# How long is 2 to 77,232,917 minus 1

The largest known prime seems to be updated about once every two years. On December 26, 2017, the newest largest known prime number was discovered. The prior discovery was in January 7, 2016 (it was found by a machine on September 17, 2015 but no human took note of it until January 7, 2016).

The first digits of 2 to 77,232,917 minus 1

Like the previous largest prime, this freshly found largest prime number is of the form $2^p-1$. That is, it is a power of 2 minus 1. In particular, the $p$ for this new largest prime is 77,232,917. So the number is obtained by multiplying 2 by itself 77,232,917 times and subtracting 1.

This number consists of over 23 million decimal digits (23,249,425 to be precise). If writing 5 digits per inch, all the digits of this prime number would cover a stretch of highway over 73 miles in length! That’s only 5 miles under the distance of three marathons.

The previous largest prime number that was discovered in January 2016 consists of only about 22 million digits (22,338,618 to be precise). These would cover about 70.5 miles. So the newly discovered largest prime number would stretch out a further distance of about 2.5 miles!

Here’s the first 120 and the last 120 digits of this new largest prime number.

4673331833592310999883355855611155212513
2110281771449579858233859356792348052117
7207484311099740208849621368090038049317 …

(the middle 23,249,185 digits omitted)

… 2853760045187860554022233766729256792821
3196546734339594539737047636927989462799
9939614659217371136582730618069762179071

Any prime number of the form $2^p-1$ is called a Mersenne prime, in honor of the French monk Marin Mersenne, who studied these numbers more than 350 years ago. The latest largest prime is a Mersenne prime. So is the one before that. In fact, most of the record largest primes are of this type. There is a large worldwide community of volunteers who devote their free time in hunting for Mersenne primes, called Great Internet Mersenne Prime Search (GIMPS for short).

The prime number $2^{77232917}-1$ is the 50th known Mersenne prime. Its discoverer, Jonathan Pace, is a GIMPS volunteer for over 14 years. The recent discovery is exciting news. The discovery of such a large prime is akin to the scaling of a new and higher Mount Everest. The even more exiting news is that there are plenty more Mount Everest waiting to be discovered and scaled. Euclid proved around 2000 years ago that there are infinitely many prime numbers. Theoretically we know there is no such thing as the largest prime. When we speak of the largest prime, it is only the largest prime number verified by the computing resource that is currently available.

For more information about the latest new largest prime, see the website for GIMPS or Google the Internet. The news of this new discovery is reported in this piece from npr.org. A piece in a companion blog has a basic discussion on Mersenne primes.

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Dan Ma math

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$\copyright$ 2018 – Dan Ma

## A Good Will Hunting Story from China

### Video

There is an surprising and interesting mathematical story from China. At the center of the story is a migrant worker in China named Yu Jianchun who is generating excitement and awe because of his work in complex and esoteric math problems despite having received no university education, much less formal mathematical training. Facts about the story are still scanty. But he is attracting a lot of attention from academics and from the general public in China and internationally. Here is an article from CNN about Yu. Many people are seeing a parallel between Yu and the character Will Hunting in the Oscar-winning movie “Good Will Hunting”.

After studying at a vocational school, Yu Jianchun became a migrant worker going from place to place working as a parcel delivery man. He always has a passion for mathematics and he spends almost all of his spare time studying it. He is also persistent (and probably stubborn as well). He has spent 8 years working on the problem that currently garners him national and international attention. Whenever he found work in a new city, he always seek out the mathematics professors at the local university in hope of finding confirmation for his math work. He was ignored until a math professor at Zhejiang University, Cai Tianxin, invited Yu in June 2016 to present his math work at a seminar.

The math work of Yu that is generating buzz involves Carmichael numbers, which are odd integers that are prime-like (the usual term is pseudoprimes). For both theoretical and practical reasons, it is critical to test whether a given large odd integer is a prime number. Carmichael numbers are integers that are not prime but yet pass the Fermat’s test for prime numbers. So being able to weed out the Carmichael numbers from the prime numbers will be critical. Testing whether a number is a prime number is more than an intellectual curiosity. Prime numbers are the back bones of encryption systems such as the RSA algorithm, which makes online shopping safe and secure. So the study of prime numbers and Carmichael numbers has implications for information security.

Carmichael numbers are named after R. D. Carmichael who discovered 15 such numbers in 1910. These numbers are very rare. For example, of all the numbers that are less than one billion, there are only 646 Carmichael numbers. In contrast, there are 50,847,534 (over 50 millions) prime numbers below one billion. R. D. Carmichael conjectured that the number of Carmichael numbers is infinite. It was finally proven in 1994 that there are infinitely many such numbers, i.e. there is no upper bound on Carmichael numbers. No matter how big the whole number $n$, it is proven that there is always a Carmichael number larger than $n$.

