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Dealing with truth anyway

Jacob Rees-Mogg had a talk for the Oxford Union, published on YouTube on 2013-11-11. The Oxford Union is a debating society. A debater’s aim is to win the audience over and not necessarily to discuss truth. Rees-Mogg had an entertaining talk but it is not targeted at discerning truth indeed. His presentation comes across as modest and forceful, with the charm of perhaps some old-fashioned style. Who closely considers his words may however be shocked by the unreasonableness and closed-mindedness.

Boris Johnson and Nigel Farage have been criticised for spreading false arguments for the June 23 2016 Brexit referendum. Obviously, these two individuals cannot be held accountable for swinging the views of some 45 million voters. I wondered since the referendum whether they had had some help. Apparently Jacob Rees-Mogg had been giving a helping hand.

To clarify Rees-Mogg’s departure from truth, we first must mention some properties of the European Parliament.

Seat-to-vote ratio’s in the EU parliament

The EU Parliament has 751 seats, distributed over 28 member states with 500 million people. The distribution over countries is not proportional to the populations, since countries are units by themselves, and it is felt that this should have some effect. Thus Germany with its population of 82 million has 96 seats (1.17 seats per million), the UK with its population of 65 million has 73 seats (1.12 seats per million), and Malta with its population of 0.5 million has 6 seats (12 seats per million). There is relatively little tension about this apportionment, since the countries fall in comparable classes (large, medium, very small), and the major political differences translate into political parties. The divisions between Christian Democrats, Social Democrats, Liberals, and what have you, apparently are dispersed over countries in similar manner, or, the political parties are able to create alliances over nations. It is part of the wonder of the EU that nationalism is being channeled and that there is more scope for civil democracy. A recent paper of mine on proportional representation is here.

Jacob Rees-Mogg’s quote on Malta

Jacob Rees-Mogg does not explain above democratic solution for dealing with Member States of different sizes. He criticises the EU that Malta is over-represented compared to the UK. It is a fact that Malta has a higher seat-to-vote ratio, but only pointing to this fact obscures the other considerations. He mentions a perhaps older figure of 15 instead of the current 11, but that is irrelevant here. The demagoguery is that many in his audience apparently are not be aware of the key notions in this apportionment, and he apparently takes advantage of their lack of knowledge to win them over to his own closed-mindedness. The demagoguery is that he creates a suggestion as if Malta has 15 times more influence than the UK, as if 6 is 15 times larger than 73 (as, indeed, 6 = 15 * 73).

The quote at the final minute starting at about 11.30 is, with the abusive “proportionally outvote” and the threat of “spectres”:

“So what is this great experiment doing ? It is helping once again the rise of the extreme right, and in some cases the extreme left. That is the threat to democracy that is there, that is coming, that is deeply destructive. But the fundamental problem, the real issue at hand tonight is that there is less democracy in this country, because of the European Union. Because, Ladies and Gentlemen, however you vote the next general election, 60% of our laws, and some say higher, is made on the basis of European agreements, where the Maltese proportionally outvote us 15 to 1. Whoever you vote for, matters less than somebody in Malta votes for, about the laws of our country. And if you are unsatisfied with that, and you want it changed, I cannot give you any redress, because the United Kingdom Parliament, the most ancient democratic Parliament in the world, has been made powerless. That is the threat to democracy. It is here, but it is on the continent as well. It is a frightening spectre. The best way to deal with it, is to deal with our relationship with the European Union, to put our own democracy first and foremost, and hope that others follow.”

Does it really require a protest ?

It is almost silly to protest to this demagoguery:

  • The situation w.r.t. the UK and Malta in the EU Parliament has been explained.
  • The UK has District Representation (DR) instead of Proportional Representation (PR), which causes that the UK is much less democratic than most countries in the EU or the EU Parliament itself. The PR Gini for the UK of 2017 is 15.6%, but there has been a lot of strategic voting, so that we don’t really know what the first preferences of UK voters are. By comparison, Holland has a PR Gini of only 3.6%, and people in Holland could vote for the party of their first choice. See this weblog text and this paper.
  • I tend to think that Rees-Mogg really worries about the state of democracy, while A.C. Grayling rather sees an elitist or even pecunary motive, see this article, as in “follow the money”. Yet Rees-Mogg doesn’t study the topic, and thus he is condemned to repeat an ideology. He studied history but not science. His voting track record apparently shows that he consistently voted against Proportional Representation. Hypocrisy apparently is part of his old-fashioned style.

