Yesterday, I had the fortunate opportunity to spend some time with one of the most unusual mathematicians I have ever met--a middle-aged man born in China who, quietly, without talking much or making statements at international conferences, just solved one of the most important long-standing enigmas in mathematics: uncovering a key property of prime numbers.

A *prime number* is one that can be divided without a remainder only by 1 or by itself. The first prime is 2 (for technical reasons, 1 is not considered a prime), the next one is 3, and then come, in succession, 5, 7, 11, 13, 17, 19, and so on. We see, for example, that 19 is not divisible (without a remainder) by any integer other than 1 or 19; as compared with the composite (meaning non-prime) number 20, which is divisible by 2, 4, 5, and 10 (in addition to 1 and 20). Prime numbers are the building-blocks of our numbers, because every positive integer is either a prime number or a product of primes. The number 666 ("The number of the beast" of the Apocalypse) is composite and is the product of the primes 2, 3, again 3, and 37. The primes are thus the basic elements of our entire number system we use for everything. And, by themselves, they are not of academic interest only. To give you an idea about how important prime numbers are, every time you use your ATM card, the bank's computer verifies that you are who you claim you are through an algorithm that breaks down a huge number into a unique product of two known prime numbers (which is something that cannot quickly be done by trial and error, as an electronically-savvy thief might try to do).

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The primes have been known since early antiquity and fascinated the ancient Greeks, who marveled at their nature. What is interesting about prime numbers is that they become *rarer* as we progress through the integers. The numbers of primes in the first five blocks of 1,000 integers are: 168, 135, 127, 120, and 119. And the block of 1,000 numbers just below and including 10 million has only 53 primes. So we see that the primes decrease in number. Do they ever end? Is there a point in the number system after which there are no longer any prime numbers, as we might expect from noticing the average decrease in the density of the primes?

The answer, given by the Greek mathematician Euclid of Alexandria, as far back as 300 B.C., is *no*. Euclid proved that there are, in fact, *infinitely many* prime numbers! The proof is so elegant and understandable by anyone who can do basic arithmetic that I feel compelled to present it here.

**Euclid of Alexandria** (Wikimedia Commons)

So Euclid says: Let's assume that there are only a *finite* number of primes. In that case, there must be a *last* prime, after which the numbers, all the way to infinity, are composite (i.e., products of prime numbers, such as 666 above). Euclid then says: Let me call that last prime *p*. Now, so cleverly, Euclid writes down the following number: 2 x 3 x 5 x 7 x 11 x...x *p* + 1, that is, the product of all the prime numbers, from the first one, 2, to the last one, *p*, *plus the number 1*. And he asks the question: Is this new number a prime or not? If it is a prime, then we have just found a prime that is greater than the assumed largest prime, *p*--which is a contradiction! And if this new number is *not* a prime, then by the definition of a composite number it must be divisible by at least one of the prime numbers, i.e., by 2, by 3, by 5, by 7, by 11,...or by *p*. Call the prime number that, without remainder, divides our number above *q*. But when we *actually do the division*, we must get a remainder of 1/*q*, because we have that "1" added to the product of primes above. So in fact *q* does *not* divide our number without a remainder. Our number, therefore, cannot be composite. So here, too, we have a contradiction! This contradiction of the assumption that there exists a "last" prime number establishes the theorem: There are *infinitely many* prime numbers!

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Now, primes like company. Often we see two primes that are "neighbors" because they are separated by only one number (the separation by one is necessary because, after 2, all primes must be odd, and after an odd number there must come an even number): 5 and 7, 11 and 13, 17 and 19, and so on. So mathematicians have asked the question: Are there only a finite set of such pairs of primes, or do pairs of primes continue to infinity? And, are there always primes separated by a length of numbers, or do primes with a given "gap" between them eventually disappear as the primes become rarer? A seminal textbook on number theory, *An Introduction to the Theory of Numbers*, by the prominent British mathematician G. H. Hardy (who brought to Cambridge the Indian genius Ramanujan, for those who know that story) and his coauthor E. M. Wright, says about such conjectures that (Oxford, 1998, p. 5): "their proof or disproof is at present beyond the resources of mathematics."

Beyond the resources of mathematics? Apparently, not everyone agreed to be defeated by numbers, even if this statement was made by one of the most famous mathematicians (now deceased).

