Testirajmo Ghoulovo znanje matematike!

[BladeRunner]

Pokušao, neslavno propao.

Može li neko da napiše rješenje? Hvala.

[mac]

Samo to sam čekao! Evo hint (hehe): umesto da gledamo ishode "pismo" i "glava" (sa četiri moguća ishoda) bolje je ako gledamo ishode "isto" i "različito", jer tada vidimo da postoje samo dva moguća ishoda. Sve što treba naša dva junaka da urade je da pokriju ova dva ishoda, i tada će uvek jedan biti u pravu.

[Albedo 0]

a kako da to pokriju?

[mac]

Pa jedan treba da gađa "isto", a drugi "različito"...

[Albedo 0]

skonto!

[BladeRunner]

Prosto me sramota kako sam ja to zakomplikovao.

Gledao sam 16 mogućih ishoda, pa su problematični parovi PP-GG, PG-PG, GP-GP i GG-PP. Svi ostali (12) daju ishod koji nam treba. I onda sam uzeo da gledam vjerovatnoću ishoda ako uvijek kažu isto i u kakvoj je to vezi sa produženjem kazne (to me nije odvelo nigdje, zato što je nemoguće predvidjeti niz ishoda bacanja novčića).

Međutim ako prvi stalno ponovi ono što je dobio bez obzira da li je bilo pismo ili glava  (PP, GG), a drugi uvijek kaže suprotno (PG, GP) dobijaju se četiri kombinacije (PP-PG, PP-GP, GG-PG, GG-GP) koje sve daju bar jedan pogodak.

Svaka čast. Da sam se ja pitao, Pera i Mika bi zaglavili doživotnu

[Midoto]

Sviđa mi se ovaj tip zadataka gde je bitno da je bar jedan u pravu - obično ispadne da je optimalno da tačno jedan pogađa, a onaj drugi greši.
@BladeRunner - I meni je u prvi mah bila misao da nema nikakve strategije. Nekako mi je bilo krajnje neintuitivno da jedan obavezno pogreši Ove nove generacije (vrlo pragmatične) rešavaju zadatak za nekoliko sekundi. Depresivno.

[Palmer]

ukucati u gugl zbog radi rešenja


exp(-((x-4)^2+(y-4)^2)^2/1000) + exp(-((x+4)^2+(y+4)^2)^2/1000) + 0.1exp(-((x+4)^2+(y+4)^2)^2)+0.1exp(-((x -4)^2+(y-4)^2)^2)

[Meho Krljic]