 

Malta enlarged some 30 x UK, dotted with 15 x UK. Spot the Real Malta

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continued from the former entry of April 4

The turnout of 32% at the April 6 2016 Dutch Referendum on the EU-Ukraine Association Agreement does not seem convincing. The 61% No vote seems impressive over the 38% Yes vote and 1% blanks, but the low turnout gives it little weight. (PM. The electorate are 13 million voters. Turnout was 4.2 million. For the treaty were 1.6 million and against the treaty were 2.5 million.)

This advisory referendum is labelled “valid” because it passed the ceiling of 30%, but the margin is so low that Mark Rutte and his government might feel excused to still ratify the treaty anyhow.

It would take only 12% to switch the No majority into a Yes majority.

Potentially, the No voters were most motivated to turn up, and the Yes voters potentially stayed at home because: (a) the referendum was advisory only, and they have reason to believe that the government would ratify the treaty anyway, (b) the referendum would be invalid when the ceiling of 30% turnout would not be reached, (c) they might believe that they were the real majority so that the outcome would be Yes anyway. Perhaps a 12% higher turnout would only have come from Yes voters.

Thus the EU might blame Mark Rutte and his government for not having tried hard enough to increase the turnout that might have caused that switch in outcome.

However, the earlier poll by Maurice the Hond suggests that a full turnout would show 53% No. We can find this figure by weighing the views of the polled votes per party with the current polled seats per party. Indeed, De Hond also concludes that the No vote would never fall below 50%. However, it is hazy what a full turnout would be. There hardly ever is 100%. Let us take the 75% turnout of the last general elections in 2012.

Assuming proportional change between these two data points {.32, .61} and {.75, .53} gives us the following sloping line that drops to 50% at 90% turnout.

Referendum result at 32% and potential at 75%

Referendum result at 32% and potential at 75%

Conclusions are:

  • With the {.75, .53} poll data point included, the referendum result is more impressive than it at first seems. Apparently the Yes vote indeed had a losing uphill battle.
  • Mark Rutte and his goverment should have tried harder to inform the electorate about the pro’s and con’s of the treaty, if they really thought that more information would have resulted in both a higher turnout and a switch to more Yes votes.
  • Mark Rutte and his goverment might not have put in this effort, since the referendum was only advisory and might not have reached the 30% ceiling.

On the sight of it, Mark Rutte and his government have blundered.

It may also be plain old Dutch deviousness: with a ploy to play into Putin’s paranoia, and a scheme to prevent a casus belli. Remember the movie Being there“.

Yesterday I had my regular lunch with Vladimir Putin, President of the Russian Federation.

We mostly talked on how the oil price inflates corruption, and we focused on which sport he wants to enter in the next Olympic games: judo, swimming, horseback riding, shooting, or some other sport that he hasn’t shown us yet that he excels in. We curtly discussed the Dutch referendum of next Wednesday April 6 2016 on the EU – Ukraine Association Agreement.

TC: “Vlad, you have been remarkably silent on this Dutch referendum. I had expected to see the Russian Ambassador on Dutch TV campaigning for a No, but I haven’t seen him.”

VP: “Well, I want to invade the Eastern Ukraine this Spring, and this treaty is a welcome cause of war. I would be dismayed when the Dutch people would say No to the treaty and thus block it. Besides, I have asked the Ambassador to undergo a sex change operation to be more agreeable to the Dutch gays.”

TC: “When you ask the Dutch to say No, they will likely become obstinate and vote Yes. Then you have your casus belli and can also claim that you have done your best to prevent it. Or am I trying to out-smart you ?”

VP: “That would also be seen as meddling in internal national affairs. Russia doesn’t meddle. We spy or invade. I trust that the Dutch government will ratify the treaty, even when the population says No. I have promised your prime minister Mark Rutte a good job at Gazprom later on, and he seemed very happy to sit up and eat the cookie.”

