Make no mistake: the US administration is very badly rattled by the revelations that they spy, at a massive scale, on all its citizens and many non-US citizens. One common response I see repeatedly in the blog/news sphere is the comment "We already knew this was happening; Snowden has done little to advance our knowledge". No. We might reasonably have suspected it. We might even have had evidence of some aspects of it. But it is a far cry from that to the now absolute certainty that it is happening and on such a vast scale. What Snowden did was heroic, informative, and very dangerous for him.
Another common response from the administration apologists is to declare that innocent people have nothing to hide. Many people have said, but it is worth repeating, that this is unbelievably naive. We have all got something to hide. We all need privacy. I do not want my government to use my private secrets against me if ever I do need to defend myself against an accusation. I do not want to use my toilet in a glass-walled room. I do not want the opinions I had thirty years ago to be used against me. Isn't that absolutely obvious? And isn't it absolutely obvious that I have a right to that protection?
I would love the consequence of the NSA's activities coming to light to be a rueful admission by the US administration that they have gone way too far in ignoring the rights of their people. Obviously that would in turn also be unbelievably naive. So what do I expect to actually happen?
Well, I don't think they will try to seize Snowden and "render" him back to the US. I believe they will be much more subtle and wage a campaign in the media to gradually discredit him. Already we see that officialdom's main response is to ignore the actual content of the leaks and concentrate on his breaking of US law. That is going to continue to be the issue that they hammer home. Unfortunately they have a huge advantage in a war of opinion. They have thousands of opinion-moulders and free access to every major news outlet. Snowden and his supporters cannot match that propaganda machine.
To begin with, many US journalists who might well have known or suspected what was going on and turned an expedient (or cowardly) blind eye have clearly felt some irritation at missing out on the scoop of the decade. Where in the US Press do we see the staunch defenders of their constitution? Where is the outrage at what has been done to their society? Instead we see nuanced discussions about how or whether Snowden has broken the law. Where is the recognition that it is clearly impossible for Snowden to get a fair hearing from the US authorities?
Furthermore the US administration can channel opinion pieces in droves to the media. Overwhelmingly we shall therefore see articles attacking Snowden personally and down-playing the surveillance issues. Oh - you didn't think that some US officials might not tow the administration line, did you? Why should they when they haven't done hitherto?
Only time will tell whether the administration can successfully distract the public attention from the real issue. If that sounds pessimistic let me close with one optimistic comment. I have read many article in the US Press about Snowden and, although most of them ignore or downplay the enormity of the NSA's trampling over civil rights, the comments of readers are much more supportive. What I hope will happen is that this evidently large supportive constituency of folk, who understand that the true issue is civil liberties not breach of employment contract, will eventually have an effect on those powerful vested interests who wish to sweep the real issue under the carpet.
Friday, 28 June 2013
Tuesday, 25 June 2013
A limerick for the NSA
Their actions are dirty and base But now they've been put in their place They cannot have known When the whistle was blown How much egg they would find on their face
Friday, 14 June 2013
American lies
The US is claiming that the Syrian government has used poison gas against their rebel opponents, and they are using this as a reason to engage in the conflict on the rebel side. Should we believe or discount their claim? I have no idea but the fact that the claim was made adds nothing to whether I think it might be true. The problem is that, in providing excuses for war, the US has absolutely no credibility. I think about the Iraq war and the lies they told then about weapons of mass destruction.
The FBI is claiming that Edward Snowden's revelations have jeopardized American lives. Should we believe or discount this claim? Again, I have no idea. Again the claim adds nothing to whether I think it might be true. This time I think about the Bradley Manning leaks about the conduct of the wars in Iraq and Afghanistan where the claims that they jeopardized American lives have not been confirmed (and you would certainly expect the State Department to be trumpeting them abroad if there was confirmation).
The problem about lies is that they destroy credibility. In the case of individual liars we don't have to be all that worldly-wise to withhold our judgement or immediately discount their statements. Which of us, when Bill Clinton proclaimed "I did not have sex with that woman", immediately accepted the statement? Or which of us, when former US Senator Todd Akin made the claim that victims of "legitimate rape" cannot become pregnant immediately accepted the statement? We are free to accept or reject the personal statements of individual politicians usually without any significant harm done. If Clinton or Akin tried to sell me a second-hand car I wouldn't buy it, nor do I given any credence to any other personal protestations they might make.
