Primes that have factors in the complex plane

Fermat's Two Square Theroem

An odd prime number p can be expressed as a sum of two squares if and only if p ≡ 1 (mod 4) ie one more that a number divisible by 4.

Here are some examples:

5 = (4 X 1) + 1 = 2² + 1²
13 = (4 X 3) + 1 = 3² + 2²
17 = (4 X 4) +1 = 4² + 1²

These p ≡ 1 (mod 4) primes are not the only prime numbers the rest are of the form p ≡ 3 (mod 4) ie three more than a number divisible by 4.

Here are some examples:

7 , 11 , 19 , 23 These p ≡ 3 (mod 4) cannot be expressed as the sum of two squares.

Proofs of the theorem are very complex as shown at this link

Here is a video that includes a visual explanation which is more easy to follow though still somewhat complex:

The Real Numbers

Real numbers are broadly split into two main groups:

Rational Numbers

Numbers that can be written as a simple fraction ^a / _b (where b is not equal to 0). This includes natural numbers (1, 2, 3 ...) , whole numbers (0, 1, 2, 3 ... ), integers (positive/negative whole numbers), terminating decimals (e.g., (0.75), and repeating decimals (e.g., (0.333...).

Irrational Numbers

Numbers that cannot be expressed as a fraction. They feature non-terminating, non-repeating decimal expansions. Common examples include π (Pi) Euler's number e , and non-perfect square roots (like √2)

Real numbers can be thought of as all points on a number line.

Here is a link to a more complete description of Real Numbers. 

Some Multiplication Rules for Real Numbers

A positive number multiplied by a positive number equals a positive number
For example: +5 X + 2 = + 10

A positive number multipleid by a negative number equals a negative number
For example: +5 X -2 = -10

A negative number multiplied by a negative number equals a positive number
For example: -5 X -2 = +10

A Personal Aside

I remember the class in year 8 when my maths teacher taught us those multiplication rules.

I thought at the time that these rules were just agreed to by mathematicians. Now I would now use the term conventions. It was only much later that I realised that there were mathematical proofs for them.

The first two are fairly obvious but the third is not as clear.

Here is a video explaining why a positive times a negative is negative and a negative times a negative is positive.

The video at this link  provides four proofs for a negative X a negative = a positive.

A consequence of the third rule (a negative X a negative = a positive) is that the square root of a negative number does not exist - though as we will see later that should be the square root of a negative number does not exist in the Real Numbers.

Before learning the rules above it might seem that √-4 = -2, but we now know that -2 X -2 equals +4.

Imaginary Numbers

In the mid 16th century, Geralomo Cardono, while working on cubic equations came across solutions that contained the square root of a negative number - √-121

Rafael Bombelli decided to treat square roots of negative numbers and found that doing so produced accurate real number results.

Mathematicians came to call these types of numbers Imaginary Numbers.

Leonhard Euler, in the 18th Century used the letter i to denote the imaginary unit - √-1.

The definition of i is i² = -1 .

Complex Numbers

Here is the video that first introduced me to the idea: https://youtu.be/EV-v-9KbJ7Y

Hinkley C

This is a post on the decision making and building of the Hinkley C Nuclear Power Station in the UK.

That the UK government was seriously considering building nuclear power plants was signalled by the then PM Tony Blair 5 at the Confederation of British Industry (CBI) Annual Conference on 28 November 2005.

Here is a report discussing the announcement

Here is another link with quotations from blair

Speaking at the CBI’s annual conference Mr Blair told delegates that energy policy was back on the agenda “with a vengeance.”

“Round the world you can sense feverish re-thinking. Energy prices have risen. Energy supply is under threat. Climate change is producing a sense of urgency”, he said. “I can today announce that we have established a review of the UK’s progress against the medium and long-term Energy White Paper goals. The Energy Minister Malcolm Wicks will be in the lead, with the aim of publishing a policy statement on energy in the early summer of 2006.”

“It will include specifically the issue of whether we facilitate the development of a new generation of nuclear power stations,” he said.

The link below is to a news conference in on June 8th 2006. (The link will display in a new tab). The section on nuclear starts at 32:06 minutes in.

