935819

13176245766935827

13176245766935891

13176245766935893

13176245766935941

13176245766935963

13176245766936019

13176245766936053

13176245766936059

13176245766936061

13176245766936163

13176245766936209

13176245766936229

13176245766936251

13176245766936281

13176245766936319

13176245766936331

13176245766936403

13176245766936407

13176245766936463

13176245766936521

13176245766936619

13176245766936643

13176245766936667

13176245766936691

13176245766936701

13176245766936709

13176245766936773

13176245766936821

13176245766936883

13176245766936953

13176245766936971

13176245766937013

13176245766937027

13176245766937069

13176245766937093

13176245766937111

13176245766937189

13176245766937217

13176245766937247

13176245766937301

13176245766937441

13176245766937553

13176245766937573

13176245766937607

13176245766937613

13176245766937691

13176245766937697

13176245766937739

13176245766937769

13176245766937777

13176245766937789

13176245766937793

13176245766937801

13176245766937831

13176245766937859

13176245766937871

13176245766937907

13176245766937927

13176245766937949

13176245766937951

13176245766937973

13176245766937987

13176245766938039

13176245766938071

13176245766938123

13176245766938131

13176245766938147

13176245766938161

13176245766938173

13176245766938183

13176245766938273

13176245766938291

13176245766938381

13176245766938447

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.

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

13176245766932411

13176245766932441

13176245766932473

13176245766932477

13176245766932497

13176245766932587

13176245766932639

13176245766932803

13176245766932819

13176245766932837

13176245766932867

13176245766932873

13176245766932887

13176245766932917

13176245766932951

13176245766932977

13176245766933017

13176245766933049

13176245766933079

13176245766933113

13176245766933121

13176245766933133

13176245766933269

13176245766933343

13176245766933407

13176245766933437

13176245766933481

13176245766933521

13176245766933523

13176245766933551

13176245766933583

13176245766933593

13176245766933749

13176245766933797

13176245766933803

13176245766933829

13176245766933883

13176245766933979

13176245766933983

13176245766934007

13176245766934027

13176245766934069

13176245766934111

13176245766934153

13176245766934177

13176245766934219

13176245766934297

13176245766934307

13176245766934361

13176245766934373

13176245766934379

13176245766934391

13176245766934457

13176245766934477

13176245766934489

13176245766934511

13176245766934543

13176245766934561

13176245766934661

13176245766934721

13176245766934769

13176245766934781

13176245766934787

13176245766934823

13176245766934841

13176245766934883

13176245766934931

13176245766934951

13176245766934961

13176245766934979

13176245766934981

13176245766935021

13176245766935039

13176245766935081

13176245766935123

13176245766935183

13176245766935323

13176245766935327

13176245766935347

13176245766935351

13176245766935357

13176245766935443

13176245766935477

13176245766935479

13176245766935501

13176245766935527

13176245766935569

13176245766935639

13176245766935663

13176245766935687

13176245766935731

13176245766935773

13176245766935819

13176245766935827

13176245766935891

13176245766935893

13176245766935941

13176245766935963

13176245766936019

13176245766936053

13176245766936059

13176245766936061

13176245766936163

13176245766936209

13176245766936229

13176245766936251

13176245766936281

13176245766936319

13176245766936331

13176245766936403

13176245766936407

13176245766936463

13176245766936521

13176245766936619

13176245766936643

13176245766936667

13176245766936691

13176245766936701

13176245766936709

13176245766936773

13176245766936821

13176245766936883

13176245766936953

13176245766936971

13176245766937013

13176245766937027

13176245766937069

13176245766937093

13176245766937111

13176245766937189

13176245766937217

13176245766937247

13176245766937301

13176245766937441

13176245766937553

13176245766937573

13176245766937607

13176245766937613

13176245766937691

13176245766937697

13176245766937739

13176245766937769

13176245766937777

13176245766937789

13176245766937793

13176245766937801

13176245766937831

13176245766937859

13176245766937871

13176245766937907

13176245766937927

13176245766937949

13176245766937951

13176245766937973

13176245766937987

13176245766938039

13176245766938071

13176245766938123

13176245766938131

13176245766938147

13176245766938161

13176245766938173

13176245766938183

13176245766938273

13176245766938291

13176245766938381

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

Let k equal (R49081 minus 1) divided by 3. This integer is

By the way, if you think that the 155839 composites between the two (above) consecutive primes constitutes a

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.
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