I spent the academic year 2007-8 teaching mathematics at the University of New Hampshire. And there was something that irritated me. Often, while I was teaching my class, a person would pace outside my door, distracting my students. He was like a ghost, an apparition suddenly materializing in the corridor, staring into the room, apparently deep in thought, turning on his heels, pacing in the corridor, then coming back, again peering into the classroom and startling my students, and then disappearing from view. It went on for months. So I discreetly asked my department chair, Eric Grinberg, about it, and he said: "Oh, that's Tom--Yitang Zhang; he is working on a very important theorem in number theory. Haven't you met?" And I realized that I had indeed met him when I joined the department for a year, but that he was so quiet and untalkative that I didn't remember him. Now, UNH is an excellent school, but it only had one mathematician who was famous for proving a major theorem, Ken Appel (he has died recently), who in 1976 co-proved the famous four-color map theorem, so I didn't think that it was likely that someone else in this department would do something similar. And I learned to ignore the distractions and just teach my class.

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**Tom Zhang of New Hampshire** (Univ. of New Hampshire)

Then last week I got an e-mail from Eric, who'd since moved to the University of Massachusetts, Boston: "Come hear Tom Zhang present his theorem Tuesday at 2 p.m." The announcement he attached said that Zhang had recently stunned the world of mathematics by proving an immensely important property of prime numbers. Tom was virtually unknown to mathematicians and had not published in many years. His proof of a big theorem, recently accepted for publication by the leading math journal *Annals of Mathematics* has taken the math world by surprise. So I came, eager to see how he did it, accompanied by my friend Marina Ville, a mathematician visiting from the University of Tours, in France.

We weren't disappointed. Tom Zhang was no longer the apparition haunting my class. He was a flesh-and-blood mathematician, full of energy and eloquence, and he gave us a brilliant presentation of his proof of what is known as the "bounded gap conjecture." He showed that there are *infinitely many* consecutive prime numbers that differ by no more than 70 million. This means that the gaps between prime numbers do not necessarily increase indefinitely as the numbers grow--as we might suspect from the fact that the primes become rarer as the numbers progress. There are infinitely many prime number pairs whose separation is at most 70,000,000. This is a huge breakthrough in mathematics, and mathematicians hope that Zhang's methods should apply to further proving that there are infinitely many prime pairs separated by one intervening number only.

Immediately, one wonders: why 70 million? Where does such a number come from?--It seems so arbitrary. In fact, Zhang's technical proof employed a bound on key mathematical terms in his equation, a number raised to the power 1/4 + 1/1168. The power 1/4 (meaning the fourth root of a number) had been employed by earlier mathematicians working on this hard problem. Zhang's great breakthrough was a mathematical tightrope walk: in order to prove a result about consecutive primes, he had to raise the power 1/4 just *slightly*, without disturbing other terms. He finally managed to do this--satisfy the competing needs of different terms in his equation--by increasing the exponent 1/4 by the magical amount 1/1168. "Why this number?" someone in the audience asked him. "I was tired," he answered, "and this number worked." We all laughed. It was the use of the number 1/1168 in an exponent that led to the maximal gap of 70,000,000. Presumably, when he is not tired, Zhang may improve the bound, lowering the 70 million to smaller gaps between primes, and perhaps even solve the more specific "twin prime" conjecture on the infinitude of pairs of primes separated by one number.

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Eric is a gracious host, and after Zhang's talk, he invited us for a tour of the UMB campus, situated on a tongue of land that juts into Massachusetts Bay and borders the John F. Kennedy Library. As we walked by the water's edge on a beautiful June day, a group of six mathematicians, we reached the Library and saw Kennedy's wooden sailboat displayed on the grass in front of it. Tom kept leaving our group to meditate alone for a few minutes at a time. "He's already working on his next theorem," Marina said to me. "Sure," I said, "and admiring JFK's pretty sailboat is certainly more inspiring than staring into a classroom."

**JFK's Sailboat**

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## FAQs

### Has the Riemann hypothesis been solved 2023? ›

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

**Has anyone solved the Riemann hypothesis? ›**

The Riemann hypothesis will probably remain at the top of mathematicians' wishlists for years to come. Despite its importance, **no attempts so far have made much progress**.

**What is the 137th prime number? ›**

← 136 137 138 → | |
---|---|

Ordinal | 137th (one hundred thirty-seventh) |

Factorization | prime |

Prime | 33rd |

Divisors | 1, 137 |

**What is the secret of the prime numbers? ›**

**A prime number is a whole number greater than 1 that have only 1 and itself as divisors**. The primes start 2, 3, 5, 7, 11,… and continue to pop up here and there to infinity. That is, there are infinitely many primes. This was proven by Euclid about 300 BC.