Link to Press Conference

From the transcript

Question from Journalist

You talked earlier about nuclear power coming back up the agenda with a vengeance, and I was wondering whether you think it is possible to build a new set of nuclear power stations with the kind of extended planning inquiries that we have had before the construction of places like Sizewell B? And secondly, how can a government set a regulatory framework that acknowledges the low carbon contribution of some nuclear power stations?

Reply from Blair

Well, I don't think I can say much more on the planning system, but I think it is very important to emphasize, because I think there is a misunderstanding maybe in parts of the public here. I am talking about replacing our existing nuclear power capability, because if we don't then over the next few years we are going to see a dramatic reduction. So sometimes this argument is put as if here am I saying right let's increase dramatically the amount of nuclear power we have. That is not the issue, the issue is over the next 15 years 20 percent of the electricity that we get in this country that comes from nuclear is going to decline to virtually nothing. So the question is, are you going to be able to make that up from other sources? And we are already planning to make a big push on renewables, we are already planning a major push on energy efficiency, and therefore to me it is very difficult to see how you are going to be able to have a secure energy supply in the future unless you are replacing at least the nuclear power stations that are going to be decommissioned. So I can't tell what the planning system is going to be yet, and that is something that obviously the Energy Review will look at, but that is the heart of the issue, it is whether you replace these ones.

It is clear from Blair's answer that he and his government supported the building of new nuclear power plants in 2006. Nuclear power was "coming back up the agenda with a vengeance" and "...to me it is very difficult to see how you are going to be able to have a secure energy supply in the future unless you are replacing at least the nuclear power stations that are going to be decommissioned."

Blair also mentioned the Energy Review which was published later in 2006.

Click on this link for the energy review

The Nuclear section starts on page 113

The Energy Review clearly supported the building of new nuclear power plants in line with the arguments of tony Blair above.

The Nuclear section of the report begins with the following paragraph

5.93 Nuclear power is a source of low carbon generation which contributes to the diversity of our energy supplies. Under likely scenarios for gas and carbon prices, new nuclear power stations would yield economic benefits in terms of carbon reduction and security of supply. Government believes that nuclear has a role to play in the future UK generating mix alongside other low carbon generating options. Evidence gathered during the Energy Review consultation supports this view.

Électricité de France (EDF) was clearly the likely builder of the Hinkley project early in the process. In February 2007 Vincent de Rivaz, chief executive of UK division EDF Energy stated “EDF will turn on its first nuclear plant in Britain before Christmas 2017 because it will be the right time”.

Sources: https://www.mub.eps.manchester.ac.uk/nuclearhitchhiker/a-drop-in-the-ocean/ and https://docplayer.net/8662131-Hinkley-point-new-nuclear-power-plant-the-story-so-far.html

A bill to authorise the building of new nuclear power plants (among other things) was introduced into the House of Commons on 10th January 2008. Here is a link to the Bill

In introducing the bill the Secretary of State for Business, Enterprise and Regulatory Reform, John Hutton stated:

Giving the go-ahead today that new nuclear power should play a role in providing the UK with clean, secure and affordable energy is in our country’s vital long-term interest. I therefore invite energy companies today to bring forward plans to build and operate new nuclear power stations. Set against the challenges of climate change and security of supply, the evidence in support of new nuclear power stations is compelling. We should positively embrace the opportunity of delivering this important part of our energy policy.

.

Here is a link to Hansard showing Mr Hutton's statement.

Rooftop Solar - A Great Investment

Introduction

A few weeks ago my wife, Margaret, asked me whether the Solar Electricity System we installed in our house was a good financial idea. I decided to investigate and answer the question.

Later in this post I will describe the methods I used to investigate this issue, but the answer can be clearly stated: our Solar Electricity System is a great investment. I calculated that we saved $813.40 in 2023. I determined that the yearly cost of electricity without the Solar System would be $1,505.02 . The sum of our bills for the year was $691.62 : $1,505.02 - $691.62 = $813.40 .

If we were to invest the money we spent on our Solar Electricity System we would need a return of 11.5% to make the same money as we saved on our electricity bills with Solar. A return of 11.5% PA in an investment that is safe is extremely difficult to find.

Our Solar Electricity System had an out of pocket cost of $7,055. It is a high quality system, see details below, and the cost includes the Victorian Government Rebate of $2,225. Percentage return on investment is calculated by amount saved divided by cost of electricity system multiplied by 100: $813.40 / 7,055 * 100 = 11.5 .

