# Mathematicians Open a New Front on an Ancient Number Problem | Quanta Magazine (2023)

As a high school student in the mid-1990s, Pace Nielsen encountered a mathematical question that he’s still struggling with to this day. But he doesn’t feel bad: The problem that captivated him, called the odd perfect number conjecture, has been around for more than 2,000 years, making it one of the oldest unsolved problems in mathematics.

Part of this problem’s long-standing allure stems from the simplicity of the underlying concept: A number is perfect if it is a positive integer, n, whose divisors add up to exactly twice the number itself, 2n. The first and simplest example is 6, since its divisors — 1, 2, 3 and 6 — add up to 12, or 2 times 6. Then comes 28, whose divisors of 1, 2, 4, 7, 14 and 28 add up to 56. The next examples are 496 and 8,128.

Leonhard Euler formalized this definition in the 1700s with the introduction of his sigma (σ) function, which sums the divisors of a number. Thus, for perfect numbers, σ(n) = 2n.

But Pythagoras was aware of perfect numbers back in 500 BCE, and two centuries later Euclid devised a formula for generating even perfect numbers. He showed that if p and 2p − 1 are prime numbers (whose only divisors are 1 and themselves), then 2p−1 × (2p − 1) is a perfect number. For example, if p is 2, the formula gives you 21 × (22 − 1) or 6, and if p is 3, you get 22 × (23 − 1) or 28 — the first two perfect numbers. Euler proved 2,000 years later that this formula actually generates every even perfect number, though it is still unknown whether the set of even perfect numbers is finite or infinite.

Nielsen, now a professor at Brigham Young University (BYU), was ensnared by a related question: Do any odd perfect numbers (OPNs) exist? The Greek mathematician Nicomachus declared around 100 CE that all perfect numbers must be even, but no one has ever proved that claim.

Like many of his 21st-century peers, Nielsen thinks there probably aren’t any OPNs. And, also like his peers, he does not believe a proof is within immediate reach. But last June he hit upon a new way of approaching the problem that might lead to more progress. It involves the closest thing to OPNs yet discovered.

## A Tightening Web

Nielsen first learned about perfect numbers during a high school math competition. He delved into the literature, coming across a 1974 paper by Carl Pomerance, a mathematician now at Dartmouth College, which proved that any OPN must have at least seven distinct prime factors.

“Seeing that progress could be made on this problem gave me hope, in my naiveté, that maybe I could do something,” Nielsen said. “That motivated me to study number theory in college and try to move things forward.” His first paper on OPNs, published in 2003, placed further restrictions on these hypothetical numbers. He showed not only that the number of OPNs with k distinct prime factors is finite, as had been established by Leonard Dickson in 1913, but that the size of the number must be smaller than 24k.

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These were neither the first nor the last restrictions established for the hypothetical OPNs. In 1888, for instance, James Sylvester proved that no OPN could be divisible by 105. In 1960, Karl K. Norton proved that if an OPN is not divisible by 3, 5 or 7, it must have at least 27 prime factors. Paul Jenkins, also at BYU, proved in 2003 that the largest prime factor of an OPN must exceed 10,000,000. Pascal Ochem and Michaël Rao have determined more recently that any OPN must be greater than 101500 (and then later pushed that number to 102000). Nielsen, for his part, showed in 2015 that an OPN must have a minimum of 10 distinct prime factors.

Even in the 19th century, enough constraints were in place to prompt Sylvester to conclude that “the existence of [an odd perfect number] — its escape, so to say, from the complex web of conditions which hem it in on all sides — would be little short of a miracle.” After more than a century of similar developments, the existence of OPNs looks even more dubious.

“Proving that something exists is easy if you can find just one example,” said John Voight, a professor of mathematics at Dartmouth. “But proving that something does not exist can be really hard.”

The main approach so far has been to look at all the conditions placed upon OPNs to see if at least two are incompatible — to show, in other words, that no number can satisfy both restriction A and restriction B. “The patchwork of conditions established so far makes it extremely unlikely that [an OPN] is out there,” Voight said, echoing Sylvester. “And Pace has, for a number of years, been adding to that list of conditions.”

Unfortunately, no incompatible properties have yet been found. So in addition to needing more restrictions on OPNs, mathematicians probably need new strategies, too.