TC: “Thus by July 2016 Russia will have occupied the region East of the Dnieper, and Kiev will be divided by a Wall. The EU can have its treaty with the West, and can start trying to feel happy with the mess they created.”

VP: “There is something very alluring to a permanent stalemate, like the division between North and South Korea. A permanent state of war will be very invigorating to Russia’s youth.”

TC: “Ah yes, I understand what you are thinking. Marches and parades. Battle songs. Soccer stadiums filled with ballet dancers in clockwork performances. Adrenalin is better dan meldonium. The European Union will not understand what hit them. You are brilliant and a real father for Russia.”

Vlad turned to me with a surprisingly modest expression: “Don’t give me credit. I got the idea from Garry Kasparov. Let me give you his Memo. I don’t mind when you publish it and let your Dutch friends read it too. Kasparov always loses from me in chess or politics, and he is a lousy house cleaner and waiter, but his Memo deserves some credit.”

I read the Memo in the plane, and was happy that there was plenty of Stolichnaya for the remaining three hours of the flight back home.

Memo by Garry Kasparov for Vladimir Putin, April 1 2016

Normally I try for a straight argument and end up into convolutions. Now I won’t even try.

The EU – Ukraine Association Agreement (henceforth treaty) text is here and on wikipedia (or in Dutch here and here and wiki). The wikipedia article shows that the treaty is ratified in all countries except that Holland has a referendum on April 6 2016 that might still block it.

Wikipedia actually has an article on this Dutch referendum. The Dutch referendum allows citizens only to advise the government. The government might still choose its own way – which likely is a Yes. Hence we are mostly discussing chimeras.

My main concern is the risk of war, whence I am against the treaty as it stands. Article 7 on Foreign and security policy involves the EU in a “timely and coherent manner” in the defence of Ukrainean sovereignty. The Ukraine might claim that the Crimea has been stolen by Russia, must be returned, and that the EU is legally obliged to help to get it back in “timely and coherent manner” . This is Article 7 with my emphasis:

1.   The Parties shall intensify their dialogue and cooperation and promote gradual convergence in the area of foreign and security policy, including the Common Security and Defence Policy (CSDP), and shall address in particular issues of conflict prevention and crisis management, regional stability, disarmament, non-proliferation, arms control and arms export control as well as enhanced mutually-beneficial dialogue in the field of space. Cooperation will be based on common values and mutual interests, and shall aim at increasing policy convergence and effectiveness, and promoting joint policy planning. To this end, the Parties shall make use of bilateral, international and regional fora.

2.   Ukraine, the EU and the Member States reaffirm their commitment to the principles of respect for independence, sovereignty, territorial integrity and inviolability of borders, as established in the UN Charter and the Helsinki Final Act of 1975 of the Conference on Security and Cooperation in Europe, and to promoting these principles in bilateral and multilateral relations.

3.   The Parties shall address in a timely and coherent manner the challenges to these principles at all appropriate levels of the political dialogue provided for in this Agreement, including at ministerial level.

Professor Richard Sakwa (page at Kent) advises a No. His book is here and reviews are here and here. He agrees that the Ukraine already is a sovereign nation and has the right to engage into the current treaty with the EU. The EU has the right to engage into such a treaty too. However, it would not be wise to do so. The EU is sleepwalking again, like the sleepwalkers of 1914-1918.

Sakwa was on Dutch TV on March 20 2016 explaining – in English with Dutch subtitles (minute 41) – that the EU should look at the whole region, including the position of Russia. He was challenged by the Dutch vice-prime-minister Lodewijk Asscher with the argument that including Russia in the picture would violate the notion that the Ukraine is a sovereign nation. Asscher is a lawyer and wields a legal argument in a discussion on geopolitics and war. When Hitler invaded Holland, Asscher would say that this was illegal. Asscher has been doing all kinds of silly things but he has a quiet and soft presentation that Dutch people seem to take for wisdom.