**What is the hardest math problem in the world? ›**

Today's mathematicians would probably agree that **the Riemann Hypothesis** is the most significant open problem in all of math. It's one of the seven Millennium Prize Problems, with $1 million reward for its solution.

**Why can't we prove the Riemann hypothesis? ›**

Importantly, the upper bound is dependent on the highest number of known zeroes of the Riemann Zeta Function; but **it's completely infeasible, and likely impossible, to calculate enough zeroes to limit the constant enough to prove RH**. If the Riemann Hypothesis is true, then it is only barely true.

**What is the holy grail of math? ›**

Abstract. A trustworthy proof for **the Riemann hypothesis** has been considered as the Holy Grail of Mathematics by several authors. The Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$.

**Could the Riemann hypothesis be false? ›**

Denote by \zeta the Riemann zeta function and let \Theta be the supremum of the real parts of its zeros. We demonstrate in this note that \Theta \geq \frac{3}{4}. This disproves the Riemann hypothesis, which asserts that \Theta = \frac{1}{2}.

**Why is 3x 1 a problem? ›**

The 3x+1 problem **concerns an iterated function and the question of whether it always reaches 1 when starting from any positive integer**. It is also known as the Collatz problem or the hailstone problem. . This leads to the sequence 3, 10, 5, 16, 4, 2, 1, 4, 2, 1, ... which indeed reaches 1.

**Is 2099 a prime number? ›**

**This number is a prime**. The next self prime year is 2099.

### Why is 9 a magic 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 is number 9 special? ›**

As the final numeral, the number nine holds special rank. **It is associated with forgiveness, compassion and success on the positive side as well as arrogance and self-righteousness on the negative**, according to numerologists. Though usually , numerologists do have a famous predecessor to look to.

**What is the illegal prime number? ›**

This number is a prime. This was the first known illegal prime. What people often forget is that a program (any file actually) is a string of bits (binary digits), so every program is a number.

**What is the rarest prime number? ›**

The new prime number, known as M77232917, is one million digits larger than the previous record. It is also a particularly rare type of prime called a **Mersenne prime**, meaning that it is one less than a power of two. Three is a Mersenne prime because it is a prime and is equal to 2^{2} – 1.

**Why is 911 not a prime number? ›**

**The number 911 is divisible only by 1 and the number itself**. For a number to be classified as a prime number, it should have exactly two factors.

**What does x3 y3 z3 k equal? ›**

In mathematics, entirely by coincidence, there exists a polynomial equation for which the answer, 42, had similarly eluded mathematicians for decades. The equation x^{3}+y^{3}+z^{3}=k is known as the **sum of cubes problem**.

**Has 3x 1 been solved? ›**

In 1995, Franco and Pom-erance proved that the Crandall conjecture about the aX + 1 problem is correct for almost all positive odd numbers a > 3, under the definition of asymptotic density. However, **both of the 3X + 1 problem and Crandall conjecture have not been solved yet**.

**What is x3 y3 z3? ›**

The equation x3+y3+z3=k is known as the **sum of cubes problem**. While seemingly straightforward, the equation becomes exponentially difficult to solve when framed as a "Diophantine equation" -- a problem that stipulates that, for any value of k, the values for x, y, and z must each be whole numbers.

**What would happen if the Riemann hypothesis was solved? ›**

If proved, **it would immediately solve many other open problems in number theory and refine our understanding of the behavior of prime numbers**.

**What is the prime number paradox? ›**

Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states that **every even natural number greater than 2 is the sum of two prime numbers**.

### What is the prize for solving the Riemann hypothesis? ›

MILLENNIUM PRIZE SERIES: The Millennium Prize Problems are seven mathematics problems laid out by the Clay Mathematics Institute in 2000. They're not easy—a correct solution to any one results in a **US$1,000,000** prize being awarded by the institute.

**Who is the archangel of math? ›**

**Foucault**, Archangel of Numbers.

**What is the divine symbol in math? ›**

golden ratio, also known as the golden section, golden mean, or divine proportion, in mathematics, the irrational number (1 + Square root of√5)/2, often denoted by the Greek letter **ϕ or τ**, which is approximately equal to 1.618.

**Who is the black hole of mathematics? ›**

word four, In figure (1) and (2). On basis of above explanation, we can conclude that the word FOUR is the „black hole word‟ in English and **four fundamentals signs (+ , - , × and ÷)** in Mathematics are „ the black hole signs‟ in Maths.

