Patterns of Prediction

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RogerE
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Re: Patterns of Prediction

Post by RogerE »

Circles within a Circle

Here are some questions that relate to expectations about circles within circles, and spatial perception more generally.

• Suppose we take a large circle and put six small circles inside it. Do you expect any special features to restrict the way the small circles can be arranged? If we impose some constraints on the configuration — for example, each small circle must touch the large circle and its two small circle neighbours — do you expect any special features will then come into play?

• Imagine such a configuration in which there are 10 identical small circles. How big do you predict the large circle must be to accommodate the configuration?

• If the large circle is three times the radius of a set of identical small circles, how many small circles do you predict can be placed, without overlap, inside the large circle? What if the large circle is four time the radius of the small circles?

I was prompted to think about some questions like this after I made a post about overlapping circles in a welcome note to a new Stampboards member:
https://www.stampboards.com/viewtopic.php?f=1&t=10277&start=26068
In follow up, I wrote the following discussion, which you might enjoy:
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Screen Shot 2021-09-20 at 1.50.54 am.png
Screen Shot 2021-09-20 at 1.51.20 am.png
Screen Shot 2021-09-20 at 1.52.13 am.png
Screen Shot 2021-09-20 at 2.42.59 am.png
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/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Circles within a Circle cont.

Without attempting to add more "results" about the circle packing questions suggested at the end of my previous post, here's a diagram to expand your thinking about possible configurations of circles in a circle. What questions and predictions can this motivate?
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images-2.png
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/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Overlapping Circles

There are various configurations involving overlapping circles. Here is a philatelic example:
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USA, 1960: Wheels of Freedom, <br />Plate block of four, lower left corner (Sc 1162)
USA, 1960: Wheels of Freedom,
Plate block of four, lower left corner (Sc 1162)
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Here is a discussion focussing on the simplest configuration, the overlap of two circles.
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Screen Shot 2021-09-25 at 6.18.57 pm.png
Screen Shot 2021-09-25 at 6.19.20 pm.png
Screen Shot 2021-09-25 at 6.19.49 pm.png
Screen Shot 2021-09-25 at 6.20.09 pm.png
Screen Shot 2021-09-25 at 6.20.39 pm.png
Screen Shot 2021-09-25 at 6.21.16 pm.png
Screen Shot 2021-09-25 at 6.21.44 pm.png
Screen Shot 2021-09-25 at 6.22.18 pm.png
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/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

More Circle Configurations

Here is a symmetrical configuration of three identical circles. It has seven "elementary" bounded regions,
of three different types. (Assume each passes through the centres of the other two. Can you determine the area of each one?)
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Symmetrical configuration of three overlapping identical circles
Symmetrical configuration of three overlapping identical circles
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Here is a symmetrical configuration of four identical circles. How many "elementary" bounded regions does it have? There are four different types. (Can you determine the area of each one?)
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Symmetrical configuration of four overlapping identical circles
Symmetrical configuration of four overlapping identical circles
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Here is a s configuration of four identical ellipses. How many "elementary" bounded regions does it have?
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Configuration of four overlapping identical ellipses
Configuration of four overlapping identical ellipses
What numbers apply for configurations of more identical circles? How about ellipses?

What can you say about this configuration? (Number of "elementary" bounded regions? Area of yellow portion? How does the area of the yellow portion compare with the are of the white portion? etc).
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Screen Shot 2021-09-25 at 7.48.56 pm.png
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/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Check/hint for the yellow and white configuration in the previous post.

Did you deduce that the yellow area in the yellow and white configuration is three times the white area?

/RogerE ;)

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Re: Patterns of Prediction

Post by RogerE »

Another check/hint for the yellow and white configuration in the recent post timestamped [Sat Sep 25, 2021 21:32:37 pm]
https://www.stampboards.com/viewtopic.php?f=6&t=93774&start=203
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Inside the large circle there are four yellow regions. Did you expect/predict that they all have the same area?
Inside the large circle there are eight white regions. Did you expect/predict that they all have the same area?

/RogerE ;)

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Re: Patterns of Prediction

Post by RogerE »

Three Identical Overlapping Circles

I recently posed the challenge to predict/determine the relative sizes of the regions determined by three identical circles, each of which passes through the centres of the other two.
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Screen Shot 2021-09-28 at 12.16.59 pm.png
Screen Shot 2021-09-28 at 12.17.43 pm.png
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/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Reminder:
Predicting Global Trends
.
I don't wish to detract from the seriousness of the Worldmeter statistics, but I suggest that Stampboards members and visitors might get a "feel" for how these numbers are growing by attempting to predict the World Population and Total Coronavirus Cases will be on 30 Sep 2021 (at midnight GST). Don't post your predictions, just keep them privately for your own reference, so you can see how your estimates compare with the numbers that eventuate on Worldmeters.