To this end, Nielsen is already considering a new plan of attack based on a common tactic in mathematics: learning about one set of numbers by studying close relatives. With no OPNs to study directly, he and his team are instead analyzing “spoof” odd perfect numbers, which come very close to being OPNs but fall short in interesting ways.

## Tantalizing Near Misses

The first spoof was found in 1638 by René Descartes — among the first prominent mathematicians to consider that OPNs might actually exist. “I believe that Descartes was trying to find an odd perfect number, and his calculations led him to the first spoof number,” said William Banks, a number theorist at the University of Missouri. Descartes apparently held out hope that the number he crafted could be modified to produce a genuine OPN.

But before we dive into Descartes’ spoof, it’s helpful to learn a little more about how mathematicians describe perfect numbers. A theorem dating back to Euclid states that any integer greater than 1 can be expressed as a product of prime factors, or bases, raised to the correct exponents. So we can write 1,260, for example, in terms of the following factorization: 1,260 = 22 × 32 × 51 × 71, rather than listing all 36 individual divisors.

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If a number takes this form, it becomes much easier to calculate Euler’s sigma function summing its divisors, thanks to two relationships also proved by Euler. First, he demonstrated that σ(a × b) = σ(a) × σ(b), if and only if a and b are relatively prime (or coprime), meaning that they share no prime factors; for example, 14 (2 × 7) and 15 (3 × 5) are coprime. Second, he showed that for any prime number p with a positive integer exponent a, σ(pa) = 1 + p + p2 + … pa.

So, returning to our previous example, σ(1,260) = σ(22 × 32 × 51 × 71) = σ(22) × σ(32) × σ(51) × σ(71) = (1 + 2 + 22)(1 + 3 + 32)(1 + 5)(1 + 7) = 4,368. Note that σ(n), in this instance, is not 2n, which means 1,260 is not a perfect number.

Now we can examine Descartes’ spoof number, which is 198,585,576,189, or 32 × 72 × 112 × 132 × 22,0211. Repeating the above calculations, we find that σ(198,585,576,189) = σ(32 × 72 × 112 × 132 × 22,0211) = (1 + 3 + 32)(1 + 7 + 72)(1 + 11 + 112)(1 + 13 + 132)(1 + 22,0211) = 397,171,152,378. This happens to be twice the original number, which means it appears to be a real, live OPN — except for the fact that 22,021 is not actually prime.

That’s why Descartes’ number is a spoof: If we pretend that 22,021 is prime and apply Euler’s rules for the sigma function, Descartes’ number behaves just like a perfect number. But 22,021 is actually the product of 192 and 61. If Descartes’ number were correctly written as 32 × 72 × 112 ×132 × 192 × 611, then σ(n) would not equal 2n. By relaxing some of the normal rules, we end up with a number that appears to satisfy our requirements — and that’s the essence of a spoof.

It took 361 years for a second spoof OPN to come to light, this one thanks to Voight in 1999 (and published four years later). Why the long lag time? “Finding these spoof numbers is akin to finding odd perfect numbers; both are arithmetically complex in similar ways,” Banks said. Nor was it a priority for many mathematicians to look for them. But Voight was inspired by a passage in Richard Guy’s book Unsolved Problems in Number Theory, which sought more examples of spoofs. Voight gave it a try, eventually coming up with his spoof, 34 × 72 × 112 × 192 × (−127)1, or −22,017,975,903.

Unlike in Descartes’ example, all the divisors are prime numbers, but this time one of them is negative, which is what makes it a spoof rather than a true OPN.

After Voight gave a seminar at BYU in December 2016, he discussed this number with Nielsen, Jenkins and others. Shortly thereafter, the BYU team embarked on a systematic, computationally based search for more spoofs. They would choose the smallest base and exponent to start from, such as 32, and their computers would then sort through the options for any additional bases and exponents that would result in a spoof OPN. Nielsen assumed that the project would merely provide a stimulating research experience for students, but the analysis yielded more than he anticipated.

## Sifting Through the Possibilities

After employing 20 parallel processors for three years, the team found all possible spoof numbers with factorizations of six or fewer bases — 21 spoofs altogether, including the Descartes and Voight examples — along with two spoof factorizations with seven bases. Searching for spoofs with even more bases would have been impractical — and extremely time-consuming — from a computational standpoint. Nevertheless, the group amassed a sufficient sample to discover some previously unknown properties of spoofs.