**Do mathematicians believe the Riemann hypothesis? ›**

**Most mathematicians believe that the Riemann hypothesis is indeed true**. Calculations so far have not yielded any misbehaving zeros that do not lie in the critical line. However, there are infinitely many of these zeros to check, and so a computer calculation will not verify all that much.

**What is the hardest math problem the Riemann hypothesis? ›**

In mathematics, the Riemann hypothesis is the **conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 12**. Many consider it to be the most important unsolved problem in pure mathematics.

**What is Riemann hypothesis in simple terms? ›**

The Riemann hypothesis **asserts that all interesting solutions of the equation**. **ζ(s) = 0**. **lie on a certain vertical straight line**. This has been checked for the first 10,000,000,000,000 solutions.

**What is the equation nobody can solve? ›**

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?

**What math problems have never been solved? ›**

**10 Math Equations That Have Never Been Solved**

- The Riemann Hypothesis. Equation: σ (n) ≤ Hn +ln (Hn)eHn. ...
- The Collatz Conjecture. Equation: 3n+1. ...
- The Erdős-Strauss Conjecture. Equation: 4/n=1/a+1/b+1/c. ...
- Equation Four. ...
- Goldbach's Conjecture. ...
- Equation Six. ...
- The Whitehead Conjecture. ...
- Equation Eight.

**What is the world's longest math problem? ›**

Andrew Wiles (UK), currently at Princeton University in New Jersey, USA, proved Fermat's Last Theorem in 1995. He showed that xn+yn=zn has no solutions in integers for n being equal to or greater than 3. The theorum was posed by Fermat in 1630, and stood for 365 years.

### How many factors of 666 are prime? ›

Factors of 666 are numbers that, when multiplied in pairs give the product as 666. There are 12 factors of 666, of which the following are its prime factors **2, 3, 37**. The Prime Factorization of 666 is 2^{1} × 3^{2} × 37^{1}.

**Why 217 is not a prime number? ›**

No, 217 is not a prime number. The number 217 is divisible by 1, 7, 31, 217. For a number to be classified as a prime number, it should have exactly two factors. Since **217 has more than two factors, i.e. 1, 7, 31, 217, it is not a prime number**.

**Why is 2027 a prime number? ›**

About the number 2027:

**2027 has only two factors which are 1 and the number itself** and hence 2027 is the prime number.

**Why do I see 1111? ›**

As previously stated, 1111 can be interpreted as **a message from your angels or the universe (or whatever higher power you believe in) that you're on the right path**. If you keep seeing 1111 everywhere, it's a sign to keep going and trust the direction you're moving in as everything is falling into place, says Kelly.

**What does 999 mean in Chinese? ›**

999 - **It has a similar meaning to '666'**. Because '9' looks like '6' upside down, so it usually mean something is even better than things we call '666'. It also has another meaning just like '99' for the same reason.

**What is the most magical number in the world? ›**

It might just be the number **137**. Those three digits, as it turns out, have long been the rare object of fascination that bridges the gulf between science and mysticism. "137 continues to fire the imagination of everyone from scientists and mystics to occultists and people from the far-flung edges of society," Arthur I.

**What number represents woman? ›**

The number 2 was symbolic of the female principle, 3 of the male; they come together in 2 + 3 = 5 as marriage. **All even numbers were female**, all odd numbers male. The number 4 represented justice.

**What is the life path of 9 in 2023? ›**

Being a number 9, you have a wealth of knowledge and are always eager to learn more. In 2023, **you will be able to put your knowledge to use in real life and achieve positive results**. People in business will have all the success; however, those in the job sector may encounter some challenges.

**Which numerology number is powerful? ›**

Since the birth of numerology in ancient Greece, the numbers **11, 22, and 33** have been revered as the master numbers – commanding an extra-strength presence in the cosmos. People with these super digits in their birth charts often rise to be high-decibel movers and shakers, spiritual leaders or community influencers.

**Why is 69 not a prime number? ›**

No, 69 is not a prime number. The number 69 is divisible by 1, 3, 23, 69. For a number to be classified as a prime number, it should have exactly two factors. **Since 69 has more than two factors, i.e. 1, 3, 23, 69, it is not a prime number**.