Thus the reasoning is:

  • There should be a peace agreement with both Russia and Ukraine.
  • There should be a trade agreement with both Russia and Ukraine.
  • This current Association Agreement does not satisfy these conditions. In fact, it reduces the freedom of the Ukraine to enter into trade agreements with Russia. The Ukraine becomes the outer border of the European trade area. Trade between the Ukraine and Russia will be hit, and this will cause unemployment in the Ukraine, similarly to the unemployment that occurred in Eastern Germany after the fall of the Berlin Wall (see here).
  • The EU should stop the policy that association with the EU means association with NATO.
  • The EU should use “Europe” as a term in geography and not as synonymous with EU.
  • Hence No to the current treaty.
Richard Sakwa on Dutch TV (March 20 2016)

Richard Sakwa on Dutch TV (March 20 2016, minute 41)

PM 1. Arno Wellens (in Dutch) finds it curious that Asscher was not at the discussion table, and was asked for his opinion while sitting in the audience. Wellens suggests that Sakwa anticipated a 20 minute focus on his analysis, but was surprised to find himself in a discussion with the vice-prime-minister. Perhaps this is the case. Still, it is useful to see the clash between geopolitics (Sakwa) and legalism (Asscher).

PM 2. Another English source is Mark Almond at Oxford, but perhaps a bit less outspoken as Sakwa.

PM 3. For Dutch viewers there is a nice discussion of March 27 between Jeroen Dijsselbloem (Yes) and Arjo Klamer (No). Klamer’s main concern is that the Ukraine has too much corruption, so that the EU’s neoliberal policies will benefit the oligarchs and be disastrous for the common people. This is partly correct. It is strange that Klamer doesn’t give a better economic analysis, but see here (in Dutch).

PM 4. There is some information on a EUR 11 bn Ukraine state building project financed by the EU. This is about EUR 250 per Ukrainean, or one-time 12.5% of the annual GDP per capita of EUR 2000. We may guess where this money is going to land. Perhaps it is an acceptable bribe to get better law and police in the long term. That said, there is still the issue of war. It would be foolish to pay a bribe to get involved in fighting.

There is Good news and there is Bad news. The Good news is that while Bashar al-Assad flew in secrecy to Moscow to meet with Putin – of which photo’s were released when he was safe at home again – that also his enemy IS leader and self-acclaimed “Caliph Ibrahim” flew in secrecy to Moscow to see Putin – though no photo’s were released. When asked, Putin will invoke plausible deniability.

The President of the Russian Federation will also show his intense irritation about questions on this – a dead give-away. I happened to be present at the occasion because of my monthly chat with him, so let me help him by reporting on the main details.

The Bad news is that self-acclaimed “Caliph Ibrahim” brought along an ancient game of chess as a present for Putin, and then destroyed it before his eyes, throwing it to shambles and beheading the pieces.

Vlad: “That was a gift ! You were going to give it to me ! You completely ruined it !”

sa”CI”: “Let this be a lesson for you !”

Vlad: “That was an ancient game of chess ! A priceless artifact !”

sa”CI”: “Yes, indeed. We found it when we ransacked the Mosul Museum. It belonged to Egyptian Pharao Ramses I when he fought the war at Armageddon around in 3000 BC. It was his gift for his ally from Babylon, Nebukadnezar the Terrible.”

Vlad: “And you barbarian just smashed this ?! I must ask Garry Kasparov to glue those heads on again. He can be smart, you know ? He won’t glue a pawn’s head onto a horse. Though these Egyptian pieces look funny.”

sa”CI”: “That isn’t a horse but a crocodile. And what you call a pawn is a scorpion.”

Queen Nefertari at the board (Source: wikimedia commons)

Queen Nefertari at the board (Source: wikimedia commons)

Vlad: “Well, in that case I am happy that you cut those heads off. Was that your lesson for me ?”

sa”CI”: “Not really. See those pieces that look like penises ? We cannot tolerate pornography.”

Vlad: “And now I am unhappy again. You Cut Those Off.”

The strong man of Russia shuddered.

sa”CI”: “I said that Ramses gave it as a present to his ally Nebukadnezar the Terrible. You must know the full story. They fought together against Alexander the Great and his Hittites, and when they had won the war, then Ramses put Nebukadnezar in a cellar, and gave him this game of chess, so that he and Alexander the Great could pass the time by trying to take each other’s penises.”