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The group observed that for any fixed number of bases, k, there is a finite number of spoofs, consistent with Dickson’s 1913 result for full-fledged OPNs. “But if you let k go to infinity, the number of spoofs goes to infinity too,” Nielsen said. That was a surprise, he added, given that he didn’t know going into the project that it would turn up a single new odd spoof — let alone show that the number of them is infinite.

Another surprise stemmed from a result first proved by Euler, showing that all the prime bases of an OPN are raised to an even power except for one — called the Euler power — which has an odd exponent. Most mathematicians believe that the Euler power for OPNs is always 1, but the BYU team showed it can be arbitrarily large for spoofs.

Some of the “bounty” obtained by this team came from relaxing the definition of a spoof, as there are no ironclad mathematical rules defining them, except that they must satisfy the Euler relation, σ(n) = 2n. The BYU researchers allowed non-prime bases (as with the Descartes example) and negative bases (as with the Voight example). But they also bent the rules in other ways, concocting spoofs whose bases share prime factors: One base could be 72, for instance, and another 73, which are written separately rather than combined as 75. Or they had bases that repeat, as occurs in the spoof 32 × 72 × 72 × 131 × (−19)2. The 72 × 72 term could have been written as 74, but the latter would not have resulted in a spoof because the expansions of the modified sigma function are different.

Given the significant deviations between spoofs and OPNs, one might reasonably ask: How could the former prove helpful in the search for the latter?

## A Path Forward?

In essence, spoof OPNs are generalizations of OPNs, Nielsen said. OPNs are a subset sitting within a broader family that includes spoofs, so an OPN must share every property of a spoof, while possessing additional properties that are even more restrictive (such as the stipulation that all bases must be prime).

“Any behavior of the larger set has to hold for the smaller subset,” Nielsen said. “So if we find any behaviors of spoofs that do not apply to the more restricted class, we can automatically rule out the possibility of an OPN.” If one could show, for instance, that spoofs must be divisible by 105 — which can’t be true for OPNs (as Sylvester demonstrated in 1888) — then that would be it. Problem solved.

So far, though, they’ve had no such luck. “We’ve discovered new facts about spoofs, but none of them undercut the existence of OPNs,” Nielsen said, “although that possibility still remains.” Through further analysis of currently known spoofs, and perhaps by adding to that list in the future — both avenues of research established by his work — Nielsen and other mathematicians might uncover new properties of spoofs.

Banks thinks this approach is worth pursuing. “Investigating odd spoof numbers could be useful in understanding the structure of odd perfect numbers, if they exist,” he said. “And if odd perfect numbers don’t exist, the study of odd spoof numbers might lead to a proof of their nonexistence.”

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Other OPN experts, including Voight and Jenkins, are less sanguine. The BYU team did “a great job,” Voight said, “but I’m not sure we’re any closer to having a line of attack on the OPN problem. It is indeed a problem for the ages, [and] perhaps it will remain so.”

Paul Pollack, a mathematician at the University of Georgia, is also cautious: “It would be great if we could stare at the list of spoofs and see some property and somehow prove there are no OPNs with that property. That would be a beautiful dream if it works, but it seems too good to be true.”

It is a long shot, Nielsen conceded, but if mathematicians are ever going to solve this ancient problem, they need to try everything. Besides, he said, the concerted study of spoofs is just getting started. His group took some early steps, and they already discovered unexpected properties of these numbers. That makes him optimistic about uncovering even more “hidden structure” within spoofs.

Already, Nielsen has identified one possible tactic, based on the fact that every spoof found to date, except for Descartes’ original example, has at least one negative base. Proving that all other spoofs must have a negative base would in turn prove that no OPNs exist — since the bases of OPNs, by definition, must be both positive and prime.

“That sounds like a harder problem to solve,” Nielsen said, because it pertains to a larger, more general category of numbers. “But sometimes when you convert a problem to a seemingly more difficult one, you can see a path to a solution.”

Patience is required in number theory, where the questions are often easy to state but difficult to solve. “You have to think about the problem, maybe for a long while, and care about it,” Nielsen said. “We are making progress. We’re chipping away at the mountain. And the hope is that if you keep chipping away, you might eventually find a diamond.”

## FAQs

### What are the 7 unsolved math problems? ›

Clay “to increase and disseminate mathematical knowledge.” The seven problems, which were announced in 2000, are the Riemann hypothesis, P versus NP problem, Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier-Stokes equation, Yang-Mills theory, and Poincaré conjecture.