### What is prime vampire number? ›

A vampire prime or prime vampire number, as defined by Carlos Rivera in 2002, is a true vampire number whose fangs are its prime factors. The first few vampire primes are: **117067, 124483, 146137, 371893, 536539**. As of 2007 the largest known is the square (94892254795 × 10^{103294} + 1)^{2}, found by Jens K.

**What is the oldest prime number? ›**

The first 25 prime numbers (all the prime numbers less than 100) are: **2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97** (sequence A000040 in the OEIS).

**What is the largest perfect number found? ›**

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).

**What is powerful prime number? ›**

**11, 17, 29, 37, 41, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 163, 179, 191, 197, 223, 227, 239, 251, 269, 277, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499** (sequence A051634 in the OEIS).

**What is the largest prime number under 10000000? ›**

the largest prime numbers less than one million is **999983**.

**Why did 999 change to 9-1-1? ›**

AT&T chose the number 9-1-1, which was simple, easy to remember, dialed easily (which, with the rotary dial phones in place at the time, 999 would not), and **because of the middle 1, which indicated a special number (see also 4-1-1 and 6-1-1), worked well with the phone systems at the time**.

**Do all prime numbers end in 1 3 7 9? ›**

Apart from 2 and 5, **all prime numbers end in 1, 3, 7 or 9** – they have to, else they would be divisible by 2 or 5 – and each of the four endings is equally likely. But while searching through the primes, the pair noticed that primes ending in 1 were less likely to be followed by another prime ending in 1.

**Is 9 a twin prime? ›**

**Two prime numbers are called twin primes if there is present only one composite number between them**. Or we can also say two prime numbers whose difference is two are called twin primes. For example, (3,5) are twin primes, since the difference between the two numbers 5 – 3 = 2.

**Did Yitang Zhang solve the Riemann hypothesis? ›**

Did Zhang Yitang prove or disprove the Riemann Hypothesis? **Yitang Zhang doesn't claim to have proven the Riemann Hypothesis**, he doesn't claim to have refuted the Riemann Hypothesis, he never did claim any of those things, and he hasn't done any of those things. Zhang posted a preprint on the arXiV a few days ago (Nov.

**Is Riemann hypothesis solved by Eswaran? ›**

**The proof to the Riemann Hypothesis enables mathematicians to exactly count the prime numbers**. The methodology used by Dr Easwaran showed the 'factorisation sequence of numbers' like a 'random walk'. “This method used was actually not just number theories.

### What happens if Riemann hypothesis is proved? ›

If proved, **it would immediately solve many other open problems in number theory and refine our understanding of the behavior of prime numbers**.

**What are the 7 math millennium 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**.

**Is Yitang Zhang's proof correct? ›**

In 2007, Zhang had published a preprint paper claiming that he had proved that L(1, Χ) was much greater than (log D)^{-}^{17}(log(log D))^{-}^{1}. But **his proof turned out to be wrong after mathematicians noticed the incorrectness of a few key ideas developed in that paper.**

**Did Michael Atiyah solve the Riemann hypothesis? ›**

One of the most famous unsolved problems in mathematics likely remains unsolved. At a hotly-anticipated talk at the Heidelberg Laureate Forum today, **retired mathematician Michael Atiyah delivered what he claimed was a proof of the Riemann hypothesis**, a challenge that has eluded his peers for nearly 160 years.

**Which Indian mathematician solved Riemann hypothesis? ›**

**Dr Kumar Eswaran** first published his solution to the Riemann Hypothesis in 2016, but has received mixed responses from peers.

**Have any of the millennium problems been solved? ›**

**The only Millennium Problem that has been solved to date is the Poincare conjecture**, a problem posed in 1904 about the topology of objects called manifolds.

**What is the holy grail of number theory? ›**

**The Riemann hypothesis** is modern math's holy grail. Although it is almost incomprehensible for people without intensive math training, it describes the distribution of prime numbers among positive integers.

**What is the biggest math problem ever? ›**

Mathematicians worldwide hold the **Riemann Hypothesis of 1859** (posed by German mathematician Bernhard Riemann (1826-1866)) as the most important outstanding maths problem. The hypothesis states that all nontrivial roots of the Zeta function are of the form (1/2 + b I).

**What is the hardest math problem not solved? ›**

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.

**What is the hardest math equation ever recorded? ›**

For decades, a math puzzle has stumped the smartest mathematicians in the world. **x ^{3}+y^{3}+z^{3}=k**, with k being all the numbers from one to 100, is a Diophantine equation that's sometimes known as "summing of three cubes."