GIMPS Project Discovers
Largest Known Prime Number: 2^74,207,281-1



http://www.mersenne.org/primes/?press=M74207281

RALEIGH, North Carolina -- On January 7th at 22:30 UTC, the Great Internet Mersenne Prime Search (GIMPS) celebrated its 20th anniversary with the math discovery of the new largest known prime number, 2^74,207,281-1, having 22,338,618 digits, on a university computer volunteered by   Curtis Cooper for the project. The same GIMPS software just uncovered a flaw in Intel's latest Skylake CPUs^[1], and its global network of CPUs peaking at 450 trillion calculations per second remains the longest continuously-running "grassroots supercomputing"^[2] project in Internet history.
The new prime number, also known as M74207281, is calculated by multiplying together 74,207,281 twos then subtracting one. It is almost 5 million digits larger than the previous record prime number, in a special class of extremely rare prime numbers known as Mersenne primes. It is only the 49th known Mersenne prime ever discovered, each increasingly difficult to find.  Mersenne primes were named for the French monk Marin Mersenne, who studied these numbers more than 350 years ago.  GIMPS, founded in 1996, has discovered all 15 of the largest known Mersenne primes. Volunteers download a free program to search for these primes with a cash award offered to anyone lucky enough to compute a new prime. Prof. Chris Caldwell maintains an authoritative web site on the largest known primes and is an excellent history of Mersenne primes.
The primality proof took 31 days of non-stop computing on a PC with an Intel I7-4790 CPU.  To prove there were no errors in the prime discovery process, the new prime was independently verified using both different software and hardware. Andreas Hoglund and David Stanfill each verified the prime using the CUDALucas software running on NVidia Titan Black GPUs in 2.3 days.  David Stanfill verified it using ClLucas on an AMD Fury X GPU in 3.5 days. Serge Batalov also verified it using Ernst Mayer's MLucas software on two Intel Xeon 18-core Amazon EC2 servers in 3.5 days.
Dr. Cooper is a professor at the University of Central Missouri. This is the fourth record GIMPS project prime for Dr. Cooper and his university. The discovery is eligible for a $3,000 GIMPS research discovery award. Their first record prime was discovered in 2005, eclipsed by their second record in 2006. Dr. Cooper lost the record in 2008, reclaimed it in 2013, and improves that record with this new prime. Dr. Cooper and the University of Central Missouri is the largest contributor of CPU time to the GIMPS project.
Dr. Cooper's computer reported the prime in GIMPS on September 17, 2015 but it remained unnoticed until routine maintenance data-mined it.  The official discovery date is the day a human took note of the result.  This is in keeping with tradition as M4253 is considered never to have been the largest known prime number because Hurwitz in 1961 read his computer printout backwards and saw M4423 was prime seconds before seeing that M4253 was also prime.
GIMPS Prime95 client software was developed by founder George Woltman. Scott Kurowski wrote the PrimeNet system software that coordinates GIMPS' computers. Aaron Blosser is now the system administrator, upgrading and maintaining PrimeNet as needed. Volunteers have a chance to earn research discovery awards of $3,000 or $50,000 if their computer discovers a new Mersenne prime.  GIMPS' next major goal is to win the $150,000 award administered by the Electronic Frontier Foundation offered for finding a 100 million digit prime number.
Credit for GIMPS' prime discoveries goes not only to Dr. Cooper for running the Prime95 software on his university's computers, Woltman, Kurowski, and Blosser for authoring the software and running the project, but also the thousands of GIMPS volunteers that sifted through millions of non-prime candidates.  Therefore, official credit for this discovery shall go to "C. Cooper, G. Woltman, S. Kurowski, A. Blosser, et al."
About Mersenne.org's Great Internet Mersenne Prime Search
The Great Internet Mersenne Prime Search (GIMPS) was formed in January 1996 by George Woltman to discover new world record size Mersenne primes. In 1997 Scott Kurowski enabled GIMPS to automatically harness the power of hundreds of thousands of ordinary computers to search for these "needles in a haystack".  Most GIMPS members join the search for the thrill of possibly discovering a record-setting, rare, and historic new Mersenne prime. The search for more Mersenne primes is already under way. There may be smaller, as yet undiscovered Mersenne primes, and there almost certainly are larger Mersenne primes waiting to be found.  Anyone with a reasonably powerful PC can join GIMPS and become a big prime hunter, and possibly earn a cash research discovery award.  All the necessary software can be downloaded for free at www.mersenne.org/freesoft.htm. GIMPS is organized as Mersenne Research, Inc., a 501(c)(3) science research charity. Additional information may be found at www.mersenneforum.org and www.mersenne.org; donations are welcome.