What is the unsolved math question? ›

One of the greatest unsolved mysteries in math is also very easy to write. Goldbach's Conjecture is, “Every even number (greater than two) is the sum of two primes.” You check this in your head for small numbers: 18 is 13+5, and 42 is 23+19. Computers have checked the Conjecture for numbers up to some magnitude.

What is the oldest unsolved math problem? ›

But he doesn't feel bad: The problem that captivated him, called the odd perfect number conjecture, has been around for more than 2,000 years, making it one of the oldest unsolved problems in mathematics.

What is the perfect number conjecture? ›

Conjecture 1. Euclid's rule gives all perfect numbers; in particular, no odd number is perfect. In addition to his conjecture about the nonexistence of odd perfect numbers, Nicomachus, along with Theon of Smyrna (ca. 70-135), distinguished between deficient and abundant numbers .

Has 3X 1 been solved? ›

After that, the 3X + 1 problem has appeared in various forms. It is one of the most infamous unsolved puzzles in the word. Prizes have been offered for its solution for more than forty years, but no one has completely and successfully solved it .

What is the hardest math ever? ›

The Riemann Hypothesis, famously called the holy grail of mathematics, is considered to be one of the toughest problems in all of mathematics.

What is the hardest math class in college? ›

Calculus is, according to Wikipedia, “ … the mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations.” BUT, don't give up all hope if you need this class for your degree.

What is the hardest easy math problem? ›

The Collatz Conjecture is the simplest math problem no one can solve — it is easy enough for almost anyone to understand but notoriously difficult to solve. So what is the Collatz Conjecture and what makes it so difficult? Veritasium investigates.

Who solved an unsolvable math problem? ›

George Dantzig
BornGeorge Bernard DantzigNovember 8, 1914 Portland, Oregon, US
DiedMay 13, 2005 (aged 90) Stanford, California, US
CitizenshipAmerican
Alma materUniversity of Maryland (BS) University of Michigan (MS) University of California, Berkeley (PhD)
11 more rows

What is the longest proof ever? ›

The Stampede supercalculator used for solving the "Boolean Pythagorean triples problem." Researchers use computers to create the world's longest proof, and solve a mathematical problem that had remained open for 35 years. It would take 10 billion years for a human being to read it.

### What are the 6 hardest math problems? ›

Millennium Prize Problems
• Birch and Swinnerton-Dyer conjecture.
• Hodge conjecture.
• Navier–Stokes existence and smoothness.
• P versus NP.
• Riemann hypothesis.
• Yang–Mills existence and mass gap.

What are the 5 unsolved math problems? ›

The problems consist of the Riemann hypothesis, Poincaré conjecture, Hodge conjecture, Swinnerton-Dyer Conjecture, solution of the Navier-Stokes equations, formulation of Yang-Mills theory, and determination of whether NP-problems are actually P-problems.

What is the most mathematically perfect number? ›

perfect number, a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3. Other perfect numbers are 28, 496, and 8,128. The discovery of such numbers is lost in prehistory.

What is the biggest perfect number? ›

At the moment the largest known Mersenne prime is 2 82 589 933 − 1 2^{82 589 933} - 1 282 589 933−1 (which is also the largest known prime) and the corresponding largest known perfect number is 2 82 589 932 ( 2 82 589 933 − 1 ) 2^{82 589 932} (2^{82 589 933} - 1) 282 589 932(282 589 933−1).

Why is 28 a perfect number? ›

A number is perfect if all of its factors, including 1 but excluding itself, perfectly add up to the number you began with. 6, for example, is perfect, because its factors — 3, 2, and 1 — all sum up to 6. 28 is perfect too: 14, 7, 4, 2, and 1 add up to 28. But perfect numbers aren't common at all.

What is the hardest math equation ever solved? ›

“There are no whole number solutions to the equation xn + yn = zn when n is greater than 2.” Otherwise known as “Fermat's Last Theorem,” this equation was first posed by French mathematician Pierre de Fermat in 1637, and had stumped the world's brightest minds for more than 300 years.

Who proved Collatz conjecture? ›

On September 8, Terence Tao posted a proof showing that — at the very least — the Collatz conjecture is “almost” true for “almost” all numbers.

