Kannan Soundararajan: Am I kidding myself, or have I discovered something?
Unexpected Biases in the Distribution of Consecutive Primes (PDF file)
Kannan Soundararajan: Am I kidding myself, or have I discovered something?
Unexpected Biases in the Distribution of Consecutive Primes (PDF file)
Mathematicians Discover Prime Conspiracy
A previously unnoticed property of prime numbers seems to violate a longstanding assumption about how they behave.
by Erica Klarreich
https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/
Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers — those numbers that are divisible only by 1 and themselves. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them.
Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9. In a paper posted online today, Kannan Soundararajan and Robert Lemke Oliver of Stanford University present both numerical and theoretical evidence that prime numbers repel other would-be primes that end in the same digit, and have varied predilections for being followed by primes ending in the other possible final digits.
“We’ve been studying primes for a long time, and no one spotted this before,” said Andrew Granville, a number theorist at the University of Montreal and University College London. “It’s crazy.”
The discovery is the exact opposite of what most mathematicians would have predicted, said Ken Ono, a number theorist at Emory University in Atlanta. When he first heard the news, he said, “I was floored. I thought, ‘For sure, your program’s not working.’”
This conspiracy among prime numbers seems, at first glance, to violate a longstanding assumption in number theory: that prime numbers behave much like random numbers. Most mathematicians would have assumed, Granville and Ono agreed, that a prime should have an equal chance of being followed by a prime ending in 1, 3, 7 or 9 (the four possible endings for all prime numbers except 2 and 5).
“I can’t believe anyone in the world would have guessed this,” Granville said. Even after having seen Lemke Oliver and Soundararajan’s analysis of their phenomenon, he said, “it still seems like a strange thing.”
Yet the pair’s work doesn’t upend the notion that primes behave randomly so much as point to how subtle their particular mix of randomness and order is. “Can we redefine what ‘random’ means in this context so that once again, [this phenomenon] looks like it might be random?” Soundararajan said. “That’s what we think we’ve done.”
Prime Preferences
Soundararajan was drawn to study consecutive primes after hearing a lecture at Stanford by the mathematician Tadashi Tokieda, of the University of Cambridge, in which he mentioned a counterintuitive property of coin-tossing: If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses.
Soundararajan wondered if similarly strange phenomena appear in other contexts. Since he has studied the primes for decades, he turned to them — and found something even stranger than he had bargained for. Looking at prime numbers written in base 3 — in which roughly half the primes end in 1 and half end in 2 — he found that among primes smaller than 1,000, a prime ending in 1 is more than twice as likely to be followed by a prime ending in 2 than by another prime ending in 1. Likewise, a prime ending in 2 prefers to be followed a prime ending in 1.
Soundararajan showed his findings to postdoctoral researcher Lemke Oliver, who was shocked. He immediately wrote a program that searched much farther out along the number line — through the first 400 billion primes. Lemke Oliver again found that primes seem to avoid being followed by another prime with the same final digit. The primes “really hate to repeat themselves,” Lemke Oliver said.
Lemke Oliver and Soundararajan discovered that this sort of bias in the final digits of consecutive primes holds not just in base 3, but also in base 10 and several other bases; they conjecture that it’s true in every base. The biases that they found appear to even out, little by little, as you go farther along the number line — but they do so at a snail’s pace. “It’s the rate at which they even out which is surprising to me,” said James Maynard, a number theorist at the University of Oxford. When Soundararajan first told Maynard what the pair had discovered, “I only half believed him,” Maynard said. “As soon as I went back to my office, I ran a numerical experiment to check this myself.”
Lemke Oliver and Soundararajan’s first guess for why this bias occurs was a simple one: Maybe a prime ending in 3, say, is more likely to be followed by a prime ending in 7, 9 or 1 merely because it encounters numbers with those endings before it reaches another number ending in 3. For example, 43 is followed by 47, 49 and 51 before it hits 53, and one of those numbers, 47, is prime.
But the pair of mathematicians soon realized that this potential explanation couldn’t account for the magnitude of the biases they found. Nor could it explain why, as the pair found, primes ending in 3 seem to like being followed by primes ending in 9 more than 1 or 7. To explain these and other preferences, Lemke Oliver and Soundararajan had to delve into the deepest model mathematicians have for random behavior in the primes.
Random Primes
Prime numbers, of course, are not really random at all — they are completely determined. Yet in many respects, they seem to behave like a list of random numbers, governed by just one overarching rule: The approximate density of primes near any number is inversely proportional to how many digits the number has.
In 1936, Swedish mathematician Harald Cramér explored this idea using an elementary model for generating random prime-like numbers: At every whole number, flip a weighted coin — weighted by the prime density near that number — to decide whether to include that number in your list of random “primes.” Cramér showed that this coin-tossing model does an excellent job of predicting certain features of the real primes, such as how many to expect between two consecutive perfect squares.
Despite its predictive power, Cramér’s model is a vast oversimplification. For instance, even numbers have as good a chance of being chosen as odd numbers, whereas real primes are never even, apart from the number 2. Over the years, mathematicians have developed refinements of Cramér’s model that, for instance, bar even numbers and numbers divisible by 3, 5, and other small primes.
