Tag: Numbers

World Of Math Aflutter Over New Proof Of 160-Year-Old Hypothesis

The 90-year-old mathematician Michael Atiyah has presented what he referred to as a “simple proof” of the Riemann hypothesis, a problem which has eluded mathematicians for almost 160 years.

The world of maths and the Twitter-sphere have been a frenzy ever since the British-Lebanese mathematician indicated he was to give a lecture on Monday at the Heidelberg Laureates Forum in Germany on what is widely regarded as the most important outstanding problem in maths.

Although it is almost incomprehensible for people without intensive maths training, the hypothesis describes the distribution of prime numbers among positive integers.

Prime numbers, very simple by definition, are the building blocks of modern mathematics, especially number theory. Achievements in prime number theory have been widely applied to computer sciences and telecommunications.

 




However prime numbers are also mysterious and inexplicable – the pattern in which prime numbers emerge in the line of positive integers has remained elusive to generations of mathematicians.

Riemann proposed a theory that can, in a way, shed light on that mystery; but he could not prove it, nor could all the brilliant minds that came after.

If a solution to the Riemann hypothesis is confirmed, it would be big news as mathematicians would be armed with a map to the location of all such prime numbers; a breakthrough with far-reaching repercussions in the field.

Atiyah, who specialises in geometry, is one of the UK’s most eminent mathematical figures, having received the two awards often referred to as the Nobel prizes of mathematics; the Fields Medal and the Abel Prize.

 

$1 million prize

As one of the six unsolved “Clay Millennium Problems”, any solution would be eligible for a $1 million prize. This has tempted many mathematicians over the years, none of which has yet been awarded the prize.

Atiyah is well aware of this history of failure. “Nobody believes any proof of the Riemann hypothesis, let alone a proof by someone who’s 90,” he said, but he hoped his presentation would convince his critics.

In it, he paid tribute to the work of two great 20th century mathematicians, John von Neumann and Friedrich Hirzebruch, whose developments he claims laid the foundations for his own proposed proof.

It fell into my lap, I had to pick it up,” he said in advance of giving his talk.

He has produced a number of papers in recent years making remarkable claims which have so far failed to convince his peers.

The Clay Millennium Prizes, announced in 2000, were conceived to record seven of the most difficult problems with which mathematicians were grappling at the turn of the second millennium.

They were also designed “to elevate in the consciousness of the general public the fact that in mathematics, the frontier is still open and abounds in important unsolved problems; to emphasise the importance of working towards a solution of the deepest, most difficult problems; and to recognise achievement in mathematics of historical magnitude”.

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Honey Bees Can Understand Nothing

Zero, zilch, nothing, is a pretty hard concept to understand. Children generally can’t grasp it until kindergarten. And it’s a concept that may not be innate but rather learned through culture and education.

Throughout human history, civilizations have had varying representations for it. Yet our closest animal relative, the chimpanzee, can understand it.

And now researchers in Australia writing in the journal Science say the humble honey bee can be taught to understand that zero is less than one.

The result is kind of astounding, considering how tiny bee brains are. Humans have around 100 billion neurons. The bee brain? Fewer than 1 million.

The findings suggest that the ability to fathom zero may be more widespread than previously thought in the animal kingdom — something that evolved long ago and in more branches of life.




It’s also possible that in deconstructing how the bees compute numbers, we could make better, more efficient computers one day.

Our computers are electricity-guzzling machines. The bee, however, “is doing fairly high-level cognitive tasks with a tiny drop of nectar,” says Adrian Dyer, a Royal Melbourne Institute of Technology researcher and co-author on the study.

Their brains are probably processing information in a very clever [i.e., efficient] way.”

But before we can deconstruct the bee brain, we need to know that it can do the complex math in the first place.

How to teach a bee the concept of zero

Bees are fantastic learners. They spend hours foraging for nectar in among flowers, can remember where the juiciest flowers are, and even have a form of communication to inform their hive mates of where food is to be found.

Researchers train bees like they train many animals: with food. “You have a drop of sucrose associated with a color or a shape, and they will learn to reliably go back to” that color or shape, Dyer explains.

With this simple process, you can start teaching bees rules. In this case, the researchers wanted to teach 10 bees the basic rules of arithmetic.

So they put out a series of sheets of paper that had differing numbers of objects printed on them. Using sugar as a reward, the researchers taught the bees to always fly to the sheet that had the fewest objects printed on it.