Who solved the million dollar math problem? ›

To date, the only Millennium Prize problem to have been solved is the Poincaré conjecture. The Clay Institute awarded the monetary prize to Russian mathematician Grigori Perelman in 2010. However, he declined the award as it was not also offered to Richard S. Hamilton, upon whose work Perelman built.

What's the answer to x3 y3 z3 K? ›

The equation x3+y3+z3=k is known as the sum of cubes problem. For decades, a math puzzle has stumped the smartest mathematicians in the world. x3+y3+z3=k, with k being all the numbers from one to 100, is a Diophantine equation that's sometimes known as "summing of three cubes." ∴ The required result will be 3xyz.

What is the answer for 3x 1? ›

In the 3x+1 problem, no matter what number you start with, you will always eventually reach 1. problem has been shown to be a computationally unsolvable problem.

### What is the easiest math in the world? ›

Easiest Math Problems Ever
1. 1 1+1=2. We (my partner and I) think this question should be number one because in Preschool, what you learn is based off of the number 1. ...
2. 2 0+0=0. ...
3. 3 What is 1? ...
4. 4 1+2=3. ...
5. 5 1=1. ...
6. 6 1+0=1. ...
7. 7 0-0=0. ...
8. 8 9-9=0.

What is the lowest math in college? ›

Entry-level math in college is considered the stepping stone to more advanced math. Algebra 1, trigonometry, geometry, and calculus 1 are the basic math classes. Once you have successfully navigated through these courses, you can trail blazed through more advanced courses.

What is the toughest college degree? ›

Engineering. A number of engineering majors are known to be extremely difficult including mechanical engineering, petroleum engineering, bioengineering, biomedical engineering, aerospace engineering, and chemical engineering (but not limited to these, either!).

What is the most failed course in high school? ›

Algebra is the single most failed course in high school, the most failed course in community college, and, along with English language for nonnative speakers, the single biggest academic reason that community colleges have a high dropout rate.

What is the most known math problem? ›

Dr. Wiles demonstrates to a group of stunned mathematicians that he has provided the proof of Fermat's Last Theorem (the equation x" + y" = z", where n is an integer greater than 2, has no solution in positive numbers), a problem that has confounded scholars for over 350 years.

Why is math the hardest? ›

Because math involves using plenty of multi-step processes to solve problems, being able to master it takes a lot more practice than other subjects. Having to repeat a process over and over again can quickly bore some children and this may make them become impatient with math.

Who was the last person to know all of math? ›

Henri Poincaré

Who is a mathematician hidden figure? ›

Katherine Johnson, pioneering NASA mathematician of 'Hidden Figures' fame, dies at 101. Katherine Johnson, whose career making vital calculations for NASA was immortalized in the 2016 book and movie "Hidden Figures," has died at 101.

What is the world largest math? ›

The problem centres around the Pythagorean formula a2 + b2 = c2, where a and b are the shorter sides of a triangle, and c is the hypotenuse, or longest side.

What's the longest math equation? ›

What is the world's longest equation? Answer – The Boolean Pythagorean Triples issue was initially introduced in the 1980s by California-based mathematician Ronald Graham is the longest arithmetic equation, according to Sciencealert, and includes roughly 200 gigabytes of text.

### What is the longest proof of 1 1 2? ›

Russell & Whitehead's 360-page proof that 1+1=2

It's all done in formal logic, and must surely be one of the longest proofs relative to the length and complexity of the statement it's proving.

What is the hardest math in high school? ›

What is the Hardest Math Class in High School? In most cases, you'll find that AP Calculus BC or IB Math HL is the most difficult math course your school offers. Note that AP Calculus BC covers the material in AP Calculus AB but also continues the curriculum, addressing more challenging and advanced concepts.

Has anyone solved the Riemann Hypothesis? ›

The Riemann hypothesis, a formula related to the distribution of prime numbers, has remained unsolved for more than a century.

Are all math problems solvable? ›

There are many unsolved problems in mathematics. Several famous problems which have recently been solved include: 1. The Pólya conjecture (disproven by Haselgrove 1958, smallest counterexample found by Tanaka 1980).

What is the most picked number between 1 and 100? ›

The most random two-digit number is 37, When groups of people are polled to pick a “random number between 1 and 100”, the most commonly chosen number is 37.

Why is 73 the best number? ›

“The best number is 73,” Cooper explained in the episode. “Why? 73 is the 21st prime number. Its mirror, 37, is the 12th, and its mirror, 21, is the product of multiplying seven and three ... and in binary, 73 is a palindrome, 1001001, which backwards is 1001001.”