What is special about Yu’s work is that he has discovered a new way to identify Carmichael numbers that is different from the classic algorithm. However, there has not been any precise mathematical statement on what Yu’s results are. So it is hard to get a sense of how special or how significant his results are. In the CNN article referenced above, William Banks, a mathematician who works with Carmichael numbers, seemed to indicate that Yu’s results are about formulas that can generate Carmichael numbers. Yet the same CNN article also indicated that Yu’s work is about an alternative method to verify Carmichael numbers. Is it a new formula for generating Carmichael numbers or is it a new test to check whether a given number is a Carmichael number?

It is likely that Yu’s results are currently being verified. Cai Tianxin, the math professor at Zhejiang University who invited Yu to give a talk, plans to publish Yu’s theory in a book on Carmichael numbers. So in time we will have a better sense of Yu’s achievement.

Another refreshing point about the story of Yu Jianchun is that Carmichael numbers are in the news! Occasionally prime numbers are in the news. For example, when someone finds a new largest prime number or when someone proves a long standing problem about prime numbers. But rarely do we see Carmichael numbers being mentioned by the major news outlets.

It is hard for me to comment on Yu’s work on Carmichael numbers since details are not available. However, Carmichael numbers are an interesting concept in number theory. Here’s an introduction to Carmichael numbers. Here’s an article on how to use Fermat’s test for prime numbers. Here’s another discussion on Carmichael numbers. Finally, here’s another discussion on Fermat’s test.

# Cheering for Yitang Zhang

If you Google the name of “Yitang Zhang” recently, you will find many entries with keywords such as “twin primes”, “twin prime conjecture”, “bounded gaps between primes”, along with “University of New Hampshire”. Here’s three of the entries: UNH professor solves ancient mathematics riddle; UNH lecturer stuns the math world; The Beauty of Bounded Gaps. Here’s a piece from New York Times: Solving a Riddle of Primes. The reason for all this attention from the mainstream media that normally don’t pay attention to math? Mr. Zhang made a huge breakthrough toward solving an ancient problem about prime numbers. This is a brief account of his achievement.

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A prime number is an integer greater than 1 that has no divisor other than 1 and the number itself. For example, 3 is a prime number because 1 and 3 are the only positive integers that evenly divide 3. However 6 is not a prime number since 3 evenly divides 6 in addition to 1 and 6. The first several prime numbers are: 2, 3, 5, 7, 11, 13, …

A pair of prime numbers are twin primes if they differ by 2, for examples, 3 and 5, 5 and 7, 11 and 13, 17 and 19 and so on.

The Greek mathematician Euclid gave the oldest known proof that there are infinitely many prime numbers around 300 BC. Euclid also conjectured that there are infinitely many pairs of twin primes.

The statement that there are infinitely many pairs of twin primes is called the Twin Prime Conjecture. Note that the Twin Prime Conjecture is saying that there are infinitely many pairs of primes such that the gap within each pair is exactly 2. Many great mathematical minds since ancient time had been trying to prove this conjecture but to no avail. But there were incremental progress throughout the twentieth century and in the first decade of the new century.

Mr. Yitang Zhang did not prove the Twin Prime Conjecture. He proved that there are infinitely many pairs of prime numbers such that the gap within each pair is bounded is at most 70,000,000 (70 millions). In other words, Mr. Yitang Zhang proved the Bounded Gaps Conjecture.

The gap of 70 million seems like a large number and may not seem all that significant to non-mathematicians. Now that there is now a proof that the gaps between pairs of primes do not have to increase without bound (as some mathematicians had suspected), mathematicians can work on narrowing the gaps (may be even reducing the gaps down to 2).

Indeed, since the announcement of Mr. Yitang Zhang’s result in May, some mathematicians had narrowed the gaps down to mere millions (in one case down to hundreds of thousands)!

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His achievement is even more impressive considering the background of Mr. Yitang Zhang. Mr. Zhang did earn a PhD in mathematics in 1991 from Purdue University. In the tough academic job market after graduation, he could not find any academic position. He worked as an accountant for a period. He even made sandwiches at a Subway shop at one point. He is currently a part-time math lecturer at the University of New Hampshire, a nice school for sure, but far from the elite club of mathematicians working at places like Harvard, Princeton and Stanford.

Some people have this notion that monumental math problems such as the Twin Prime Conjecture are the domain of young math whiz. Mr. Zhang is over the age of 50.

Another handicap for Mr. Zhang is that he is not in the math specialty of number theory, which would be the specialty of the mathematicians who work on problems like Twin Prime Conjecture.

The last academic publication of Mr. Yitang Zhang is from the year 2001. So he is considered an inactive researcher by many in the math community.

Given his background, no one in the math community expects great results from Mr. Yitang Zhang. Apparently he was oblivious of his supposedly “handicaps”. He certainly did not make any excuses for himself. He just kept plugging away, building upon the advances made by other mathematicians.

The story of Yitang Zhang is remarkable in two fronts. One is his mathematical work. The other is the human dimension of the story, which makes his achievement all the more remarkable. Mr. Yitang Zhang is an inspiration to us all.

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More information about Yitang Zhang can be found on the Internet. Here’s a few more entries from an Internet search.