Vlad – flushing: “Ah. Oh. So. Ah. Yes. You have got me completely freaked now, I must admit. Am I really getting grateful that you destroyed that … priceless artifact ?”

sa”CI”: “There is no need to thank me. I enjoyed doing it. I only brought it along to show something about the Middle East. We have been at each other’s throats since the beginning of civilisation. Actually, civilisation began because we have been at each other’s throats. Don’t think that you can win. Don’t think that you can invent any new power ploy that we haven’t practiced to perfection for millennia already. Just give us weapons. Then I will give you my advice.”

Courtesy of Alma-Tadema.org

Courtesy of Alma-Tadema.org

Post scriptum

Below is the ancient chess game between Nebukadnezar the Terrible and Alexander the Great from around 3000 BC, restored by Garry Kasparov. The crocodiles were too much damaged and have been replaced by Staunton horses. On insistence by Putin the penises have been replaced by the sun symbol, now the king. Because of his mortal fear of scorpions, Kasparov replaced those by penises again, but Putin selected the Staunton pawns again. The sitting figure is the queen, scarabs are rooks, and three feathers are the bishop. It has been lost in history who took what side and whose turn it was, and why they didn’t finish the game. President Putin’s main goal now is to get peace in the Middle East so that he can start digging to find out.

Ancient game of chess, restored

Ancient game of chess, restored

The discussion of Putin’s proof gave me an email from Alexis Tsipras, who just resigned as prime minister of Greece and is busy with the general elections of September 20 soon. Rather than reporting on it, I might as well fully quote it.

To: Thomas
From: Alexis@formerprimeministerofgreece.org
Subject: My proof of Fermat’s Last Theorem
Date: Fri, 28 Aug 2015 11:58:03 +0100
Google Unique Message Identifier: 23DFGA@671

Dear Thomas,

Thank you very much for your discussion of President Putin’s proof when he was a youngster of Fermat’s Last Theorem. I know his mother Vera Putina very well. The Putin family has a vacation home here in Greece, and she can stay there on the condition that she immediately leaves when Putin himself comes down. She has shown me his proof too. I can only agree with your conclusion that it shows how smart President Putin was when he was young.

Putin’s proof inspired me to find a proof too. I am sometimes exhausted by the tough negotiations with the European Heads of State and Government, if not with members of my own party. Thus I resort often to a sanatorium for recuperation. Thinking about such issues like Fermat’s Last Theorem helps to clear my mind from mundane thoughts. I was very happy last Spring to indeed find a much shorter and more elegant proof. 

For the theorem and notation I refer to your weblog. My proof goes as follows.

Theorem. No positive integers n, a, b and can satisfy an + bn = cn for n > 2.

Proof. (Alexis Tsipras, April 31 2015)

Let us assume that an + bn = cn holds, and derive a contradiction.

There are two possibilities: (1) n is even, or (2) n is uneven.

(1) If n is even, then we can write A = an/2 and B = bn/2 and C = cn/2 such that A, B and C are still integers. Then we get the following equation:

A2 + B2 = C2

This equation satisfies the condition that n = 2, and thus it doesn’t satisfy the condition n > 2.

(2) If n is uneven, then we can write A = a(n-1)/2 and B = b(n-1)/2 and C = c(n-1)/2 such that A, B and C are still integers. Then we get the following equation:

a A2 + b B2 = c C2

This equation does not satisfy the form of an + bn = cn so that it falls outside of Fermat’s Last Theorem.

In both cases the conditions of the theorem are no longer satisfied. We thus reject the hypothesis that an + bn = cn holds. Q.E.D.

This is much shorter that President Putin’s proof. And, I prove it while he only came close. I have been hesitating to tell him, fearing that he might become jealous, and be no longer willing to support Greece as he does in these difficult times for my country. Now that you have confirmed how wonderful his proof at only age 12 was, I feel more assured. Will you please publish this proof of mine too, like you did with President Putin’s proof ? I have put my best efforts in this proof, just like at the negotiations with the European Heads of State and Government. Thus I hope that it will be equally convincing, if not more.