The Data

I used two sources for the data required for the calculation - our Fronius Smart Meter and the Electricity Bills.

Fronius Smart Meter.

The Smart Meter is installed in our electricity switchboard as shown in the video below:



The Smart Meter sends data to the Fronius Site and can be accessed via an app on my phone.

The app shows instantaneous data as shown in the video below:



The app also supplies full day data as shown in the photo below:

There are three pieces of information, reported in Kilowatt Hours (kWh). Click on this link for an explanation of the meaning of the term Kilowatt Hour. Production: The amount of electricity produced on the day by the Solar Electricity System in kWh. Consumption: The amount of electricity used by the electrical appliances in the home. Self-consumption: The amount of electricity supplied by the Solar Electricity System to power appliances in the house. The app also generates a graph showing the levels of these amounts during the day. The date is also included at the bottom the output, in this case 19 Jan 2024.

Electricity Bills.

The graphic below shows the section of our electricity bills that provides the information about electricity usage and the consequent bill charge:

The start and end dates of the bill and the number of days covered.

The supply Charge - $1.0767 per day (28 days equals $30.15)

The electricity Usage (imported from the grid) in kWh (113) and cost per kWh ($0.2436) and total cost $27.53)

Solar Credit (the amount of electricity our system exported to the grid) in kWh (433) payment per kWh ($0.08) and total amount subtracted from the bill ($34.64)

GST (calculated as 10% of sum of Supply Charge and Electricity Usage - $5.77)

Total charge (Supply Charge plus Electricity Usage plus GST minus Solar Credit - $28.81)

The Calculations

The Solar Electricity System reduces the cost of Electricity paid to the supplier in three ways

1) importing less electricity from the grid because some of the electricity used in the house is supplied from Solar, 2) payment of the Solar Credit which as shown above is a direct reduction of the bill and 3) reducing the GST paid as less electricity is imported

Importing Less Electricity

The cost of the amount of electricity we consume (Consumption in the app) can be calculated by multiplying Consumption by the cost of importing per kWh - $976.29

The amount of electricity imported can be calculated from the data in App (Consumption minus Self-consumption) and from the Bills. Both methods arrive at similar amounnts to within a 99% accuracy.
I calculated this saving as $340.22 .

Solar Credit

The value of the electricity exported can be calculated by summing the values of the Solar Credit on the bills. This results in a figure of $415.59 .

Using this method the saving would be $340.22 plus $415.59 which equals $755.81 .

Reducing the GST Paid

Another method of calculating the saving is by calculating the total cost of the electricity used and then subtracting the actual cost from the bill.

The bill is calculated by summing the value of the electricity imported and the supply charge and calculating the GST from that sum (which is 10%) then subtracting the Solar Credit.

The value of the total amount of electricity consumed (calculated from the app) can be determined by multiplying that total amount by the cost (per kWh) of the electricity. The total amount of the bill if we did not have solar can be calculated by summing the value of the total amount of electricity consumed ($976.29) the total supply charge ($391.92) and the total GST ($102.31). This gives a total of $1470.52 . As we are trying to calculate the total bill cost without solar, that GST value is too low. Without solar more electricity would be imported which would increase the GST value by $34.50 .

Consequently I calculate the total cost of our electricity without solar as $1,505.02 .

Given that the total cost of electricity from the bills was $691.62 the saving is $813.40

The Solar Electricity System

We purchased the Solar in two installments.

The initial purchase was from EnviroGroup. Usually the size of household solar electricity systems is 5 to 6 Kilo Watts. We initially purchases a smaller one as at that time we did not have access to a Solar Credit (sometimes called a Feed In Tarrif).

We purchased 8 solar panels that had a maximum output of 2,560 Kilowatts. We purchased a Fronius 5 Kilowatt inverter, as Fronius is one of the best brands and so we could expand the number of panels in the future. The cost of this system was: $5789.99 .

When a Feed In Tarrif became available we added another 8 panels from SolarGain with a maximum output of 2,600 Kilowatts. Giving a maximum output from the panels of 5,160 Kilowatts. We included the Fronius Smart Meter in this purchase. Total price: $3490.00 .

These two purchases totalled $9279.99 which might seem pricey, but we decided that a system with top quality components was the best solution.