# Who Wants to be a Millionaire?

It is rare to find reportings of math in the mainstream media that explain an actual algebra equation. It is rarer still to find such reportings that involve a math problem attached with serious prize money. The other day I found just such a piece in time.com – a math problem with an award of $1 million (US dollars). I would like to discuss this math problem and its significance. It turns out this is a number theory problem that is intimately connected to the famous math problem called Fermat’s Last Theorem. So that is where we start. Fermat’s Last Theorem There are no positive integers $A$, $B$, $C$ and $x$ that satisfy the following equation $\displaystyle A^x+B^x=C^x$ where $x$ is greater than 2. A bit of perspective. The requirement that $x>2$ is necessary. If $x$ = 2, then the equation $\displaystyle A^2+B^2=C^2$ would have many solutions. For examples, $\displaystyle 3^2+4^2=5^2$ and $\displaystyle 5^2+12^2=13^2$ (note that this is the familiar Pythagorean Theorem). Fermat’s Last Theorem was stated as a true mathematical fact by Pierre de Fermat in 1637 (at least he implied that he had a proof). He scribbled in the margin of his own copy of the Greek text Arithmetica by Diophantus commenting that a proof of this statement was too large to fit in the margin. No one knows whether Fermat really had a proof of his conjecture. But we do know that no one could prove Fermat's last Theorem until 1995. For several hundred years (until 1995), Fermat's Last Theorem was regarded as one of the most difficult mathematical problems. In 1993 Andrew Wiles, a British mathematician, had formulated a proof of Fermat's Last Theorem after working for 6 years in secret. But a flaw was discovered in one part of the proof. In 1995, after working with a former student for over a year, Wiles finally had patched the hole that was in the original proof. Thus Fermat’s Last Theorem had ceased to be a conjecture and became a true mathematical statement. Apparently the successful resolution of Fermat’s Last Theorem was not the end of the story. Soon after, people began to hunt for more research opportunities by generalizing Fermat’s Last Theorem. Someone proposed a new problem, which is now called Beal’s Conjecture (after Andrew Beal, the person who proposed it). Consider the following statement. Beal’s Conjecture There are no positive integers $A$, $B$, $C$, $x$, $y$ and $z$ that satisfy the following equation $\displaystyle A^x+B^y=C^z$ where $x$, $y$ and $z$ are greater than 2, and $A$, $B$ and $C$ share no common prime factor. Note the similarity between this new statement and Fermat’s Last Theorem. In fact, Fermat’s Last Theorem is a specific case of Beal’s Conjecture. If the exponents are made the same (and greater than 2), we come back to Fermat’s Last Theorem and we know that the equation $\displaystyle A^x+B^x=C^x$ has no integer solutions (as proven by Andrew Wiles). However, when the exponents are not equaled, the equation $\displaystyle A^x+B^y=C^z$ does have integer solutions. To illustrate, note that $\displaystyle 3^3+6^3=3^5$. Here, $A$, $B$ and $C$ share the common prime factor 3. Another example: $\displaystyle 7^6+7^7=98^3$. Now $A$, $B$ and $C$ share the common prime factor 7. Beal’s Conjecture is essentially saying that if the exponents $x$, $y$ and $z$ are greater than 2, the equation $\displaystyle A^x+B^y=C^z$ has integer solutions only when $A$, $B$ and $C$ share a common prime factor (just like in the 2 examples just given). Andrew Beal, the one who proposed the Beal’s Conjecture in 1993, is a Texas billionaire banker who is also an enthusiastic amateur mathematician. After unsuccessful attempts at resolving his conjecture, he put up prize money to spur interest –$5,000 in 1997 and $100,000 in year 2000. He upped the prize money just recently to$1 million in June 2013.

Just in case there is any reader who still thinks the prize is a joke or a hoax, keep in mind that Andrew Beal is a prominent banker with financial worth in the billions. In fact, the \$1 million prize money is already in safe keeping at the American Mathematical Society (AMS). Here’s is more information about the Beal’s Prize. Mr. Beal would also like to use this prize to draw young people into the wonderful world of mathematics. According to AMS, “one of Andrew Beal’s goals is to inspire young people to think about the equation, think about winning the offered prize, and in the process become more interested in the field of mathematics“.

Beal’s conjecture is no doubt a significant math problem because of its connection with Fermat’s Last Theorem. The serious prize money attached to it will give it exposure and generate some buzz. It is no doubt a difficult problem, clearly not a one-weekend project and clearly not a problem that can be solved by just using elementary math techniques (but applied in a clever way). It took over 350 years to resolve Fermat’s Last Theorem. So it will likely take years before we see any progress on Beal’s Conjecture (the conjecture is only 16 years old).

Math problems like Beal’s Conjecture are not a fool’s errand. Many new mathematical ideas and methods will often be created in the process of resolving monumental problems such as Fermat’s Last Theorem and hopefully in the case of Beal’s Conjecture, thus pushing forward the frontier of mathematics. No doubt this is one of the goals of Mr. Andrew Beal in setting up the Beal’s prize.