But when a state lies, especially one as powerful as the US, we all have a problem. Of course, states have always lied but nowadays the stakes are higher.
To begin with, if we live in a democracy, we are part of a social contract. We pay our taxes and we observe the laws of our country. In return we are entitled to be honestly governed by the people we have chosen in a plebiscite where we have weighed the pros and cons of competing candidates. If governments violate their side of this bargain they endanger this social contract. If they betray our trust they lose our cooperation and our goodwill. This is a downside for a government who, presumably, wish to be re-elected. The problem is that damage is being done to the political fabric in the longer term. When politicians see that the only downside is to be kicked out in favour of the next lot they may, when their personal economic advantage is at stake, simply accept this downside. Worse still is that the lying precedent that has been set can all the more easily followed by the next lot. In this way, the social contract continues to be degraded.
In my opinion much has gone wrong with American governance over the last generation and especially since the 9/11 attacks. The implosion of the former USSR removed a check on the behaviour of the US - and their greater power led to greater corruption. Then, in the wake of 9/11, when (understandably) many Americans looked for very strong leadership, that leadership abused its power in the most shocking ways. Internationally, they went to war for the flimsiest of reasons. Domestically, they instigated a regime of surveillance and harassment against their internal enemies in which many innocent citizens became victims. All of this is possible only because liars are not held to account. They can lie with impunity and now a climate of scepticism shrouds all governmental announcements. Unfortunately this is bound to lead to a rise in those who believe in conspiracy theories. How ironic that the lies over the Benghazi attack might encourage people to believe that the US government brought down the Twin Towers.
So what should we as individuals do? Obviously, it is a rare person who can make a great difference. But we shouldn't just leave it to others and we mustn't be cowed. At every opportunity we must denounce the lies - and not care that this will lead to much repetition. Remember: governments should fear the people, not people fear the government.
Don't be afraid to say: Clinton was a liar over Monica Lewinsky, Bush was a liar over Iraq's weapons of mass destruction, Obama lied when he said that Mitt Romney planned to raise taxes by $2000 on middle-income tax-payers.
I am not saying disbelieve anything the government says. No, let's not encourage conspiracy-mongering. But I am saying don't believe anything until you have some corroborative evidence. Politicians need to win back their credibility - it's in all our interests.
Wednesday, 22 May 2013
A surprise in the world of prime numbers
The world of mathematics is buzzing with the news of a breakthrough in the theory of prime numbers by a virtually unknown mathematician named Yitang Zhang. Prime numbers are numbers like 3, 5, 23, 97 which cannot be expressed as a product of two smaller numbers. Prime numbers have been studied since the time of the Ancient Greeks and about 23 centuries ago the famous geometer Euclid discovered that there are an infinite number of them. Since that time the study of prime numbers has produced some of the deepest results in the whole of mathematics yet these results are often very easy to understand (but the reasons the results are true are another matter).
It is easy to see that 2 and 3 are the only consecutive prime numbers (because if you have two consecutive numbers one of them must be even but 2 is the only even prime number). What about prime numbers differing by 2? Here there are more: (3, 5), (5, 7), (11, 13) for example and there are many other so-called twin primes. But are there are an infinite supply of twin primes? Despite hundreds of years of research we don't know the answer to this innocuous question.
What about pairs of prime numbers that differ by 4 (like (3,7) or (19, 23))? We don't know if there are infinitely many such pairs either. And if you replace 4 by any other even number at all we still don't know. For example, we don't know whether there are infinitely many prime pairs that differ by 30 say (like 31 and 61).
Enter Yitang Zhang. In his 50's he is definitely beyond the age where most mathematicians make their mark and until now he has been practically unknown. After he got his PhD in 1992 he had found it difficult to get an academic job, working for several years as an accountant and even in a Subway sandwich shop. But he never gave up doing mathematics and eventually was appointed at the University of New Hampshire. There he pursued an unremarkable research career with no publication since 2001 but he was loved by his students apparently because he set easy exams. Now he has burst onto the world's mathematical stage with a result that has surprised all the experts. He has proved that there is some number k that "works" for prime pairs. We don't know that k=2 or k=4; all we know is that k is less than 70 million. And for this k, whatever it is, Zhang has proved that there are infinitely many pairs of prime numbers that differ by k.