For More Information on Mersenne Primes
Prime numbers have long fascinated amateur and professional mathematicians. An integer greater than one is called a prime number if its only divisors are one and itself. The first prime numbers are 2, 3, 5, 7, 11, etc. For example, the number 10 is not prime because it is divisible by 2 and 5. A Mersenne prime is a prime number of the form 2^P-1. The first Mersenne primes are 3, 7, 31, and 127 corresponding to P = 2, 3, 5, and 7 respectively. There are only 49 known Mersenne primes.
Mersenne primes have been central to number theory since they were first discussed by Euclid about 350 BC. The man whose name they now bear, the French monk Marin Mersenne (1588-1648), made a famous conjecture on which values of P would yield a prime. It took 300 years and several important discoveries in mathematics to settle his conjecture.
Previous GIMPS Mersenne prime discoveries were made by members in various countries.
   In January 2013, Curtis Cooper et al. discovered the 48th known Mersenne prime in the U.S.
   In April 2009, Odd Magnar Strindmo et al. discovered the 47th known Mersenne prime in Norway.
   In September 2008, Hans-Michael Elvenich et al. discovered the 46th known Mersenne prime in Germany.
   In August 2008, Edson Smith et al. discovered the 45th known Mersenne prime in the U.S.
   In September 2006, Curtis Cooper and Steven Boone et al. discovered the 44th known Mersenne prime in the U.S.
   In December 2005, Curtis Cooper and Steven Boone et al. discovered the 43rd known Mersenne prime in the U.S.
   In February 2005, Dr. Martin Nowak et al. discovered the 42nd known Mersenne prime in Germany.
   In May 2004, Josh Findley et al. discovered the 41st known Mersenne prime in the U.S.
   In November 2003, Michael Shafer et al. discovered the 40th known Mersenne prime in the U.S.
   In November 2001, Michael Cameron et al. discovered the 39th Mersenne prime in Canada.
   In June 1999, Nayan Hajratwala et al. discovered the 38th Mersenne prime in the U.S.
   In January 1998, Roland Clarkson et al. discovered the 37th Mersenne prime in the U.S.
   In August 1997, Gordon Spence et al. discovered the 36th Mersenne prime in the U.K.
   In November 1996, Joel Armengaud et al. discovered the 35th Mersenne prime in France.
Euclid proved that every Mersenne prime generates a perfect number. A perfect number is one whose proper divisors add up to the number itself. The smallest perfect number is 6 = 1 + 2 + 3 and the second perfect number is 28 = 1 + 2 + 4 + 7 + 14. Euler (1707-1783) proved that all even perfect numbers come from Mersenne primes. The newly discovered perfect number is 2^74,207,280 x (2^74,207,281-1). This number is over 44 million digits long! It is still unknown if any odd perfect numbers exist.
There is a unique history to the arithmetic algorithms underlying the GIMPS project. The programs that found the recent big Mersenne finds are based on a special algorithm. In the early 1990's, the late Richard Crandall, Apple Distinguished Scientist, discovered ways to double the speed of what are called convolutions -- essentially big multiplication operations. The method is applicable not only to prime searching but other aspects of computation.  During that work he also patented the Fast Elliptic Encryption system, now owned by Apple Computer, which uses Mersenne primes to quickly encrypt and decrypt messages.  George Woltman implemented Crandall's algorithm in assembly language, thereby producing a prime-search program of unprecedented efficiency, and that work led to the successful GIMPS project.
School teachers from elementary through high-school grades have used GIMPS to get their students excited about mathematics.  Students who run the free software are contributing to mathematical research. David Stanfill's verification computation for this discovery was donated by Squirrels (airsquirrels.com) which services K-12 education and other customers.
^[1] [url=http://hardwareluxx.de]http://hardwareluxx.de[/url] ; [url=http://arstechnica.com/gadgets/2016/01/intel-skylake-bug-causes-pcs-to-freeze-during-complex-workloads]http://arstechnica.com/gadgets/2016/01/intel-skylake-bug-causes-pcs-to-freeze-during-complex-workloads[/url]
^[2] Science (American Association for the Advancement of Science), May 6, 2005 p810.