These simple coin-tossing models tend to be very useful rules of thumb about how prime numbers behave. They accurately predict, among other things, that prime numbers shouldn’t care what their final digit is — and indeed, primes ending in 1, 3, 7 and 9 occur with roughly equal frequency.
Yet similar logic seems to suggest that primes shouldn’t care what digit the prime after them ends in. It was probably mathematicians’ overreliance on the simple coin-tossing heuristics that made them miss the biases in consecutive primes for so long, Granville said. “It’s easy to take too much for granted — to assume that your first guess is true.”
The primes’ preferences about the final digits of the primes that follow them can be explained, Soundararajan and Lemke Oliver found, using a much more refined model of randomness in primes, something called the prime k-tuples conjecture. Originally stated by mathematicians G. H. Hardy and J. E. Littlewood in 1923, the conjecture provides precise estimates of how often every possible constellation of primes with a given spacing pattern will appear. A wealth of numerical evidence supports the conjecture, but so far a proof has eluded mathematicians.
The prime k-tuples conjecture subsumes many of the most central open problems in prime numbers, such as the twin primes conjecture, which posits that there are infinitely many pairs of primes — such as 17 and 19 — that are only two apart. Most mathematicians believe the twin primes conjecture not so much because they keep finding more twin primes, Maynard said, but because the number of twin primes they’ve found fits so neatly with what the prime k-tuples conjecture predicts.
In a similar way, Soundararajan and Lemke Oliver have found that the biases they uncovered in consecutive primes come very close to what the prime k-tuples conjecture predicts. In other words, the most sophisticated conjecture mathematicians have about randomness in primes forces the primes to display strong biases. “I have to rethink how I teach my class in analytic number theory now,” Ono said.
At this early stage, mathematicians say, it’s hard to know whether these biases are isolated peculiarities, or whether they have deep connections to other mathematical structures in the primes or elsewhere. Ono predicts, however, that mathematicians will immediately start looking for similar biases in related contexts, such as prime polynomials — fundamental objects in number theory that can’t be factored into simpler polynomials.
And the finding will make mathematicians look at the primes themselves with fresh eyes, Granville said. “You could wonder, what else have we missed about the primes?”
Maths experts stunned as they crack a pattern for prime numbers
‘I was floored’
by Adam Lusher
http://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html
Two academics have shocked themselves and the world of mathematics by discovering a pattern in prime numbers.
Primes – numbers greater than 1 that are divisible only by themselves and 1 – are considered the ‘building blocks’ of mathematics, because every number is either a prime or can be built by multiplying primes together – (84, for example, is 2x2x3x7).
Their properties have baffled number theorists for centuries, but mathematicians have usually felt safe working on the assumption they could treat primes as if they occur randomly.
Now, however, Kannan Soundararajan and Robert Lemke Oliver of Stanford University in the US have discovered that when it comes to the last digit of prime numbers, there is a kind of pattern.
Apart from 2 and 5, all prime numbers have to end in 1, 3, 7 or 9 so that they can’t be divided by 2 or 5. So if the numbers occurred randomly as expected, it wouldn’t matter what the last digit of the previous prime was. Each of the four possibilities – 1, 3, 7, or 9 – should have an equal 25 per cent (one in four) chance of appearing at the end of the next prime number.
But after devising a computer programme to search for the first 400 billion primes, the two mathematicians found prime numbers tend to avoid having the same last digit as their immediate predecessor – as if, in the words of Dr Lemke Oliver they “really hate to repeat themselves.”
A prime ending in 1 was followed by a prime ending in 1 only 18.5 per cent of the time, significantly less often than the expected 25 per cent. And, the pair found, primes ending in 3 tended be followed by primes ending in 9 more often than in 1 or 7.
The pattern – already being referred to as ‘the conspiracy among primes’ – has left mathematicians amazed that it could have remained undiscovered for so long. “I was floored,” Ken Ono, a number theorist at Emory University in Atlanta, told Quanta Magazine. “I have to rethink how I teach my class in analytic number theory now.”
Professor Soundararajan himself admitted to New Scientist: “It was very weird. It’s like some painting you are very familiar with, and then suddenly you realise there is a figure in the painting you’ve never seen before.”
Professor Soundararajan’s friend Professor Andrew Granville, of University College, London, told The Independent that the Stanford man’s surprise was such that he initially doubted his own discovery. “He asked me to ‘look over’ his paper four weeks ago, which is code for ‘Am I kidding myself, or have I discovered something?’” Professor Granville added: “He did ask me in November whether I would believe anything like this. Apparently, I looked at him as if he was crazy.”
The two US-based academics argue that the last-digit pattern they have discovered could be explained by the k-tuple conjecture, devised in the early 20th Century to predict how groupings of primes will appear. They also point out that as primes stretch to infinity, they eventually lose the last-digit pattern and start to appear in a random manner.