The reason that this has excited mathematicians is that no result like this has ever been proved before and it gives some hope that the original prime twin problem might eventually be solved. It is also a surprise result in that it hardly ever happens that a giant step like this is taken by such an obscure mathematician. When it has happened before (such as for the Indian genius Ramanajuan) the newcomer is usually much younger since mathematics at its most creative is usually a young person's metier.
Zhang's theorem probably won't have much practical use. If you like problems about ages and birthdays here's a consequence that might be appealing. Somewhere, sometime, there were two mammals of different ages and there will be an infinite number of years when both their ages are prime numbers. Might there be two human beings with this property? We don't know because humans haven't been around for 70 million years whereas mammals have.
Sunday, 19 May 2013
Freedom of information: a window?
You may not agree that the glory days of newspaper journalism are long gone. Perhaps you think that the Washington Post's Watergate investigation which led to the eventual resignation of US President Nixon was less than stellar reporting. Maybe you think that the meek way almost the entire mainstream media accepted the US and UK lies about Iraqi weapons of mass destruction as an excuse to go to war was an isolated collective error of judgement. If so, you and I have quite different opinions on the way that newspaper reporting has changed over the years.
But perhaps we agree on something else: that the Wikileaks revelations told us things about how the US conducted its wars in Afghanistan and Iraq, and how its emissaries around the world conducted business with foreign powers gave us an unprecedented grandstand view of events that most of us never hear about. Perhaps you think, along with Senator Joe Lieberman, that the leaks were "outrageous, reckless and despicable" but I hope you would fall short of Sarah Palin's call to pursue Julian Assange with the same urgency that Al-Qaida leaders were pursued, or Congressman Mike Rogers' threat to have foreign national Assange executed for treason against the US. Nevertheless the scale of what we learnt via Bradley Manning and Julian Assange cannot be denied.
The point of this post is not to persuade you to my view (that Manning and Assange are among the great heros of our time). It is more to alert you to the fact that, if you care about knowing what is going on in the world, you have just lived through a short period where we the people had information about how great powers operate of a magnitude that we might not see again for a very long time.
The intense efforts that the US and UK governments went to in order to suppress events in the Afghan and Iraqi wars, and the publication of US diplomatic cables tell us how much they were embarrassed by the Wikileaks collaboration with the Guardian and other newspapers. I have no doubt that they have ramped up their security to prevent a repeat. In any case the US retaliation against Manning has been so severe that other potential leakers of his persuasion might well think again (certainly Julian Assange himself is in no hurry to be extradited to Sweden in case he is handed over to US authorities). Furthermore the US authorities are now pursuing the Associated Press organisation by subpoena-ing phone records that might bear on the CIA successfully thwarting a plot by al-Qaeda in Yemen to blow up a U.S. jetliner (this is not the first such aggressive subpoena act).
So don't expect anything as informative as the Wikileak bonanza to strike again for a very long time.
Therefore my take home message is this. Have a good luck at what we already have. You can go to the Wikileaks web site itself. However you might find this is more than you can handle! It is vast. Instead you might try the Guardian site which is very well-organised and will tell you also a lot about the politics associated with the reaction of the US and the UK.
But perhaps we agree on something else: that the Wikileaks revelations told us things about how the US conducted its wars in Afghanistan and Iraq, and how its emissaries around the world conducted business with foreign powers gave us an unprecedented grandstand view of events that most of us never hear about. Perhaps you think, along with Senator Joe Lieberman, that the leaks were "outrageous, reckless and despicable" but I hope you would fall short of Sarah Palin's call to pursue Julian Assange with the same urgency that Al-Qaida leaders were pursued, or Congressman Mike Rogers' threat to have foreign national Assange executed for treason against the US. Nevertheless the scale of what we learnt via Bradley Manning and Julian Assange cannot be denied.
The point of this post is not to persuade you to my view (that Manning and Assange are among the great heros of our time). It is more to alert you to the fact that, if you care about knowing what is going on in the world, you have just lived through a short period where we the people had information about how great powers operate of a magnitude that we might not see again for a very long time.