[Ygg]

GIMPS Project Discovers
Largest Known Prime Number: 2^74,207,281-1

Evo i nekoliko zanimljivih videa o tome:

http://youtu.be/tlpYjrbujG0

http://youtu.be/lEvXcTYqtKU

http://youtu.be/jNXAMBvYe-Y

[Meho Krljic]

Kad smo već kod prostih brojeva:



Mathematicians Discover Prime Conspiracy

A previously unnoticed property of prime numbers seems to violate a longstanding assumption about how they behave.

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?"

[Meho Krljic]

Academics Make Theoretical Breakthrough in Random Number Generation

Two University of Texas academics have made what some experts believe is a breakthrough in random number generation that could have longstanding implications for cryptography and computer security. David Zuckerman, a computer science professor, and Eshan Chattopadhyay, a graduate student, published a paper in March that will be presented in June at the Symposium on Theory of Computing. The paper describes how the academics devised a method for the generation of high quality random numbers. The work is theoretical, but Zuckerman said down the road it could lead to a number of practical advances in cryptography, scientific polling, and the study of other complex environments such as the climate.


"We show that if you have two low-quality random sources—lower quality sources are much easier to come by—two sources that are independent and have no correlations between them, you can combine them in a way to produce a high-quality random number," Zuckerman said. "People have been trying to do this for quite some time. Previous methods required the low-quality sources to be not that low, but more moderately high quality. "We improved it dramatically," Zuckerman said. The technical details are described in the academics' paper "Explicit Two-Source Extractors and Resilient Functions." The academics' introduction of resilient functions into their new algorithm built on numerous previous works to arrive at landmark moment in theoretical computer science. Already, one other leading designer of randomness extractors, Xin Li, has built on their work to create sequences of many more random numbers. "You expect to see advances in steps, usually several intermediate phases," Zuckerman said. "We sort of made several advances at once. That's why people are excited." In fact, academics worldwide have taken notice. Oded Goldreich, a professor of computer science at the Weizmann Institute of Science in Israel, called it a fantastic result. "It would have been great to see any explicit two-source extractor for min-entropy rate below one half, let alone one that beats Bourgain's rate of 0.499," Goldreich said on the Weizmann website. "Handling any constant min-entropy rate would have been a feast (see A Challenge from the mid-1980s), and going beyond that would have justified a night-long party." MIT's Henry Yuen, a MIT PhD student in theoretical computer science, called the paper "pulse-quickening." "If the result is correct, then it really is — shall I say it — a breakthrough in theoretical computer science," Yuen said. The study of existing random number generators used in commercial applications has intensified since the Snowden documents were published; sometimes random numbers aren't so random. Low quality random numbers are much easier to predict, and if they're used, they lower the integrity of the security and cryptography protecting data, for example. Right now, Zuckerman's and Chattopadhyay's result is theoretical and work remains in lowering the margins of error, Zuckerman said. Previous work on randomness extractors, including advances made by Zuckerman, required that one sequence used by the algorithm be truly random, or that both sources be close to random. The academics' latest work hurdles those restrictions allowing the use of sequences that are only weakly random. Their method requires fewer computational resources and results in higher quality randomness. Today's random number systems, for example, are fast, but are much more ad-hoc. "This is a problem I've come back to over and over again for more than 20 years," says Zuckerman. "I'm thrilled to have solved it."

[lilit]

i need help.

imam tri nezavisne grupe u eksperimentu (6 tačaka po grupi) a rezultati mi ne ukazuju na normalnu distribuciju, što bi valjda značilo da treba da koristim neki neparametrijski test za poređenje? pitanje: da li je kruskall wallis OK?
a da za post-poređenje između grupa koristim dunn's multiple comparisons test?

[Father Jape]

A da možda nađeš nekoga pouzdanijeg od Sagite da to pitaš?

Mada me teši da i naučnici evropskog ranga imaju poteškoća sa statistikom. <3

[Mica Milovanovic]

Ako ti nije kasno, mogu da pitam sutra moje matematičare u Institutu...

[lilit]

A da možda nađeš nekoga pouzdanijeg od Sagite da to pitaš?

Mada me teši da i naučnici evropskog ranga imaju poteškoća sa statistikom. <3

ma pitala sam al je rano
mislim, 99.99% sam sigurna da može ovako kako sam pitala al konsultacije su uvek bolje nego one man show

Ako ti nije kasno, mogu da pitam sutra moje matematičare u Institutu...

nije kasno. hvala. ako te ne mrzi preterano

[Mica Milovanovic]

Nije mi teško. Pošalji mi na privatni email detalje o problemu. Ovo nije dovoljno da bih pitao.

[tomat]

i need help.

imam tri nezavisne grupe u eksperimentu (6 tačaka po grupi) a rezultati mi ne ukazuju na normalnu distribuciju, što bi valjda značilo da treba da koristim neki neparametrijski test za poređenje? pitanje: da li je kruskall wallis OK?
a da za post-poređenje između grupa koristim dunn's multiple comparisons test?

odma da se ogradim i kažem da mi je poznavanje statistike vrlo vrlo ograničeno.

ako se dobro sećam, prema ovome kako si opisala problem (a što reče Mića, ne bi bilo zgoreg da pošalješ više detalja), trebalo bi da može. Mann-Whitney U je valjda za poređenje dve grupe (kada nema normalne raspodele), a Kruskall-Wallis za višestruko poređenje, odnosno tri ili više grupa.

kad sam ja nešto prčkao oko toga, naleteo sam da Kruskall-Wallis takođe zahteva proveru homogenosti varijanse, samo se koristi neparametarska verzija Levenovog testa, gde se provera homogenost rangova ili tako nešto, ne mogu tačno da se setim.

[Meho Krljic]

Computer generated math proof is largest ever at 200 terabytes

(Phys.org)—A trio of researchers has solved a single math problem by using a supercomputer to grind through over a trillion color combination possibilities, and in the process has generated the largest math proof ever—the text of it is 200 terabytes in size. In their paper uploaded to the preprint server arXiv, Marijn Heule with the University of Texas, Oliver Kullmann with Swansea University and Victor Marek with the University of Kentucky outline the math problem, the means by which a supercomputer was programmed to solve it, and the answer which the proof was asked to provide.