After the next elections I will probably be exhausted again. I would like to work on another problem then. Do you have any suggestions ?

Sincerely yours,

Alexis

Fermat and Tsipras (source: wikimedia commons)

Fermat and Tsipras (source: wikimedia commons)

The door rang. I was surprised to see Vera Putina. It appeared that Putin’s mother was visiting her granddaughter’s fiancé’s family in Holland. “It is not safe for me to go to Moscow,” she explained, expressing the sentiment of many.

When she was settled in the safety in my living room with a good cup of Darjeeling, it also appeared that there was more.

VP: “I am upset. The Western media depict my son only as a sportsman. They show him doing judo, riding horses, fighting bears, and the last week they featured him as a diver in a submarine. Of course he is very athletic, but he is also a smart man. I want you to look at his intellectual side too.”

Me: “It is fine of you to ask, because there indeed is a general lack of awareness about that.”

VP: “Let me tell you ! When my father had his love affair with Grand Duchess Kira Kirillovna of Russia [see the Putin family tree here], one of their secret meeting places was in the royal archives of the Romanovs. Sometimes my father, because he was unemployed, took some of the old documents to sell on the market. You know, my mother Kira had an expensive taste.”

VP said this without blinking an eye. The unperturbedness when taking other people’s possessions and territories must have a family origin.

VP: “My father once found the application letter by Pierre de Fermat for membership of the St. Petersburg mathematical society. It detailed his proof of his Last Theorem.”

Me: “Ah, that might explain why he never published it ! He used it for his application, and this got lost in the archives ?!”

VP: “I don’t know about that. My father sent it in for the Wolfskehl Prize of 100,000 gold marks, but it was rejected for it didn’t satisfy the criterion of having been published in a peer-reviewed journal.”

Me: “This is historically very interesting. If you still have that letter by Fermat, no doubt a historical journal will gladly publish it. It doesn’t matter on content since Andrew Wiles now proved it.”

VP: “No, no, no !” She gestured with passion as an Russian woman can do. “Fermat is not important ! It is what Vladimir did ! When he was twelve, he also looked at Fermat’s letter, and he found an omission ! Moreover, he worked on the problem himself, and almost solved it. Here, I brought along the papers to prove it to you.”

She delved into the bag that she had brought along and produced a stack of papers. I also saw a wire bound notebook such as children use in school.

Me: “Almost solving means not solving. Mathematics is rather strict on this, gospodina Putina. But it is historically interesting that you have Fermat’s original proof and that your son worked on it.”

VP: “For this, he had to learn Latin too !’

She gave me the stack. There was a great deal of difference between her nonchalant and triumphant handing over of the papers and my hesitant and rather reverent accepting of them.

VP: “You look it over, and inform the Western media that my son almost solved Fermat’s Last Theorem when he was only twelve ! If I hadn’t told him that he had to go to his judo lessons, he would have finished it for sure !”

She said the latter as proof that she had been a good mother, but also with a touch of regret.

Confronted with such motherly compassion I could only respond that I would oblige. Hence, below is Vladimir Putin’s proof. First I translate Fermat’s own proof from Latin (also using the Russian transcript that Putin made) and then give Putin’s correction.

Fermat (1601-1665) and Putin (1952+)

Fermat (1601 – 1665) and Putin (1952 – ∞) (Source: wikimedia)

Fermat’s Last Theorem, using middle school algebra

Theorem. No positive integers n, a, b and can satisfy the equation an + bn = cn for n > 2.

Proof. (Pierre de Fermat, April 31 1640, letter to czar Michael I of Russia)

Without loss of generality b. Take k = n – 2 > 0. We consider two cases:

(1) a2 + b2c2

(2) a2 + b2 > c2.

(1) When a2 + b2c2 then a2 + b2 + d = c2 for d 0

Then a < c and b < c. Then also ak < ck and bk < ck for k > 0.

If the theorem doesn’t hold, then there is a k > 0 such that:

ak+2 + bk+2ck+2

ak a2 + bk b2 = ck c2 = ck (a2 + b2 + d)

a2 (akck) + b2 (bkck) = d ck ≥ 0

negative + negative ≥ 0

Impossible. Thus the theorem holds for (1).