To make your predictions a realistic exercise, I suggest you make the predictions several times, say at midnight GST tomorrow (17 Aug), then a week later (24 Aug), and a week after that (31 Aug).
Then wait for 30 Sep to see what the Worldmeters actually report.
[/centre]

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Reminder:
Predicting Global Trends
.
Recall the predicition invitation made mid-August:
https://www.stampboards.com/viewtopic.php?f=6&t=93774&start=179
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I don't wish to detract from the seriousness of the Worldmeter statistics, but I suggest that Stampboards members and visitors might get a "feel" for how these numbers are growing by attempting to predict the World Population and Total Coronavirus Cases will be on 30 Sep 2021 (at midnight GST). Don't post your predictions, just keep them privately for your own reference, so you can see how your estimates compare with the numbers that eventuate on Worldmeters.

To make your predictions a realistic exercise, I suggest you make the predictions several times, say at midnight GST tomorrow (17 Aug), then a week later (24 Aug), and a week after that (31 Aug).
Then wait for midnight GST on 30 Sep to see what the Worldmeters actually report.
/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Worldmeters Predictions

Happy October!
Good health and optimism to all Stampboarders and other readers :D

Although it is 2am on 1 Oct 2021 in Eastern Australia (except Qld), there are still 8 hours remaining before it is midnight GMT on 30 Sept.

For the record, my predictions for Worldmeters are:
World population: 7 896 730 388
World Coronavirus cases: 238 209 356
World Coronavirus deaths: 4 829 290

/RogerE

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Re: Patterns of Prediction

Post by RogerE »

The Worldmeters report for midnight GST 30 Sep 2021:
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World Population
... at 25 sec after midnight:
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Screen Shot 2021-10-01 at 10.00.25 am.png
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... and 6 min after that:
Screen Shot 2021-10-01 at 10.06.27 am.png
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World Coronavirus Pandemic
... at 3 min before midnight:
Screen Shot 2021-10-01 at 10.01.36 am.png
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Screen Shot 2021-10-01 at 10.04.34 am.png
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... and 15 min after midnight:
Screen Shot 2021-10-01 at 10.16.44 am.png
Commentary:

During September:

• World Population increased by six and two-thirds million people (6.68M).
That is almost a quarter million per day.

• Total number of Coronavirus cases increased by sixteen million people (16.05M).
That is over half a million per day.

• Number of deaths from Coronavirus increased by over a quarter million people (264K), almost 8,800 per day.

Footnote: My predictions were higher than the actual numbers in all three of these statistics.
My World Population prediction was almost 5000 too high, a trivial over-estimate (0.00006%).
My Coronavirus case prediction was 3.69M too high, an over-estimate of 1.6%.
My Coronavirus deaths prediction was 33K too high, and over-estimate of 0.7%.
These observations show that the rate of growth of world population is quite steady, while the rate of growth of Coronavirus infections and deaths are both decreasing slightly (a good trend!).

/RogerE

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Re: Patterns of Prediction

Post by RogerE »

Another Look at Worldmeters
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The actual title of the website is Worldometer, but I find it simpler to call its screens worldmeters
https://www.worldometers.info/
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Here is an update on the World Population worldmeter, with times in AEST = Australian Eastern Standard Time:
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World population at 3:22:46pm AEST
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Screen Shot 2021-10-01 at 3.22.46 pm.png
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World population at 10:00:25am AEST
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Screen Shot 2021-10-01 at 10.00.25 am.png
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Current rate of growth of World Population:
Comparison of the two screens shows a growth in World Population of 49,846 in 5hrs 22min.
That is a rate of 155 people per minute, so, in round numbers,
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the World Population is growing by 1000 every 6½ minutes

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Populations of the 20 most populous countries
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Screen Shot 2021-10-01 at 3.21.56 pm.png
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/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Prime Numbers with Digit Patterns

In the "counting up by prime numbers" thread, I just made a calculation which led me to notice that 2357 is a prime number. Because its digits can be parsed into the first four prime numbers, in order, that's a nice coincidence.
Of course, 23 is a prime which can be parsed into the first two prime numbers.