What is the hardest math problem with numbers? ›

For decades, a math puzzle has stumped the smartest mathematicians in the world. x3+y3+z3=k, with k being all the numbers from one to 100, is a Diophantine equation that's sometimes known as "summing of three cubes."

Why is 9 the perfect number? ›

The number 9 is revered in Hinduism and considered a complete, perfected and divine number because it represents the end of a cycle in the decimal system, which originated from the Indian subcontinent as early as 3000 BC.

Why 7 is a perfect number? ›

In the Bible, scholars claim that God created the world in six days and used the seventh day to rest. Because of this, the number seven is used to illustrate an idea of completeness throughout the Bible. In both Islam and Judaism, there are seven heavens.

Why is 12 a perfect number? ›

It is the smallest of two known sublime numbers, which are numbers that have a perfect number of divisors whose sum is also perfect. Twelve is the number of divisors of 60 and 90, the second and third unitary perfect numbers (6 is the first).

### Why is 1089 a magic number? ›

In magic. 1089 is widely used in magic tricks because it can be "produced" from any two three-digit numbers. This allows it to be used as the basis for a Magician's Choice.

What birthday is the perfect number? ›

It just so happens that June 28th, or the 28th day of the 6th month of the year, is the only day/month combination that involves two mathematically perfect numbers: 6 and 28. The next “perfect” number doesn't occur until 496, and you won't find the fourth until you get all the way up to 8128.

Is 496 a perfect number? ›

496 is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime 31, 25 − 1, with 24 (25 − 1) yielding 496.

What is the most famous unsolved math problem? ›

The Collatz conjecture is one of the most famous unsolved mathematical problems, because it's so simple, you can explain it to a primary-school-aged kid, and they'll probably be intrigued enough to try and find the answer for themselves.

How many of the 7 millennium problems have been solved? ›

One of the seven problems has been solved, and the other six are the subject of a great deal of current research. The timing of the announcement of the Millennium Prize Problems at the turn of the century was an homage to a famous speech of David Hilbert to the International Congress of Mathematicians in Paris in 1900.

What is 3x 1 answer? ›

In the 3x+1 problem, no matter what number you start with, you will always eventually reach 1. problem has been shown to be a computationally unsolvable problem.

Who was the No 1 mathematician? ›

1. Pythagoras. The life of the famous Greek Pythagoras is somewhat mysterious. Probably born the son of a seal engraver on the island of Samos, Pythagoras has been attributed with many scientific and mathematical discoveries in antiquity.

Who is the most badass mathematician? ›

Srinivasa Ramanujan: The Man Who Knew Infinity. Greatest Indian Mathematician. Age 0: Born to a poor family in Madras, India.

What is the largest number in the world? ›

A "googol" is the number 1 followed by 100 zeroes. The biggest number with a name is a "googolplex," which is the number 1 followed by a googol zeroes.

What is the longest math proof? ›

The Stampede supercalculator used for solving the "Boolean Pythagorean triples problem." Researchers use computers to create the world's longest proof, and solve a mathematical problem that had remained open for 35 years. It would take 10 billion years for a human being to read it.

### Have any of the Millennium Prize Problems been solved? ›

To date, the only Millennium Prize problem to have been solved is the Poincaré conjecture. The Clay Institute awarded the monetary prize to Russian mathematician Grigori Perelman in 2010.

Has the Navier Stokes equation been solved? ›

Due to their complexity, it is natural to wonder how they can be solved. The reality is that no analytical solutions exist to the Navier-Stokes equations in their most general form.

Why can't 3x 1 be solved? ›

Multiply by 3 and add 1. From the resulting even number, divide away the highest power of 2 to get a new odd number T(x). If you keep repeating this operation do you eventually hit 1, no matter what odd number you began with? Simple to state, this problem remains unsolved.

What is the hardest math problem to solve? ›

“There are no whole number solutions to the equation xn + yn = zn when n is greater than 2.” Otherwise known as “Fermat's Last Theorem,” this equation was first posed by French mathematician Pierre de Fermat in 1637, and had stumped the world's brightest minds for more than 300 years.

What is the answer to 3x 1 13? ›

3x+1=13 One solution was found : x = 4 Rearrange: Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation : ...

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