The math problem has been named the boolean Pythagorean Triples problem and was first proposed back in the 1980's by mathematician Ronald Graham. In looking at the Pythagorean formula: a2 + b2 = c2, he asked, was it possible to label each a non-negative integer, either blue or red, such that no set of integers a, b and c were all the same color. He offered a reward of $100 to anyone who could solve the problem.

To solve this problem the researchers applied the Cube-and-Conquer paradigm, which is a hybrid of the SAT method for hard problems. It uses both look-ahead techniques and CDCL solvers. They also did some of the math on their own ahead of giving it over to the computer, by using several techniques to pare down the number of choices the supercomputer would have to check, down to just one trillion (from 102,300). Still the 800 processor supercomputer ran for two days to crunch its way through to a solution. After all its work, and spitting out the huge data file, the computer proof showed that yes, it was possible to color the integers in multiple allowable ways—but only up to 7,824—after that point, the answer became no.

While technically, the team, along with their computer did create a proof for the problem, questions remain, the first of which is, is the proof really a proof if it does not answer why there is a cut-off point at 7,825, or even why the first stretch is possible? Strictly speaking, it is, the team used another computer program to verify the results, and the proof did give a definitive answer to the original question—which caused Graham to make good on his offer by handing over the $100 to the research team—but, nobody can read the proof (or other similar but smaller proofs also generated by computers but which are still too large for a human to read) which begs the philosophical question, does it really exist?

Explore further: Computer generated math proof is too large for humans to check

More information: Solving and Verifying the boolean Pythagorean Triples problem via Cube-and-Conquer, arXiv:1605.00723 [cs.DM] arxiv.org/abs/1605.00723

Abstract
The boolean Pythagorean Triples problem has been a longstanding open problem in Ramsey Theory: Can the set N = {1,2,...} of natural numbers be divided into two parts, such that no part contains a triple (a,b,c) with a2+b2=c2 ? A prize for the solution was offered by Ronald Graham over two decades ago.
We solve this problem, proving in fact the impossibility, by using the Cube-and-Conquer paradigm, a hybrid SAT method for hard problems, employing both look-ahead and CDCL solvers. An important role is played by dedicated look-ahead heuristics, which indeed allowed to solve the problem on a cluster with 800 cores in about 2 days.
Due to the general interest in this mathematical problem, our result requires a formal proof. Exploiting recent progress in unsatisfiability proofs of SAT solvers, we produced and verified a proof in the DRAT format, which is almost 200 terabytes in size. From this we extracted and made available a compressed certificate of 68 gigabytes, that allows anyone to reconstruct the DRAT proof for checking.

via Nature




Read more at: http://phys.org/news/2016-05-math-proof-largest-terabytes.html#jCp

[Meho Krljic]

Mathematicians finally starting to understand epic ABC proof 

It has taken nearly four years, but mathematicians are finally starting to comprehend a mammoth proof that could revolutionise our understanding of the deep nature of numbers.
The 500-page proof was published online by Shinichi Mochizuki of Kyoto University, Japan in 2012 and offers a solution to a longstanding problem known as the ABC conjecture, which explores the fundamental relationships between numbers, addition and multiplication beginning with the simple equation a + b = c.
Mathematicians were excited by the proof but struggled to get to grips with Mochizuki's "Inter-universal Teichmüller Theory" (IUT), an entirely new realm of mathematics he had developed over decades in order to solve the problem. A meeting held last year at the University of Oxford, UK with the aim of studying IUT ended in failure, in part because Mochizuki doesn't want to streamline his work to make it easier to comprehend, and because of a culture clash between Japanese and western ways of studying mathematics.
Now a second meeting, held last month at his home ground in Kyoto, has proved more successful. "It definitely went better than expected," says Ivan Fesenko of the University of Nottingham, UK, who helped organise the meeting.
  The breakthrough seems to have come from Mochizuki explaining his theory in person. He refuses to travel abroad, only speaking via Skype at the Oxford meeting, which had made it harder for mathematicians outside Japan to get to grips with his work. "It was the key part of the meeting," says Fesenko. "He was climbing the summit of his theory, and pulling other participants with him, holding their hands."
Glimmer of understanding At least 10 people now understand the theory in detail, says Fesenko, and the IUT papers have almost passed peer review so should be officially published in a journal in the next year or so. That will likely change the attitude of people who have previously been hostile towards Mochizuki's work, says Fesenko. "Mathematicians are very conservative people, and they follow the traditions. When papers are published, that's it."
"There are definitely people who understand various crucial parts of the IUT," says Jeffrey Lagarias of the University of Michigan, who attended the Kyoto meeting, but was not able to absorb the entire theory in one go. "More people outside Japan have incentive to work to understand IUT as it is presented, all 500 pages of it, making use of new materials at the various conferences."
But many are still not willing devote the time Mochizuki demands to understand his work. "The experts are still on the fence," says Lagarias. "They are waiting for someone else to read the proof and asking why it cannot be made easier to understand."
It is likely that the IUT papers will be published in a Japanese journal, says Fesenko, as Mochizuki's previous work has been. That may affect its reception by the wider community. "Certainly which journal they are published in will have something to do with how the math community reacts," says Lagarias.
The glimmer of understanding that has started to emerge is well worth the effort, says Fesenko. "I expect that at least 100 of the most important open problems in number theory will be solved using Mochizuki's theory and further development."
But it will likely be many decades before the full impact of Mochizuki's work on number theory can be felt. "The magnitude of the number of new structures and ideas in IUT will take years for the math community to absorb," says Lagarias.