_______________
Genda Benda:
Gosh, and I thought I had important things to lose sleep over.
gary271:
Excuse my ignorance, but this is what springs to my mind:
– Does the base in which the number is expressed make the “last digit” a spurious concept ? A number is prime in whichever base it’s expressed, but the last digit varies.
– I know 400 billion is a big number, but it’s exactly 0.0000000% (you get the idea) of all numbers that exist, so can one say there’s a pattern in prime numbers ?
A-t-on découvert une loi ordonnant les nombres premiers?
par Pascal Gavillet
http://pascalgavillet.blog.tdg.ch/
Depuis quelques jours, la communauté mathématique est en ébullition. Deux chercheurs de l’université de Stanford, en Californie – Kannan Soundararajan et Robert Lemke Oliver –, aidés d’énormes processeurs, ont découvert une propriété inédite concernant les nombres premiers (représentés ci-dessus dans un graphique circulaire), qui pour rappel, ne sont divisibles que par eux-mêmes et par 1. Le pire, c’est que cette découverte est d’une simplicité affolante, au point de se demander pourquoi personne n’y a songé avant. Elle consiste à observer les chiffres terminant les nombres premiers. Ceux-ci ne peuvent en effet se terminer que par 1, 3, 7 ou 9. Pour des raisons qu’il n’est nul besoin de démontrer, une terminaison par 2, 4, 6 ou 8 est exclue – tous les nombres de cette forme étant des multiples de 2. Raisonnement identique pour 5 ou 0 et les multiples de 5. Restent donc ces quatre chiffres, seule possibilité de terminaison pour un candidat à la primalité : 1, 3, 7 et 9. Nos deux chercheurs ont donc observé tous les nombres premiers jusqu’à un milliard et ont remarqué que la fréquence d’apparition de certaines terminaisons après d’autres n’était pas équiprobable. Prenons l’exemple d’un nombre premier se terminant par 1. En toute logique, la probabilité que le premier suivant, son successeur, se termine par 1, 3, 7 ou 9 devrait être la même. Or non, justement. Il n’en est rien. Ainsi, un premier se finissant par 1 n’a que 18% de chances d’être suivi par un premier de même forme. En revanche, il y a 30% de chances qu’il soit suivi par un premier se terminant par 3 ou 7. Et 22% par un premier se terminant par le chiffre 9. Et ainsi de suite.
Le problème, c’est que ces écarts probabilistes ne sont pas minimes. Ils sont importants, conséquents. Suffisamment en tout cas pour poser question et surtout remettre en cause l’ordre a priori aléatoire de l’apparition des premiers dans la suite des entiers. D’autant plus que l’écart se creuse encore plus lorsqu’on débute la chaîne par un premier se terminant par 9. Il a alors 65% de chances supplémentaires d’être suivi par un premier se terminant par 1 que par un autre se terminant par 9. La logique voudrait que toutes ces probabilités s’équilibrent, comme je l’ai dit avant. C’est loin d’être le cas, ce qui laisse supposer l’existence d’une loi cachée ordonnant la succession des nombres premiers et leur apparition selon un critère moins aléatoire qu’on le pensait. A moins de redéfinir la notion d’aléatoire, autrement dit de l’élargir. Nous en sommes loin.
Enfin, pour réfuter cette découverte, on pourrait affirmer que ces fréquences d’apparitions ne sont pas si illogiques lorsqu’on observe tous les nombres, premiers ou pas, se terminant par 1, 3, 7 ou 9. Prenons l’exemple de la chaîne 41, 43, 47 et 49. 41 est premier. Il est suivi par 43 (ce sont en l’occurrence des jumeaux), puis par 47. La probabilité qu’il soit suivi par un premier se terminant à son tour par 1 est donc plus faible que les autres. C’est empirique et imparable, et valable pour n’importe quelle chaîne analogue. Sauf que les deux chercheurs de Stanford (photo ci-dessous) ont envisagé ce cas de figure (raisonnement) et constaté qu’il ne tenait pas la route par rapport à la magnitude des biais découlant de leur observation des nombres dans d’autres bases (exemple en base 3, où tous les nombres se terminent par 1 ou 2). En d’autres termes, les mathématiciens ont du pain sur la planche pour plusieurs décennies.
This man is the next Tesla.
You should find out more, in particular about his evidence for his claim that the Nazca (and Palpa) lines are in fact an interstellar fractal interferometer (interstellar radio antenna) He released this information on his radio show around the turn of the year.
http://www.jmccanneyscience.com/JamesMcCanneyScienceHour_December_31_2015.mp3
He archives them for free on his website.
http://www.jmccanneyscience.com/SalesPgBkCOverFINALThe%20Nazca%20and%20Palpa%20Lines%20Peru-page-001.jpg
Here’s his bio:
http://www.jmccsci.com/NewVisitorSUB-PAGE.HTM
In any case, if finding out the pattern in prime numbers matters to you, you should find out what this mathematician has discovered and why it’s such a challenge to find accurate information about him.
http://www.jmccanneyscience.com/CalculatePrimesCoversandTableofContents.HTM
Thank you.