The intense efforts that the US and UK governments went to in order to suppress events in the Afghan and Iraqi wars, and the publication of US diplomatic cables tell us how much they were embarrassed by the Wikileaks collaboration with the Guardian and other newspapers. I have no doubt that they have ramped up their security to prevent a repeat. In any case the US retaliation against Manning has been so severe that other potential leakers of his persuasion might well think again (certainly Julian Assange himself is in no hurry to be extradited to Sweden in case he is handed over to US authorities). Furthermore the US authorities are now pursuing the Associated Press organisation by subpoena-ing phone records that might bear on the CIA successfully thwarting a plot by al-Qaeda in Yemen to blow up a U.S. jetliner (this is not the first such aggressive subpoena act).
So don't expect anything as informative as the Wikileak bonanza to strike again for a very long time.
Therefore my take home message is this. Have a good luck at what we already have. You can go to the Wikileaks web site itself. However you might find this is more than you can handle! It is vast. Instead you might try the Guardian site which is very well-organised and will tell you also a lot about the politics associated with the reaction of the US and the UK.
Saturday, 20 April 2013
A serious case of poor economic analysis
In this post I want to give some more publicity to a story that has, so far, only been run in specialised economic news columns. It is a dramatic one that suggests much of the world has taken a wrong financial turn partly because of one influential academic paper, around which there are now some serious questions.
Two economists, Carmen Reinhart and Kenneth Rogoff at Harvard University, released a paper in 2008, “Growth in a Time of Debt.” whose main conclusion was that countries whose public debt was over 90% of their Gross Domestic Product have below average growth rates, slightly negative in fact. Their paper has been widely cited (454 Google Scholar citations as of 21 April, 2013). In addition it has been used by government exchequers to justify severe austerity measures. The most significant such example is when Paul Ryan, the Republican chair of the US House of Representatives budget committee, a notable economic hawk who has pushed for rapid fiscal tightening, cited the Reinhart-Rogoff paper as “conclusive empirical evidence that total debt exceeding 90 per cent of the economy has a significant negative effect on economic growth”. In addition, as reported in the Guardian, British Chancellor George Osborne has repeatedly said that his monetary policy was highly influenced by the paper.
Last week there was a bombshell. Three researchers from the University of Massachusetts, Thomas Herndon, Michael Ash, and Robert Pollin, published a paper "Does high public debt consistently stifle economic growth" which seriously undermined (some have said "tore to shreds") the Reinhart-Rogoff conclusions. The new paper challenges the Reinhart-Rogoff methodology in two ways: by criticising the ways in R&R excluded some data that did not accord with their conclusion, and by criticising how they weighted their summations. You can read the actual details in the source articles. More sensationally, Herndon et al found an error in the spreadsheet that R&R had used to construct their main summative table.
In their response R&R who were obviously deeply embarrassed by the spreadsheet error nevertheless argued that their main conclusions were still sound and a debate is now beginning between the two sides. So far Herndon et al robustly maintain that the R&R study is not only flawed in the way that they originally claimed but that the results are almost the opposite of those claimed by R&R.
I hope that the debate does not simply recede into exchanges between academics guarding their reputations. This issue is of fundamental importance to the way in which we regulate our economies. I am so far dismayed by the entrenched attitudes of those who originally espoused the R&R conclusions. In particular the response from George Osborne's office has been abysmal - just a claim that their economic policy does not rest on one academic paper and that "the majority of economists" back Osborne's strategy.
It seems to me that economists, and the entire discipline of Economics, faces a challenge. They need to sort out for policy-makers just what the situation is vis a vis debt versus growth. I know this might be a big ask but they must do their level best. If we see two camps emerging divided by political affiliations we would be tempted to kiss goodbye to any pretence that Economics is a science.