We applied for the Victorian Solar Rebate and received $2225.00.

The out of pocket cost of our solar electricity system was: $7054.99

Netflix Recommendations

Here are some shows that I have watched on Netflix that I enjoyed and am happy to recommend to other people. I have attempted to make my comments as spoiler free as possible but still make them informative. I have included trailers for the shows, which will give you a sense of the look and feel of the show but might contain some spoilers. The trailers are all for season 1 of the show. They are in no particular order. There are many shows on Netflix that I want to explore but the ones below are those that I have watched. Other recommendations welcome in the comments.

Travellers

An excellent time travel show. Only three seasons but with a satisfying conclusion.



The Umbrella Academy

An interesting and quirky look at time travel and dystopia. I am looking forward to the next season



Sense8

Eight people across the world have a connection. Unusual Sci-Fi with a satisfying resolution. It has strong and positive LGBTIQ themes. Series complete.



The Last Kingdom

Set at the same time as the show Vikings with English Kingdoms battling Viking invasions. I found it more interesting and satisfying than the Vikings TV show. So far with four seasons with I think more to come



Van Helsing

The Vampire apocalypse. Avoid if you don’t like horror shows, but this one has a very interesting story line and a satisfying resolution. Series complete



Shadow and Bone

Fantasy – chosen one / good vs evil, but done well. Covers the first book in the series very well. The introduction of very interesting characters from the follow-up series “Six of Crows” was a masterstroke. I am looking forward to the next season.



Outlander

Time travel / historical drama, based on the best selling series by Diana Gabaldon. So far 5 seasons. I am looking forward to season 6.



Anne with an E

An interesting adaption of Anne of Green Gables. I have one friend who hated it because she loved the book so much, but I enjoyed the show. Show completed with a satisfying conclusion



Gotham

Set in the Batman universe this tells the story of Commissioner Gordon from when he was a rookie detective. Back stories include Batman, The Penguin, Catwoman, The Riddler, The Joker, Bane, Poison Ivy and Scarecrow. Show completed satisfactorily with 100 episodes



Bridgerton

The loves, loves and rivalries of upper class families in Regency England (1813). Looking forward to the second season.



The Cook of Castamar

Set in Spain in 1720 a story of family rivalries and politics but with the central theme of the relationship between the Duke and his cook. A Spanish production. We watched it dubbed in English which was very well done, but there is the option to watch it in the original Spanish with English subtitles. Series completed with a satisfying conclusion.



Longmire

A modern day police procedural set in contempory Wyoming, USA. Six satisfactorily completed seasons.



The Witcher

A fantasy series based on the stories by Andrzej Sapkowski. I really enjoyed the first season and am looking forward to season 2. Note there are multiple timelines in the story.



The Witcher: Nightmare of the Wolf

An excellent one off animated prequel to the Witcher.



Godless

Set in the American west. This is an excellent limited series with 7 episodes



Blindspot

A FBI crime show with an interesting twist. There are currently 5 seasons and we have watched the first two. We will continue on with series 3 soon.



Love, Death & Robots

A collection of animated short stories spans several genres, including science fiction, fantasy, horror and comedy. I have watched the first four of the shows and will come back to it later. The animation is excellent.



Lucifer

Based on characters created by Neil Gaiman, Sam Kieth and Mike Dringenberg, this is a redemption story for the original villain - the Devil. Six seasons with a satisfying conclusion.



Line of Duty

A great contemporary police drama with 5 seasons.



The Crown

A series following the life of Queen Elizabeth II. Excellent show currently with four seasons with season 5 coming next year.



Knightfall An excellent series on the Knights Templar. Currently two seasons with no plans for a season 3.



Rake

This great Australian show is on Netflix if you haven’t already seen it on the ABC.



Peaky Blinders

A notorious crime gang in 1919 Birmingham UK. An excellent show but I find it hard to find anyone in this show to barrack for as there are so many villains.



Marcella

A contemporary British Police procedural, made interesting by the personal, psychological struggles of its main character. Complete after three series.



Jack Irish

A great contemporary Australian crime series. Netflix has two seasons of the TV show and three movies.



Babylon Berlin

A German series set in Berlin in the late 1920s and early 1930s. A great story and highly atmospheric, made doubly so because we know what is coming in 1933. The dubbing was excellent so we watched it that way. Currently three seasons. The German Production company is filming the fourth season, so hopefully that season will appear on Netflix next year.