Once the bees learned this rule, they could reliably figure out that two shapes are less than four shapes, that one shape is smaller than three. And they’d keep doing this even when a sugary reward was not waiting for them.

And then came the challenge: What happens when a sheet with no objects at all was presented to the bees? Would they understand that a blank sheet — which represented the concept of zero in this experiment — was less than three, less than one?

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Most Precise Measurement Of The Proton’s Mass

What is the weight of a proton? Scientists from Germany and Japan have made an important step toward better understanding this fundamental constant.

By means of precision measurements on a single proton, they were able to improve the precision by a factor of three and also correct the existing value.

To determine the mass of a single proton more accurately, the group of physicists from the Max Planck Institute for Nuclear Physics in Heidelberg and RIKEN in Japan performed an important high-precision measurement.

In a greatly advanced Penning trap system, designed by Sven Sturm and Klaus Blaum from MPI-K, using ultra-sensitive single particle detectors that were partly developed by RIKEN’s Ulmer Fundamental Symmetries Laboratory.




The proton is the nucleus of the hydrogen atom and one of the basic building blocks of all other atomic nuclei. Therefore, the proton’s mass is an important parameter in atomic physics: it is one of the factors that affect how the electrons move around the atomic nucleus.

This is reflected in the spectra, i.e., the light colours (wavelengths) that atoms can absorb and emit again. By comparing these wavelengths with theoretical predictions, it is possible to test fundamental physical theories.

Further, precise comparisons of the masses of the proton and the antiproton may help in the search for the crucial difference – besides the reversed sign of the charge – between matter and antimatter.

Penning traps are well-proven as suitable “scales” for ions. In such a trap, it is possible to confine, nearly indefinitely, single charged particles such as a proton, for example, by means of electric and magnetic fields.

Inside the trap, the trapped particle performs a characteristic periodic motion at a certain oscillation frequency. This frequency can be measured and the mass of the particle calculated from it.

In order to reach the targeted high precision, an elaborate measurement technique was required.

The carbon isotope 12C with a mass of 12 atomic mass units is defined as the mass standard for atoms. “We directly used it for comparison,” says Sven Sturm.

First we stored each one proton and one carbon ion (12C6+) in separate compartments of our Penning trap apparatus, then transported each of the two ions into the central measurement compartment and measured its motion.

From the ratio of the two measured values the group obtained the proton’s mass directly in atomic units. The measurement compartment was equipped with specifically developed purpose-built electronics.

Andreas Mooser of RIKEN’s Fundamental Symmetries Laboratory explains its function: “It allowed us to measure the proton under identical conditions as the carbon ion despite its about 12-fold lower mass and 6-fold smaller charge.”

The resulting mass of the proton, determined to be 1.007276466583(15)(29) atomic mass units, is three times more precise than the presently accepted value.

The numbers in parentheses refer to the statistical and systematic uncertainties, respectively.

Intriguingly, the new value is significantly smaller than the current standard value.

Measurements by other authors yielded discrepancies with respect to the mass of the tritium atom, the heaviest hydrogen isotope (T = 3H), and the mass of light helium (3He) compared to the “semiheavy” hydrogen molecule HD (D = 2H, deuterium, heavy hydrogen).

Our result contributes to solving this puzzle, since it corrects the proton’s mass in the proper direction,” says Klaus Blaum.

Florian Köhler-Langes of MPIK explains how the researchers intend to further improve the precision of their measurement: “In the future, we will store a third ion in our trap tower. By simultaneously measuring the motion of this reference ion, we will be able to eliminate the uncertainty originating from fluctuations of the magnetic field.”

The work was published in Physical Review Letters.

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The Largest Prime Number Was Discovered By A FedEx Employee

A FedEx employee in Tennessee has discovered the largest known prime number.

Germantown, Tenn., resident John Pace found the number through his volunteer work with the Great Internet Mersenne Prime Search (GIMPS), a project that crowd sources computing power to search for a subset of prime numbers called Mersenne primes.

Like a normal prime number, these can only be divided by themselves and one. What sets them apart is that they can all be expressed as the number 2 raised to a given power minus one.




The newly discovered Mersenne prime, called M77232917, can be expressed as 2 to the 77,232,917 power minus one. It’s the 50th Mersenne prime to be discovered and it’s more than 23 million digits long.

Pace might be the only person in history who went into math for the money.

He told NPR, “There was a $100,000 prize attached to finding the first prime that had a 10 million digit result, and I was like, ‘Well you know, I’ve got as much chance as anybody else.’