Two economists, Carmen Reinhart and Kenneth Rogoff at Harvard University, released a paper in 2008, “Growth in a Time of Debt.” whose main conclusion was that countries whose public debt was over 90% of their Gross Domestic Product have below average growth rates, slightly negative in fact. Their paper has been widely cited (454 Google Scholar citations as of 21 April, 2013). In addition it has been used by government exchequers to justify severe austerity measures. The most significant such example is when Paul Ryan, the Republican chair of the US House of Representatives budget committee, a notable economic hawk who has pushed for rapid fiscal tightening, cited the Reinhart-Rogoff paper as “conclusive empirical evidence that total debt exceeding 90 per cent of the economy has a significant negative effect on economic growth”. In addition, as reported in the Guardian, British Chancellor George Osborne has repeatedly said that his monetary policy was highly influenced by the paper.
Last week there was a bombshell. Three researchers from the University of Massachusetts, Thomas Herndon, Michael Ash, and Robert Pollin, published a paper "Does high public debt consistently stifle economic growth" which seriously undermined (some have said "tore to shreds") the Reinhart-Rogoff conclusions. The new paper challenges the Reinhart-Rogoff methodology in two ways: by criticising the ways in R&R excluded some data that did not accord with their conclusion, and by criticising how they weighted their summations. You can read the actual details in the source articles. More sensationally, Herndon et al found an error in the spreadsheet that R&R had used to construct their main summative table.
In their response R&R who were obviously deeply embarrassed by the spreadsheet error nevertheless argued that their main conclusions were still sound and a debate is now beginning between the two sides. So far Herndon et al robustly maintain that the R&R study is not only flawed in the way that they originally claimed but that the results are almost the opposite of those claimed by R&R.
I hope that the debate does not simply recede into exchanges between academics guarding their reputations. This issue is of fundamental importance to the way in which we regulate our economies. I am so far dismayed by the entrenched attitudes of those who originally espoused the R&R conclusions. In particular the response from George Osborne's office has been abysmal - just a claim that their economic policy does not rest on one academic paper and that "the majority of economists" back Osborne's strategy.
It seems to me that economists, and the entire discipline of Economics, faces a challenge. They need to sort out for policy-makers just what the situation is vis a vis debt versus growth. I know this might be a big ask but they must do their level best. If we see two camps emerging divided by political affiliations we would be tempted to kiss goodbye to any pretence that Economics is a science.
Wednesday, 17 April 2013
Prime conjunctions
I have 4 children and around 10 years ago their ages were 17, 23, 29 and 31. This period lasted only for a few weeks until one of them had a birthday but I made much of it at the time for these ages are all prime numbers. One day they will be 47, 53, 59 and 61 - another prime conjunction and I hope I shall still be around to celebrate it. I will definitely miss the next (and so will they): 167, 173, 179, 181.
Of course I am extraordinarily fortunate in having so wonderful a quartet of children. Am I fortunate in another way:- that their ages allow such prime conjunctions? We shall soon see.
Let's consider a set of n children (think of n as being 2, 3 or 4 if you like). How likely is it that one day they will all have prime ages? I'm going to simplify matters in three ways. The first simplification is really unnecessary in practice: it is that we'll assume no two children were born at exactly the same time of day on the same day of the year. The second simplification is that we'll only worry about odd primes. Ignoring the prime 2 is not such a big deal. This special case is actually quite significant if you are just interested in whether the children have a prime conjunction at all but not if you are more concerned with whether they have lots of them. The third simplification is apparently quite bizarre but at the end of the post I will explain that it is not as bizarre as all that - so bear with me. This simplification is that rather than thinking about prime conjunctions we think about occasions (odd conjunctions) when all the children's ages are odd.
We want to think about their ages and we have to write down these ages in some order. Rather than that order being highest to lowest or lowest to highest I shall write them down in order of birthday throughout the year. For example three children born on 2 March 1991, 6 November 1994, 4 May 1997 will be ordered with the March birthday coming first, then the May birthday, then the November birthday. So, at the beginning of this year, their ages were 22, 15, 18. These remain their ages until 2 March when the 22 becomes 23. Another change (from 15 to 16 occurs on 4 May) and another change occurs on 6 November.
In terms of evenness and oddness it was Even, Odd, Even at the beginning of the year and then it went
Odd, Odd, Even
Odd, Even, Even
Odd, Even, Odd
and, continuing to track the changes in 2014,
Even, Even, Odd
Even, Even, Odd
Even, Odd, Odd
and then we come back to
Even, Odd, Even
Put more concisely we begin with a list of E's and O's. We change the first one, then we change the second, then we change the third, and then we go back to the start of the list and change the first (then the second, then the third).
