A nice article on recent progress on solving the twin prime conjecture

The twin prime conjecture (see here, here and here for more information) asserts that there are infinitely many primes that have a difference of 2. For example:

3 and 5 are twin primes;

5 and 7 are twin primes;

11 and 13 are twin primes;

17 and 19 are twin primes;

29 and 31 are twin primes; etc.

While most mathematicians believe the twin prime conjecture is correct, an explicit proof has not been found. Indeed, this has been one of the most popular unsolved problems in mathematics — not necessarily because it’s important, but for the curiosity that a conjecture so simply stated has eluded conquest by the world’s best mathematicians.

Still, research continues, and some major progress has been made in the past few years. (I like sharing this story with my students to convince them that not everything that can be known about mathematics has been figure out yet — a misconception encouraged by the structure of the secondary curriculum — and that research continues to this day.) Specifically, it was recently shown that, for some integer N that is less than 70 million, there are infinitely many pairs of primes that differ by N.

http://video.newyorker.com/watch/annals-of-ideas-yitang-zhang-s-discovery-2015-01-28

http://www.newyorker.com/magazine/2015/02/02/pursuit-beauty

For more on recent progress:

 

 

 

 

 

 

15-square puzzle

From the category “This Is Completely Useless”: here’s what a 15-square puzzle looks like when you arrange the tiles in order of how many factors they have.

puzzle15

What I Learned by Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post.

When I was researching for my series of posts on conditional convergence, especially examples related to the constant \gamma, the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites along with the page numbers in the book — while giving the book a very high recommendation.

Part 1: The smallest value of n so that 1 + \frac{1}{2} + \dots + \frac{1}{n} > 100 (page 23).

Part 2: Except for a couple select values of m<n, the sum \frac{1}{m} + \frac{1}{m+1} + \dots + \frac{1}{n} is never an integer (pages 24-25).

Part 3: The sum of the reciprocals of the twin primes converges (page 30).

Part 4: Euler somehow calculated \zeta(26) without a calculator (page 41).

Part 5: The integral called the Sophomore’s Dream (page 44).

Part 6: St. Augustine’s thoughts on mathematicians — in context, astrologers (page 65).

Part 7: The probability that two randomly selected integers have no common factors is 6/\pi^2 (page 68).

Part 8: The series for quickly computing \gamma to high precision (page 89).

Part 9: An observation about the formulas for 1^k + 2^k + \dots + n^k (page 81).

Part 10: A lower bound for the gap between successive primes (page 115).

Part 11: Two generalizations of \gamma (page 117).

Part 12: Relating the harmonic series to meteorological records (page 125).

Part 13: The crossing-the-desert problem (page 127).

Part 14: The worm-on-a-rope problem (page 133).

Part 15: An amazingly nasty formula for the nth prime number (page 168).

Part 16: A heuristic argument for the form of the prime number theorem (page 172).

Part 17: Oops.

Part 18: The Riemann Hypothesis can be stated in a form that can be understood by high school students (page 207).

 

 

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 17

Let \pi(n) denote the number of positive prime numbers that are less than or equal to n. The prime number theorem, one of the most celebrated results in analytic number theory, states that

\pi(x) \approx \displaystyle \frac{x}{\ln x}.

This is a very difficult result to prove. However, Gamma (page 172) provides a heuristic argument that suggests that this answer might be halfway reasonable.

Consider all of the integers between 1 and x.

  • About half of these numbers won’t be divisible by 2.
  • Of those that aren’t divisible by 2, about two-thirds won’t be divisible by 3. (This isn’t exactly correct, but it’s good enough for heuristics.)
  • Of those that aren’t divisible by 2 and 3, about four-fifths won’t be divisible by 5.
  • And so on.

If we repeat for all primes less than or equal to \sqrt{x}, we can conclude that the number of prime numbers less than or equal to x is approximately

\pi(x) \approx \displaystyle x \prod_{p \le \sqrt{x}} \left(1 - \frac{1}{p} \right).

From this point, we can use Mertens product formula

\displaystyle \lim_{n \to \infty} \frac{1}{\ln n} \prod_{p \le n} \left(1 - \frac{1}{p} \right)^{-1} = e^\gamma

to conclude that

\displaystyle \frac{1}{\ln n} \prod_{p \le n} \left(1 - \frac{1}{p} \right) \approx \displaystyle \frac{e^{-\gamma}}{\ln n}

if n is large. Therefore,

\pi(x) \approx x \displaystyle \frac{e^{-\gamma}}{\ln \sqrt{x}} = 2 e^{-\gamma} \displaystyle \frac{x}{\ln x}.