He has been participating in the program for 14 years and this is his first discovery.

The previous longest-known prime number was discovered in January of 2016 at the University of Central Missouri. It contains 22 million digits and is also a Mersenne prime.

Large prime numbers are important for the future of computing and cyber security, and the search is already on for larger numbers.

The Electronic Frontier Foundation is offering a prize of $150,000 for finding the first prime number with one hundred million digits and $250,000 for finding the first prime with one billion digits.

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Largest Known Prime Number Discovered With Over 23 Million Digits

A collaborative computational effort has uncovered the longest known prime number.

At over 23 million digits long, the new number has been given the name M77232917 for short.

Prime numbers are divisible only by themselves and one, and the search for ever-larger primes has long occupied maths enthusiasts.

However, the search requires complicated computer software and collaboration as the numbers get increasingly hard to find.




M77232917 was discovered on a computer belonging to Jonathan Pace, an electrical engineer from Tennessee who has been searching for big primes for 14 years.

Mr Pace discovered the new number as part of the Great Internet Mersenne Prime Search (GIMPS), a project started in 1996 to hunt for these massive numbers.

Mersenne primes – named after the 17th century French monk Marin Mersenne who studied them – are calculated by multiplying together many twos and then subtracting one.

Six days of non-stop computing in which 77,232,917 twos were multiplied together resulted in the latest discovery.

The number is the 50th Mersenne prime to be discovered, and the 16th to be discovered by the GIMPS project.

It is nearly one million digits longer than the previous record holder, which was identified as part of the same project at the beginning of 2016.

Mersenne primes are a particular focus for prime aficionados because there is a relatively straightforward way to check whether a number is one or not.

Nevertheless, the new prime has to be verified using four different computer programs on four different computers.

The process also relies on thousands of volunteers sifting through millions of non-prime candidates before the lucky individual chances upon their target.

Professor Caldwell runs an authoritative website on the largest prime numbers, with a focus on the history of Mersenne primes.

He emphasised the pure excitement that searching for prime numbers brings, describing the latest discovery as “a museum piece as opposed to something that industry would use”.

Besides the thrill of discovery, Mr Pace will receive a $3,000 (£2,211) GIMPS research discovery award.

GIMPS uses the power of thousands of ordinary computers to search for elusive primes, and the team behind it state that anybody with a reasonably powerful PC can download the necessary software and become a “big prime hunter”.

The next Mersenne prime discovery could be smaller or larger than the existing record holder, but the big target for the GIMPS team is to find a 100 million digit prime number.

The person who discovers such a number will be awarded $150,000 by the Electronic Frontier Foundation for their efforts.

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The 500-Page Proof That Only One Mathematician Can Understand

Nearly four years after Shinichi Mochizuki unveiled an imposing set of papers that could revolutionize the theory of numbers, other mathematicians have yet to understand his work or agree on its validity.

Although they have made modest progress.

Some four dozen mathematicians converged last week for a rare opportunity to hear Mochizuki present his own work at a conference on his home turf, Kyoto University’s Research Institute for Mathematical Sciences (RIMS).

Mochizuki is “less isolated than he was before the process got started”, says Kiran Kedlaya, a number theorist at the University of California, San Diego.

Although at first Mochizuki’s papers, which stretch over more than 500 pages, seemed like an impenetrable jungle of formulae.




Experts have slowly discerned a strategy in the proof that the papers describe, and have been able to zero in on particular passages that seem crucial, he says.

Mochizuki’s theorem aims to prove the important abc conjecture, which dates back to 1985 and relates to prime numbers — whole numbers that cannot be evenly divided by any smaller number except by 1.

The conjecture comes in a number of different forms, but explains how the primes that divide two numbers, a and b, are related to those that divide their sum, c.

If Mochizuki’s proof is correct, it would have repercussions across the entire field, says Dimitrov.

When you work in number theory, you cannot ignore the abc conjecture,” he says.

This is why all number theorists eagerly wanted to know about Mochizuki’s approach.”

For example, Dimitrov showed in January how, assuming the correctness of Mochizuki’s proof, one might be able to derive many other important results, including a completely independent proof of the celebrated Fermat’s last theorem.

But the purported proof, which Mochizuki first posted on his webpage in August 2012, builds on more than a decade of previous work in which Mochizuki worked in virtual isolation and developed a novel and extremely abstract branch of mathematics.

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