EOE, OOE, OEE, OEO, EEO, EOO, EOE
and now the cycle repeats.
For this particular set of birthdays we never find OOO so these children are never all of an odd age (and therefore never all odd prime ages (as it happens, it doesn't matter in this case whether we allow the prime 2)).
OK. Let's jump to the general case. Now we have n children. As before we list their evenness or oddness of age by order of their birthday. Suppose at the beginning of the year that gives us a bunch of evens followed by a bunch of odds. At some stage in the year those initial evens will all have changed to odds and all of them will be an odd age. Instead suppose at the beginning of the year we have a bunch of odds followed by a bunch of evens. Part way through the year everyone will now be even and so, a year later, everyone will be odd.
But what if we don't have one of these types of initial odd/even lists. If that happens then either we begin with an even, later in the list we have some odd, and still later we have another even (or we might have a similar case with the roles of evens and odds exchanged):
E ... O ... E ...
We start our process of changing the evens and odds one by one from the beginning. It is easy to see that, no matter where we are in the process, the three symbols at these places cannot all be equal; so we can never get all odds.
It is not too difficult to work out that 2n even/odd lists have the property that they consist of a bunch of one type (even or odd) followed by a bunch of the other type (odd or even). But there are 2n sequences in all. So what that means is that, with n children, the chance of their having an odd conjunction is n/2n-1. And notice that if they do have an odd conjunction then they will have others at two-yearly intervals - so many of them.
Therefore, given that I have 4 children, the chance of them having an odd conjunction is exactly 50%.
But, wait a minute, I began by asking about prime conjunctions! All I have done is analyse the chance of getting an odd conjunction. The passage from odd to prime leads us to some deep unsolved questions in number theory.
Suppose I have two children with the younger born between 1 and 3 years after the older. Then some of the time their ages will differ by 2 and every two years there will be a period when their ages are both odd. These children will have many prime conjunctions, the first few being (3, 5), (5, 7), (11, 13), (17, 19). The Twin Prime Conjecture states that there are infinitely prime conjunctions in this case (prime numbers differing by 2). It is a very long-standing unsolved problem and most mathematicians believe it is true.
What if I have two children born more than a year apart? Then every two years their ages will differ by an even number k. Now we are in the realm of Polignac's Conjecture: whether there are infinitely many primes that differ by some specific even integer k. With more than two children we come to an even more general conjecture by Dickson.
What all this means is that if your children do not have an odd conjunction (and we have seen above how this may be tested, and how likely it is) then you will never see a prime conjunction. But, more usefully, if you are looking for a prime conjunction, you should first find an odd conjunction (if any) and then starting adding two to it repeatedly hoping to find a set of odd numbers that are indeed all prime. This is not guaranteed (e.g. 7, 9, 11 - all increments include a multiple of 3). However Dickson's theorem (which I will write about in a future blog) provides some refined help if needed. Finally, unless a large number of mathematicians are going to be very surprised indeed, if your children have "several" prime conjunctions, they will have infinitely many prime conjunctions.
Of course I am extraordinarily fortunate in having so wonderful a quartet of children. Am I fortunate in another way:- that their ages allow such prime conjunctions? We shall soon see.
Let's consider a set of n children (think of n as being 2, 3 or 4 if you like). How likely is it that one day they will all have prime ages? I'm going to simplify matters in three ways. The first simplification is really unnecessary in practice: it is that we'll assume no two children were born at exactly the same time of day on the same day of the year. The second simplification is that we'll only worry about odd primes. Ignoring the prime 2 is not such a big deal. This special case is actually quite significant if you are just interested in whether the children have a prime conjunction at all but not if you are more concerned with whether they have lots of them. The third simplification is apparently quite bizarre but at the end of the post I will explain that it is not as bizarre as all that - so bear with me. This simplification is that rather than thinking about prime conjunctions we think about occasions (odd conjunctions) when all the children's ages are odd.