What do you think/predict? Will there be more prime numbers like this?

I predict that there will be infinitely many more. Let's look at the next few candidates:
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235711 = 7 x 151 x 223
23571113 = 23 x 29 x 35339
2357111317 = 11 x 214282847
235711131719 = 7 x 4363 x 7717859
23571113171923 = 61 x 478943 x 806801
2357111317192329 = 3 x 4243 x 185176472401
That list makes my prediction look "unlikely". So far we have only found two primes which can be parsed into an initial sequence of at least two of the primes. Interestingly, each of the above candidates is composite, with three prime factors. The next candidates are also composites, with five prime factors!
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235711131719232931 = 17 x 283 x 1787 x 76753 x 357211
23571113171923293137 = 7 x 67 x 67 x 151 x 4967701595369
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In fact, I searched quite a bit further, but every candidate I tested is composite.
So, at this stage my prediction has not even one more supporting example!
Does any reader have more insight to offer?

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Prime Numbers with Digit Patterns cont

As a different digit pattern to search for primes, let's look at a string of 9s followed by a singe 1.
What do you think/predict? Will there be prime numbers like this?

I predict that there will be infinitely many. Let's look at the first few candidates:

91 = 7 x 13
991 = prime
9991 = 97 x 103
99991 = prime
999991 = 17 x 59 x 997
9999991 = prime
99999991 = 7 x 13 x 769 x 1429
999999991 = 67 x 14925373
9999999991 = 19 x 19 x 277 x 100003
99999999991 = 83 x 1289 x 934693

In this case there are three early examples, but I have searched beyond without locating another instance.
For example: 9999999999999999991 = 5441 x 1837897445322551.

Note that in this pattern every candidate with an even number of digits is composite.
Here is a demonstration of that fact:
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Screen Shot 2021-10-11 at 5.12.54 am.png
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So, the only possible primes with this pattern have an odd number of digits. I have only found three here (with 3, 5, and 7 digits). I have predicted that there will be infinitely many more... Can any reader offer more insights?

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

Prime Numbers with Digit Patterns cont

As a different digit pattern to search for primes, let's now look at a countup string of numbers, starting with 1, and then followed by a single 1.
What do you think/predict? Will there be prime numbers like this?

I predict that there will be infinitely many. Let's look at the first few candidates:

11 = prime
121 = 11 x 11
1231 = prime
12341 = 7 x 41 x 43
123451 = 41 x 3011
1234561 = 211 x 5851
12345671 = 13 x 949667
123456781 = 7 x 41 x 149 x 2887
1234567891 = prime
123456789101 = 9091 x 13580111
12345678910111 = prime

In this case there are already four examples, not all clustered at the very earliest terms in the sequence, so there is a feeling that this case is more supportive of my conjecture than the previous two.

Note that in this pattern no number is a multiple of 2 or 3 or 5.
(This time I will not give my proof here, so any "diligent reader" can have the pleasure of finding a proof.)


In fact I have found two more primes in this sequence:
123456789101112131415161 = prime :D
12345678910111213141516171819202122232425261 = prime :D

Can any reader offer more insights?

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

The Crystal Ball has been rather cloudy most of the time recently, but only at the "last minute" did it clear up and predict a countup total number of posts in another thread. Sure enough:
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Countup number of posts: 3456<br />Sequence of Palindromic {a, b, c} Strings
Countup number of posts: 3456
Sequence of Palindromic {a, b, c} Strings
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https://www.stampboards.com/viewtopic.php?f=11&t=90910&start=3456
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I started that thread on Wed Jun 10, 2020 22:01:17 pm, so it has been running for just under 500 days...

/RogerE :D

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Re: Patterns of Prediction

Post by RogerE »

An even larger countup number to report, from a different thread.
This time it's the number of "replies" in the Counting randomly thread —
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Screen Shot 2021-10-23 at 5.57.23 am.png
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/RogerE :D

Footnote: Not so "pure" an example of a patterned number, but still attention-grabbing, was this "session id" I noted recently in an online activity (don't forget that 16 is 4 squared and 64 is 4 cubed):
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Screen Shot 2021-09-24 at 1.34.39 am.png
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Re: Patterns of Prediction

Post by RogerE »