Though not a formal proof, it’s a fast way to convince students that the unusual fraction \displaystyle \frac{x}{\ln x} ought to appear someplace in the prime number theorem.

green line

When I researching for my series of posts on conditional convergence, especially examples related to the constant \gamma, the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 16

Let \pi(n) denote the number of positive prime numbers that are less than or equal to n. The prime number theorem, one of the most celebrated results in analytic number theory, states that

\pi(x) \approx \displaystyle \frac{x}{\ln x}.

This is a very difficult result to prove. However, Gamma (page 172) provides a heuristic argument that suggests that this answer might be halfway reasonable.

Consider all of the integers between 1 and x.

  • About half of these numbers won’t be divisible by 2.
  • Of those that aren’t divisible by 2, about two-thirds won’t be divisible by 3. (This isn’t exactly correct, but it’s good enough for heuristics.)
  • Of those that aren’t divisible by 2 and 3, about four-fifths won’t be divisible by 5.
  • And so on.

If we repeat for all primes less than or equal to \sqrt{x}, we can conclude that the number of prime numbers less than or equal to x is approximately

\pi(x) \approx \displaystyle x \prod_{p \le \sqrt{x}} \left(1 - \frac{1}{p} \right).

From this point, we can use Mertens product formula

\displaystyle \lim_{n \to \infty} \frac{1}{\ln n} \prod_{p \le n} \left(1 - \frac{1}{p} \right)^{-1} = e^\gamma

to conclude that

\displaystyle \frac{1}{\ln n} \prod_{p \le n} \left(1 - \frac{1}{p} \right) \approx \displaystyle \frac{e^{-\gamma}}{\ln n}

if n is large. Therefore,

\pi(x) \approx x \displaystyle \frac{e^{-\gamma}}{\ln \sqrt{x}} = 2 e^{-\gamma} \displaystyle \frac{x}{\ln x}.

Though not a formal proof, it’s a fast way to convince students that the unusual fraction \displaystyle \frac{x}{\ln x} ought to appear someplace in the prime number theorem.

green line

When I researching for my series of posts on conditional convergence, especially examples related to the constant \gamma, the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 15

I did not know — until I read Gamma (page 168) — that there actually is a formula for generating nth prime number by directly plugging in n. The catch is that it’s a mess:

p_n = 1 + \displaystyle \sum_{m=1}^{2^n} \left[ n^{1/n} \left( \sum_{i=1}^m \cos^2 \left( \pi \frac{(i-1)!+1}{i} \right) \right)^{-1/n} \right],

where the outer brackets [~ ] represent the floor function.

This mathematical curiosity has no practical value, as determining the 10th prime number would require computing 1 + 2 + 3 + \dots + 2^{10} = 524,800 different terms!

green line

When I researching for my series of posts on conditional convergence, especially examples related to the constant \gamma, the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 10

Suppose p_n is the nth prime number, so that p_{n+1} - p_n is the size of the nth gap between successive prime numbers. It turns out (Gamma, page 115) that there’s an incredible theorem for the lower bound of this number:

\displaystyle \limsup_{n \to \infty} \frac{(p_{n+1}-p_n) (\ln \ln \ln p_n)^2}{(\ln p_n)(\ln \ln p_n)(\ln \ln \ln \ln p_n)} \ge \displaystyle \frac{4 e^{\gamma}}{c},

where \gamma is the Euler-Mascheroni constant and c is the solution of c = 3 + e^{-c}.

Holy cow, what a formula. Let’s take a look at just a small part of it.

Let’s look at the amazing function f(x) = \ln \ln \ln \ln x, iterating the natural logarithm function four times. This function has a way of converting really large inputs into unimpressive outputs. For example, the canonical “big number” in popular culture is the googolplex, defined as 10^{10^{100}}. Well, it takes some work just to rearrange \displaystyle f \left(10^{10^{100}} \right) in a form suitable for plugging into a calculator:

\displaystyle f \left(10^{10^{100}} \right) = \displaystyle \ln \ln \ln \left( \ln 10^{10^{100}} \right)

= \displaystyle \ln \ln \ln \left( 10^{100} \ln 10 \right)

= \displaystyle \ln \ln \left[ \ln \left(10^{100} \right) + \ln \ln 10 \right]

= \displaystyle \ln \ln \left[ 100 \ln 10 + \ln \ln 10 \right]

= \displaystyle \ln \ln \left[ 100 \ln 10 \left( 1 + \frac{\ln \ln 10}{100 \ln 10} \right) \right]

= \displaystyle \ln \left( \ln [ 100 \ln 10] + \ln \left( 1 + \frac{\ln \ln 10}{100 \ln 10} \right)\right)

\approx 1.6943

after using a calculator.