We want to think about their ages and we have to write down these ages in some order. Rather than that order being highest to lowest or lowest to highest I shall write them down in order of birthday throughout the year. For example three children born on 2 March 1991, 6 November 1994, 4 May 1997 will be ordered with the March birthday coming first, then the May birthday, then the November birthday. So, at the beginning of this year, their ages were 22, 15, 18. These remain their ages until 2 March when the 22 becomes 23. Another change (from 15 to 16 occurs on 4 May) and another change occurs on 6 November.
In terms of evenness and oddness it was Even, Odd, Even at the beginning of the year and then it went
Odd, Odd, Even
Odd, Even, Even
Odd, Even, Odd
and, continuing to track the changes in 2014,
Even, Even, Odd
Even, Even, Odd
Even, Odd, Odd
and then we come back to
Even, Odd, Even
Put more concisely we begin with a list of E's and O's. We change the first one, then we change the second, then we change the third, and then we go back to the start of the list and change the first (then the second, then the third).
EOE, OOE, OEE, OEO, EEO, EOO, EOE
and now the cycle repeats.
For this particular set of birthdays we never find OOO so these children are never all of an odd age (and therefore never all odd prime ages (as it happens, it doesn't matter in this case whether we allow the prime 2)).
OK. Let's jump to the general case. Now we have n children. As before we list their evenness or oddness of age by order of their birthday. Suppose at the beginning of the year that gives us a bunch of evens followed by a bunch of odds. At some stage in the year those initial evens will all have changed to odds and all of them will be an odd age. Instead suppose at the beginning of the year we have a bunch of odds followed by a bunch of evens. Part way through the year everyone will now be even and so, a year later, everyone will be odd.
But what if we don't have one of these types of initial odd/even lists. If that happens then either we begin with an even, later in the list we have some odd, and still later we have another even (or we might have a similar case with the roles of evens and odds exchanged):
E ... O ... E ...
We start our process of changing the evens and odds one by one from the beginning. It is easy to see that, no matter where we are in the process, the three symbols at these places cannot all be equal; so we can never get all odds.
It is not too difficult to work out that 2n even/odd lists have the property that they consist of a bunch of one type (even or odd) followed by a bunch of the other type (odd or even). But there are 2n sequences in all. So what that means is that, with n children, the chance of their having an odd conjunction is n/2n-1. And notice that if they do have an odd conjunction then they will have others at two-yearly intervals - so many of them.
Therefore, given that I have 4 children, the chance of them having an odd conjunction is exactly 50%.
But, wait a minute, I began by asking about prime conjunctions! All I have done is analyse the chance of getting an odd conjunction. The passage from odd to prime leads us to some deep unsolved questions in number theory.
Suppose I have two children with the younger born between 1 and 3 years after the older. Then some of the time their ages will differ by 2 and every two years there will be a period when their ages are both odd. These children will have many prime conjunctions, the first few being (3, 5), (5, 7), (11, 13), (17, 19). The Twin Prime Conjecture states that there are infinitely prime conjunctions in this case (prime numbers differing by 2). It is a very long-standing unsolved problem and most mathematicians believe it is true.
What if I have two children born more than a year apart? Then every two years their ages will differ by an even number k. Now we are in the realm of Polignac's Conjecture: whether there are infinitely many primes that differ by some specific even integer k. With more than two children we come to an even more general conjecture by Dickson.
What all this means is that if your children do not have an odd conjunction (and we have seen above how this may be tested, and how likely it is) then you will never see a prime conjunction. But, more usefully, if you are looking for a prime conjunction, you should first find an odd conjunction (if any) and then starting adding two to it repeatedly hoping to find a set of odd numbers that are indeed all prime. This is not guaranteed (e.g. 7, 9, 11 - all increments include a multiple of 3). However Dickson's theorem (which I will write about in a future blog) provides some refined help if needed. Finally, unless a large number of mathematicians are going to be very surprised indeed, if your children have "several" prime conjunctions, they will have infinitely many prime conjunctions.
Update 22 May 2013
The mathematical world is buzzing with the news that a virtually unknown mathematician Yitang Zhang at the University of New Hampshire has proved that there is a number k<70,000,000 for which Polignac's conjecture is true. So we know now that somewhere sometime there must have been two mammals (because mammals go back more than 70 million years) who enjoy an infinite number of prime conjunctions.
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