(2) If a2 + b2 > c2 then obviously (see the diagram) for higher powers too: an + bn > cn.

Fermat's drawing for his proof (right rewrites left)

Fermat’s drawing for his proof (RHS re-orders LHS)

Since (1) and (2) cover all possibilities, the theorem holds.

Q.E.D.

Putin’s correction, age 12

The comment by schoolboy Vlad on this proof is:

“While (2) is obvious, you cannot rely on diagrams, and you need to fully develop it. At least I must do so, since I find the diagram not so informative. I also have problems reading maps, and seeing where the borders of countries are.”

Hence, young Putin proceeds by developing the missing lemma for (2).

Lemma. For positive integers n, a, b and c: if a2 + b2 > c2 then an + bncn for n > 2.

Proof. (Vladimir Putin, October 7 1964)

Without loss of generality a b. Take k = n – 2 > 0.

If a2 + b2 > c2 then a + b > c. (Assume the contrary: a + b c then a2 + b2 < (a + b)2c2, which contradicts a2 + b2 > c2.)

Expression an + bn > cn is equivalent to (an + bn)1/n > c. The LHS can be written as:

f[n] = a (1 + (b / a)n)1/n  with a b.

This Lemma has the Pythagorean value f[2] = √(a2 + b2) > c. The function has limit f[n → ∞] = a. (See a deduction here.) Thus f[n] is downward sloping from f[2] >  to limit value a. We have two cases, drawn in the diagram below.

Case (A) Diagram LHS: c ≤ a, so that there will never be an intersection f[n] = c.

Case (B) Diagram RHS: a < c < f[2] = √(a2 + b2). There can be an intersection f[n] = c, but possibly not at an integer value of n. Observe that this case also provides a counterexample to Fermat’s claim that “obviously” f[n] > c, for, after the intersection f[n] < c. Young Putin already corrects the great French mathematician ! This is a magnificent result of the future President of the Russian Federation, at such a young age. His grandfather’s Marinus van der Lubbe’s submission to the Wolfskehl Prize would also have failed on this account.

a (1 + (b/a)^n)^(1/n) and parameter cases

f[n] = a (1 + (b / a)^n)^(1/n) and parameter cases

At this point, young Putin declares that Case (A) on the LHS is proven, based upon above considerations. He adds:

“I accept this proof on the LHS, even though I have difficulty understanding that limits or borders should not be transgressed.”

As so often happens with people who are not entirely sure of their case, the schoolboy then develops the following simple case, just to make certain.

Case (≤ b). Use numerical succession from a2 + b2 > c2.

Given an + bn > cn then prove an+1 + bn+1 > cn+1.

a an + b bnb an + b bn = b (an + bn) > b cnc cn

Thus the Lemma holds for this case.

To be really, really, sure, Putin adds an alternative proof that assumes the contrary:

ak a2 + bk b2ck c2

ak a2bk a2 + bk a2 + bk b2bk c2ck c2bk c2

a2  (akbk) + bk (a2 + b2c2) ≤ c2 (ck bk)

nonnegative + positive  ≤  nonpositive

Impossible. Thus the Lemma holds for (≤ b).

It would have been better when he had looked at Case (B) on the RHS, notably by proving that f[n] = c cannot hold for only integers.

At this point in his notebook, young Putin writes:

“I have to go to judo training. Perhaps I will continue tomorrow.”

I have looked in the remainder of the notebook but did not find further deductions on Fermat. Apparently the next day young Putin continued with what was more on his mind. It appears that he had a fantasy land called Dominatia in which he played absolute master, and it took much of his time to determine what was happening there. Something of the unruly nature of the natural numbers however must have stuck in his mind. In a perfect fantasy land everything is already as wished, but in young Putin’s Dominatia land he fantasizes about unruly citizens who must be put under control.