The Alienist

A psychological thriller set in New York in the late 19th Century. Two seasons. A third season has not been confirmed but no announcement for cancellation.



Happy

“Nick Sax is a corrupt, intoxicated, ex-cop turned hit man who is adrift in a twilight world of casual murder, soulless sex, and betrayal.” It is dark, violent and with plenty of graphic language but there are wonderfully comic and uplifting elements as well. Nick ends up with the most interesting sidekick! Only two seasons unfortunately.



Altered Carbon

A great Cyberpunk detective show unfortunately cancelled after two seasons. The first series follows fairly closely the first book, but the second series is completely different. I am a sucker for Cyberpunk and if you are too then you will really enjoy this show. Warning it is quite violent.



Russian Doll

“Russian Doll” is spiky, funny, devastating, and downright bizarre. It follows simmering New York City misanthrope Nadia through the most horrific, life-changing, revelatory night of her life.



Sweet Tooth

“Sweet Tooth is part fantasy, part sci-fi, part whimsy, part cold-eyed realism and most points in between. It is either warmly eccentric or hysterically crazy, perfect entertainment or a horrifying attempt to parlay the pandemic into a commercially palatable mashup.” The Guardian. But I liked it.



Dirk Gently’s Holistic Detective Agency

“As a freewheeling comedy-thriller, Dirk Gently's Holistic Detective Agency is agreeably daft entertainment even if it doesn't crib heavily from Douglas Adams' novels, serving as an incredibly loose adaptation. Quirky sci-fi comedy is fun but can be violent.”
I loved it. Unfortunately cancelled after two seasons.

Cloud Atlas - the Book

Introduction

The author of Cloud Atlas is David Mitchell.

I first became aware of Mitchell in a YouTube interiew/conversation at this link.

Book Structure

During the interview Mitchell described whis writing style in the video excerpt below.



In the video Mitchell states that his natural style is as a novella writer. He assembles a group of novellas connected together by "footbridges and worm holes" so it looks like that he has "handed in a Sir Christopher Wren sized thing but it is actually just leggo blocks".

Cloud Atlas is written in this style. It consists of six nested stories (of novella length); each is read or observed by a main character of the next story, thus they progress in time through the central sixth story. The first five stories are each interrupted at a pivotal moment. After the sixth story, the others are resolved in reverse chronological order.

The diagram below graphically displays the structure of the novel. Source: https://albrooke.com/2016/01/19/cloud-atlas/

In the second story Robert Frobisher the musician is writing a musical composition - Cloud Atlas Sextet. His description of the structure of the piece is reminiscent of the structure of the book. His comment "Revolutionary of gimmicky" is maybe the author asking the same question about the book. “Spent the fortnight gone in the music room reworking my year's fragments into a 'sextet for overlapping soloists': piano, clarinet, 'cello, flute, oboe, and violin, each in its own language of key, scale, and color. In the first set, each solo is interrupted by its successor; in the second, each interruption is recontinued, in order. Revolutionary or gimmicky? Shan't know until it's finished, and by then it'll be too late.”

Here is the music from the film adaption. It is not like the description in the book but is still very pleasant to listen tol



Major Themes

Reincarnation of souls is an important theme in the book. Zachry, the protagonist in the sixth story, makes it an explicit claim and even uses the name of the book in his argument.

“Souls cross ages like clouds cross skies, an' tho' a cloud's shape nor hue nor size don't stay the same, it's still a cloud an' so is a soul. Who can say where the cloud's blowed from or who the soul'll be 'morrow? Only Sonmi the east an' the west an' the compass an' the atlas, yay, only the atlas o' clouds.”

Mitchell does guard the secular hypothesis when he has Meronym reply that she does not believe in souls.

Each of the main characters has a comet birth mark and Mitchell does think that this signifies reincarnation. Unfortunately two of the characters with this birth mark Luise Rey and Timothy Cavenish have overlapping lives. When asked about this Mitchell admits that it was a mistake but goes on to give a "rationalisation" which includes that "...things are more interesting when they are fuzzy and don't quite fit ..." which can work in a novel and goes on to describe such probolems using the wonderful phrase "portable textual temples of inexactitude" and "its kind of OK for novels to not make complete sense, this is ofcause a ratioisation but its kind of a true rationalisation as well."