[Aco Popara Zver]

[Meho Krljic]

This ancient Babylonian tablet may contain the first evidence of trigonometry

[Meho Krljic]

Mathematicians Race to Debunk German Man Who Claimed to Solve One of the Most Important Computer Science Questions of Our Time

[Meho Krljic]

Mathematicians Race to Debunk German Man Who Claimed to Solve One of the Most Important Computer Science Questions of Our Time

Comments:

The proof is wrong. I shall elaborate precisely what the mistake is. For doing this, I need some time. I shall put the explanation on my homepage

[Truman]

https://courses.edx.org/courses/course-v1:Microsoft+DEV262x+1T2018/progress

Топло препоручујем овај курс из логике и рачунарског размишљања.

[Васа С. Тајчић]

У којој се школи уче Диофантове квадратне једначине?

[Truman]

https://www.nytimes.com/crosswords/game/set

Мало занимације за сваки дан, није лоше да се вежба...

[mac]

Jednom kad skapiraš sistem pretrage malo izgubi draž.

[Truman]

Мислим да сам га већ скапирао. Данас сам овај први проблем решио за једно два минута. Ал зато крећем од сутра да вежбам напредније проблеме. Него прочитах на блогу једне програмерке да она свако јутро вежба вијуге тако па рек'о да пробам.

[mac]

Bugari napravili par sajtova sa dovoljno glavolomki da ti traju ceo dan. Svaka glavolomka ima posebnu adresu, što mi je neobično, ali možda ima veze sa većom vidljivošću s gugla. Najlepše je što povremeno dodaju nove tipove glavolomki. Trude se ljudi svojski. Ako vam neki problem bude pretežak dajte ovde tip problema i identifikator problema, pa možemo zajedno da se mučimo. Evo adrese jedne od glavolomki, a do ostalih možete da dođete sa ove stranice:

https://www.puzzle-bridges.com/

[Meho Krljic]

Walter Bradley Center Fellow Discovers Longstanding Flaw in an Aspect of Elementary Calculus         

For those interested in the technical details, the second derivative of y with respect to x has traditionally had the notation "d² y/dx ²". While this notation is expressed as a fraction, the problem is that it doesn't actually work as a fraction.  The problem is well-known but it has been generally assumed that there is no way to express the second derivative in fraction form.  It has been thought that differentials (the fundamental "dy" and "dx" that calculus works with) were not actual values and therefore they aren't actually in ratio with each other. Because of these underlying assumptions, the fact that you could not treat the second derivative as a fraction was not thought to be an anomaly.  However, it turns out that, with minor modifications to the notation, the terms of the second derivative (and higher derivatives) can indeed be manipulated as an algebraic fraction.  The revised notation for the second derivative is "(d ² y/dx ²) – (dy/dx)(d ² x/dx ²)".

Rešenje je ovde gore od problema, rekli bi cinici 

Ali opet, pošto ima mnogo ludaka koji samo čekaju da pokažu prstom da "matematika nije tačna" i obruše se na nauku kao na izmišljotinu novog sveckog poretka, masona, Jevreja, feministkinja i pedera, onda je dobro da se ovakve stvari rešavaju.

[Truman]

Што каже Веља Абрамовић, сва та описна математика не вреди ништа!

[Meho Krljic]

Šta je opisna matematika i kako izvodi spadaju u nju?