Conclusions

The above supports the following conclusions:

  • The theorem & lemma are not yet proven for Case (B) on the RHS. We must still rely on Andrew Wiles.
  • Nevertheless, Vladimir Putin doesn’t do just sports but also has amazing intellectual powers, at least when he was at age twelve.
  • Fermat’s original own proof of his theorem seems to have had a serious error, but it is not precluded that it was only chance that it did not get published (with or without corrections).
  • Fermat’s Last Theorem has dubious value for education. It seems more important to develop the notion of limits, and in particular the notion that you should not transgress borders. When students do not understand this properly at a younger age then this may cause problems later on.
Appendix 1. Case (c > a ≥ b)

It may be nice to see how f[n] = c is sandwiched, when a + b > f[2] > c > a ≥ b.

Case (c > a ≥ b)  There is a point f[n] = c or an + bn = cn for reals but perhaps not for integers.

(i) At the intersection:

ak a2 + bk b2 = ck c2

Take ak c2 + bk c2 and substract the above on both sides:

(c2 – a2) ak + (c2 – b2) bk = (ak + bkck) c2

positive + positive = ?

The latter must be positive too, and hence: ak + bk > ck

Thus, assuming that the theorem doesn’t hold for n requires that it holds for k = n – 2.

(ii) After the intersection: Since f[n] is downward sloping we have f[n+1] < f[n] = c. Reworking gives:

an+1 + bn+1 < cn+1

Another way to show this is:

(a – can + (b cbn < 0

a  an+ b bn  < c (an + bn)

an+1 + bn+1cn+1

Comparing (i) and (ii) we see the switch from > to <.

Appendix 2. Parameter restrictions in general

Assume that an + bn = cn holds. There are restrictions for this to occur, notably by the remarkable product:

(an bn) (an + bn) = (an bn) cn

a2n b2n = (an bn) cn

an (an cn) = bn (bncn)             (*)

For example: when c = a, then an + bn = cn is only possible in (*) if c = a = b, but this is actually also impossible because it requires that cn + cn = cn. The table collects the findings, with the LHS and RHS now referring to equation (*).

an + bn = cn c <(LHS +) c =(LHS 0)
c > a  (LHS -)
c < (RHS +) (=),
but Case (c b)
impossible opposite signs
c b  (RHS 0) impossible impossible
cn + cn = cn
impossible
c > b  (RHS -) opposite signs impossible (=) the only risk

This table actually also proves Case (A) that Putin took for granted. Only Case (B) remains, and requires proof that f[n] = c cannot hold for only integers.

A Russian submarine took me from the beach in Scheveningen to the destroyer MS Ghost of Kursk, that brought me to the aircraft carrier Admiral Kuznetsov on the Atlantic. “I could take an plane from Schiphol to Moscow,” I suggested, but they put me on an intercontinental missile, and one half hour later the parachute landed me in Sotchi, where I was detailed to Vladimir Putin’s mansion.

Vladislav Surkov was yelling at him.

“You, bastard son of the Romanovs, you are not worthy of Mother Russia,” Surkov screamed at the top his voice.

The professional science fiction author and amateur demagogue was beyond control and was throwing all that he could find at the President of the Russian Federation.

(See the earlier weblog on Putin’s ancestral line.)

“I am a father. I have a son !” Vlad replied. “Your insults are worthless if you don’t see this elementary fact.”

“History dictates that we follow Barbara Tuchman on the Guns of August,” Surkov screamed. “The world is a stage. Try to remember your Shakespeare ! Only those stories work that refer to earlier stories that worked. We must repeat history if we want to write history. In 1914 the first World War started by a string of incompetent decisions. We can only succeed in our plans for world domination when we repeat those incompetent decisions. When we start our war on the Ukraine now in August then people in Europe will be terrified. When we wait another month then they will laugh. We must strike now !” 

Surkov yelled and pointed to the diagrams that he had hung on the walls.

“Yes, of course,” Putin replied, calmer than ever. “But I am a father now. I must take care of my son.”

Surkov launched himself at the President of the Russian Federation. Fortunately for world history, president Putin has a black belt, and the attack by the science fiction author and stage director was deflected efficiently.

Putin in a judo exercise

Putin in a judo exercise (source: screenshot youtube)

“So, what am I doing here ?” I managed to ask.

“My son needs a good education,”  Vlad said. “I am willing to postpone my war on the Ukraine till he has mastered arithmetic. So perhaps you have some suggestions.”

Vlad apparently had read this earlier weblog.