See his full response to the issue in the video below:



Dystopia

Each of the main characters in the stories were trying to improve their world.
Adam Ewing decides to join the anti-slavery movement.
Robert Frobisher composes a musical masteriece.
Luisa Rey stops the construction of a dangeroug nuclear power station.
Timothy Cavendish writes a film script of his comic adventures. The film is produced as it is watched by Somni-451 in the next story.
Sonmi-451 publishes her declarations which are revered in by later generations.
Zachry assists Meronym and resists the temptation to attack her.

But the attempts by individuals to improve the workd are ineffective as the two stories in the future are very dystopic.

In An Orison of Sonmi~451 set in the year 2144, in a country called Nea So Copros: " ... a savage consumer-driven society gone amok. Each citizen has a spending quota; all-seeing eyes watch from the street corners; and each citizen has a "soul" or a computer chip implanted in his finger, without which travel and other basic freedoms are impossible. Several stratas of society exist, ranging from Xecutives (who appear to run Nea So Copros as part of the "Juche") to Fabricants, mass-produced slave clones treated as nothing more than property." Source of quote at this link

Somni described Nea So Copros as follows: "Nea So Copros is poisoning itself to death. Its soil is polluted, its rivers lifeless,its air tosloaded, its food supplies riddled with rogue genes. the downstrata can't buy the drugs necessary to couonter these privations. Melanoma and maleria belts advance northwards at forty kilometers per year." People escape the surrounding coutryside around NeoSeoul because of: ".. malaria, flooding, drought, rogue crop-genomes, parasites, encroaching deadlands...".

“I believe there is another world waiting for us. A better world. And I'll be waiting for you there.” "Human hunger birthed the Civ’lize, but human hunger killed it too." —Meronym

Denier nonsense

Links

Global Cooling in Telegraph

The Temperature of the Earth

The following calculation assumes that the energy into the Earth system and the energy out are not affected by the atmospheric gasses. When the system is in equilibrium, that is neither warming nor cooling, Energy In must equal Energy Out.

Energy In

Energy enters the Earth system from the sun mainly in the form of visible light. This solar flux is represented by the letter L. The value is 1361 Watts per square metre (W/m²). About 30% of the light from the sun is reflected out of the Earth system, mainly by clouds and ice. The reflectivity of the planet is called the albedo and is represented by the letter α.

Each square metre then absorbs energy from the sun according to the following formula:
L(1 – α)

We are interested in the total energy absorbed. The solar flux varies over the surface of the Earth, strongest at the equator where the surface is perpendicular to the incoming light and weakest at the poles where the surface is at a large angle to the sun’s rays. Adding up all of these variable amounts is tricky but there is a simple method to calculate it. The light waves from the sun are parallel to each other and the Earth intercepts them and casts a shadow. This shadow is a circle which has a simple formula - πr². So the formula for the total energy from the Sun absorbed by the earth is:
L(1 – α) πr²

Energy out

The formula for the energy that is radiated away from the surface is:
ϵσT⁴ 4 πr²
ϵ is emissivity, how close the source is to a black body (0 not a back body to 1 a black body), σ the Stefan-Boltzman constant, 5.67 X 10^-8 (Watts/square metre), T⁴ is temperature in Kelvin raised to the fourth power, 4πr² is area of a sphere.
If the system is in equilibrium Energy In equals Energy Out
So
L(1 – α) πr² = ϵσT⁴ 4 πr²

Calculate Temperature, ie solve for T

πr² appears on both sides so it can be cancelled

So
L(1 – α) = 4 ϵσT⁴



With L = 1361 W/m² α = 0.3 σ = 5.67 X 10^-8 and ϵ = 1 (given that Earth is relatively close to being a black body)
T = 254.5 K,
(K is degrees Kelvin. The Kelvin Scale starts at absolute zero ie -273⁰ Celsius. Note Kelvin temperatures are indicated by K, and do not by convention use a degree (⁰) sign.)

Consequently 254.5 K equals -18.5⁰ C

Given that the global temperature is about 15⁰ C, according to this calculation the planet should be about 33⁰ C below current temperatures.

This calculation gives a very incorrect answer because it ignores the effect of Greenhouse Gasses.