[Truman]

Описна је оно што описује неку математичку форму. Нпр. Интеграли описују површ графикона ако се добро сећам. И то су људи измислили
Веља сматра да је права математика она генеричка, тј. она која одражава законе природе попут златног пресека.

[Meho Krljic]

Pošto sam ja izrazito slab s matmatikom, priznajem da ne razumem tu distinkciju. Šta bi bio izraz za zapreminu lopte? On opisuje geometrijsko telo ali to telo je sveprisutno u prirodi.

[Truman]

И Веља у својој физици бави запремином лопте ( https://www.youtube.com/watch?v=PES6G9a2UZE ) тако да верујем да је то генеричка математика. Ја сам рецимо изразито слаб у физици тако да не могу да проценим колико су његови прорачуни тачни. Веља тврди да су интеграли, диференцијали, статистика описна математика нижег реда која нема везе са природним законима.

[tomat]

kako onda gomila prirodnih zakona opisana izvodima i integralima?

[Truman]

Питај Вељу!!

[Васа С. Тајчић]

До формуле (1) на 219 страни коју аутори зову екстремно неинтуитивном, долази се простом заменом променљивих

Вреди приметити замену диференцијала у оператору диференцирања.

[Truman]

Немој Васо, подсећа ме на Теорије цена!

[Truman]

радим неки курс из дата сајнс па би ми користила помоћ. Имате две четворостране коцке ( личи највише на пирамиду ). Задатак је следећи:
Assume we roll 2 four sided dice. What is P({first roll larger than second roll})? Answer in reduced fraction form - eg 1/5 instead of 2/10.P је вероватноћа изражена у разломку.

[mac]

Napravi matricu gde je jedna strana rezultat jedne kocke, a druga rezultat druge. Ostatak rešenja sam izbelio:

Ćelije matrice su verovatnoće da se desi taj rezultat, i trebalo bi da svaka ćelija ima verovatnoću 1/16. Slučajevi kada je rezultat jedne kocke veći od rezultata druge su svi slučajevi iznad (ili ispod, svejedno) glavne dijagonale. Tih slučajeva ima 6, pa je ukupna verovatnoća 6/16=0.375

[Truman]

хехе, браво мек! Зна се који главни форумски мозак. Иначе сам мало варао, тј. у међувремену сам нашао одговор на овом сајту:
https://stackexchange.com/Овај сајт му дође као мајка stackoverflow. Регистровао сам се истим налогом који имам на stackoverflow.

[Truman]

мек, твој супер мозак ће ми опет требати!
Given two decks of 52 playing cards, you flip one over from each deck. Assuming fair decks, what is P({the two cards are the same suit})?
решење је 1/4 али га не капирам. То је шанса да у једном шпилу извучеш једну боју. А за два шпила да је иста зар не би требало да је вероватноћа 1/16?

[mac]

Verovatnoća da dve karte budu pik je 1/16, a toliko je i da budu herc, tref, i karo. Kad sabereš ovo dobiješ četvrtinu.

[Truman]

Видиш, ово је једини задатак који нисам умео да решим. Капирам да је то зато што делује контраинтуитивно. И даље ми делује након твог објашњења.

[Onaj stari Sendmen]

Može i jednostavnije do istog rešenja: verovatnoća da druga karta bude neka od četiri boje je 1/4. A to je upravo i verovatnoća da se složi s bojom prve karte, pošto je ona u nekoj od četiri boje. Trivijalno.

[Truman]

Тривијално, али треба наштеловати мозак да размишља на тај начин.

[mac]

Da nisi rekao rešenje ja bih opet krenuo od tabele. Tako je manja šansa da pogrešiš, ali se malo više pomučiš. I to bih možda krenuo od tabele 54x54, za svaki slučaj. Redovi su ishod prvog izvlačenja a kolone ishod drugog. Ćelije su verovatnoće da se taj ishod desi. Pošto su izvlačenja međusobno nezavisna u svakoj ćeliji je ista vrednost, proizvod verovatnoća prvog i drugog, 1/(52x52). U zbiru daju 1, kao što i treba da bude. Nas interesuju ishodi sa istim bojama, a to su 4 regije u tabeli veličina 13x13. Zbir njihovih verovatnoća je (4x13x13)x(1/(52x52))=1/4.

[Truman]

хехе, ипак ми је простије оно претходно решење. Само да не мора да се узима оловка у руке.