This function grows extremely slowly. What value of x gives an output of 0? Well:

\ln \ln \ln \ln x = 0

\ln \ln \ln x = 1

\ln \ln x = e

\ln x = e^e

x = e^{e^e} \approx 3,814,279.1

What value of x gives an output of 1? Well:

\ln \ln \ln \ln x = 1

\ln \ln \ln x = e

\ln \ln x = e^e

\ln x = e^{e^e}

x = e^{e^{e^e}}

\approx e^{3,814,279.1}

\approx 10^{3,814,279.1 \log_{10} e}

\approx 10^{1,656,420.367636}

\approx 2.3315 \times 10^{1,656,420}

That’s a number with 1,656,421 digits! At the rapid rate of 5 digits per second, it would take over 92 hours (nearly 4 days) just to write out the answer by hand!

Finally, how large does x have to be for the output to be 2? As we’ve already seen, it’s going to be larger than a googolplex:

\displaystyle f \left(10^{10^{x}} \right) = 2

\displaystyle \ln \ln \ln \left( \ln 10^{10^{x}} \right) = 2

\displaystyle \ln \ln \ln \left( 10^{x} \ln 10 \right) = 2

\displaystyle \ln \ln \left[ \ln \left(10^{x} \right) + \ln \ln 10 \right] = 2

\displaystyle \ln \ln \left[ x\ln 10 + \ln \ln 10 \right] = 2

\displaystyle \ln \ln \left[ x\ln 10 \left( 1 + \frac{\ln \ln 10}{x\ln 10} \right) \right] = 2

\displaystyle \ln \left( \ln [ x\ln 10] + \ln \left( 1 + \frac{\ln \ln 10}{x \ln 10} \right)\right) = 2

Let’s simplify things slightly by letting y = x \ln 10:

\displaystyle \ln \left( \ln y + \ln \left( 1 + \frac{\ln \ln 10}{y} \right)\right) = 2

\displaystyle \ln y + \ln \left( 1 + \frac{\ln \ln 10}{y} \right) = e^2

This is a transcendental equation in y; however, we can estimate that the solution will approximately solve \ln y = e^2 since the second term on the left-hand side is small compared to \ln y. This gives the approximation y = e^{e^2} \approx 1618.18. Using either Newton’s method or else graphing the left-hand side yields the more precise solution y \approx 1617.57.

Therefore, x \approx 1617.57 \ln 10 \approx 3725.99, so that

f \left(10^{10^{3725.99}} \right) \approx 2.

One final note: despite what’s typically taught in high school, mathematicians typically use \log to represent natural logarithms (as opposed to base-10 logarithms), so the above formula is more properly written as

\displaystyle \limsup_{n \to \infty} \frac{(p_{n+1}-p_n) (\log \log \log p_n)^2}{(\log p_n)(\log \log p_n)(\log \log \log \log p_n)} \ge \displaystyle \frac{4 e^{\gamma}}{c}.

And this sets up a standard joke, also printed in Gamma:

Q: What noise does a drowning analytic number theorist make?

A: Log… log… log… log…

green line

When I researching for my series of posts on conditional convergence, especially examples related to the constant \gamma, the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.

A curious non-randomness in the distribution of primes

I found this article extremely interesting. From https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/

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 [March 13, 2016], 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…

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… 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).

New world record for largest prime number

As of this week, we have a new world record for the largest known prime number:

2^{74,207,281}-1

The adjective known is important, because there are an infinite number of prime numbers (but not all of them are known). A good video describing this finding is below.

A good article is here:

https://www.washingtonpost.com/news/speaking-of-science/wp/2016/01/20/the-newest-prime-number-is-more-than-22-million-digits-long/?tid=sm_tw

A Mathematical Horror Story

One of my guilty pleasures in the 1990s, when I was far too old to be watching children’s cartoons, was the fantastic show “Pinky and the Brain.” In the clip below, the Brain tells a scary campfire story. (Well, it’s about math, and what’s scarier than math?)