From the Happy Day thread:
https://www.stampboards.com/viewtopic.php?f=6&t=82193&start=10584
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RogerE wrote:
19 Oct 2021 02:16
On 11 Oct the Covid constraints on New South Wales residents were eased somewhat after lockdown for over 9 weeks. To summarise the numbers in my area, I shared the following screenshots:
https://www.stampboards.com/viewtopic.php?f=6&t=82193&p=7455516&hilit=Newcastle#p7455516
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Four week totals (on 11 Oct 2021):
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Image
Image
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New South Wales data (on 11 Oct 2021):
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Image
https://www.health.nsw.gov.au/Infectious/covid-19/Pages/stats-nsw.aspx
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We see

Here are the comparative numbers one week later:
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Four week totals (on 18 Oct 2021):
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Image
Image
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New South Wales data (on 18 Oct 2021):
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Image
https://www.health.nsw.gov.au/Infectious/covid-19/Pages/stats-nsw.aspx
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For the state as a whole, the numbers are nicely down. For Newcastle and Lake Macquarie, the numbers have increased by about 10%.

/RogerE :D
_______________________
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What trends do you predict from one week after that report?

Here are the data, for you to check your prediction:
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Four week totals (on 25 Oct 2021):
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Screen Shot 2021-10-26 at 2.28.16 am.png
Screen Shot 2021-10-26 at 2.28.43 am.png
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New South Wales data (on 25 Oct 2021):
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Screen Shot 2021-10-26 at 2.26.14 am.png
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https://www.health.nsw.gov.au/Infectious/covid-19/Pages/stats-nsw.aspx
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For ease of comparison, here are those numbers gathered into a compact table:
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Screen Shot 2021-10-26 at 2.46.08 am.png
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We see downward trend in totals Hospital, ICU and Ventilation numbers for NSW, but a small uptick in Cases. As hospitalisation follows but lags case numbers next week can be expected to show an uptick in Hospital, ICU and Ventilation numbers for NSW.

Case numbers in Newcastle are growing, while in Lake Macquarie they have started to decrease. Both these Local Government Areas are seeing testing numbers trend downwards.

/RogerE :D

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Re: Patterns of Prediction

Post by Frank_King »

So, the only possible primes with this pattern have an odd number of digits. I have only found three here (with 3, 5, and 7 digits). I have predicted that there will be infinitely many more... Can any reader offer more insights?
Hi Roger,

My computer program says that N33=999999999999999999999999999999991 and N45 are both prime. However, it uses a probalistic technique (to which no counterexamples are known), so this is not 100% certain.

Jan

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Re: Patterns of Prediction

Post by RogerE »

Thank you Jan = Frank_King, excellent news!

The numbers with decimal representation like 91, 991, 9991, 99991, ... are visually striking, and it's nice that we can say that those with an odd number of digits are "potentially" primes.
https://www.stampboards.com/viewtopic.php?f=6&t=93774&start=213
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Of course, when a number N is composite, an easy proof is to display two smaller numbers with product equal to N. However, when N is prime, the proof that this is so can be much more demanding. As you correctly and cautiously point out, probabilistic methods have been developed that are capable of telling us, in a "reasonable" amount of time, that N is (probably) prime. When the consequences of this being wrong are not too important, that is a satisfactory state of knowledge, and in the present case your report about the 33 digit and 45 digit numbers lends support to my conjecture that there are infinitely many primes with the 99...91 pattern. On the other hand, if a particular "probable prime" N turning out to be composite will risk a bridge collapsing or national security being breached, then that is a much higher order of seriousness... Of course, ultimately there are tests which in principle can determine definitively whether a given N is prime, but the time taken to complete such a test might be impractically long (greater than the expected lifetime of our Sun, for example). So, for practical reasons, we might have to be content with probabilistic tests and a conclusion that N is (probably) prime. ;)

I'm sure you know all this Jan = Frank_King, but it seems worthwhile to say it for others who might not be familiar with such ideas...

News Flash!

Inspired by the report from Jan = Frank_King, I used a probabilistic prime test (perhaps the same test as Jan used) to verify that the k digit number N_k = 99..91 is (probably) prime when k = 33 and k = 45, and there are no further examples with 9 ≤ k ≤ 99. However, the test shows that, in fact, the number with k = 105 is (probably) prime! :D

And that's not all! The range 9 ≤ k ≤ 225 has FIVE examples of (probable) primes:
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k = 33
k = 45
k = 105
k = 197
k = 199
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The last two are not twins in the usual sense, but they are consecutive members in the sequence of numbers we are considering ;) This continues to support my conjecture, but it's good to remember that the larger k is, the greater the chance that the probabilistic test for primality fails to identify that N_k is actually composite...

/RogerE :D

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