Earlier today, I tweeted about my exciting new Pi search website, which lets you search for any string of digits within the infinite decimal expansion of π. If you haven’t seen it, go and check it out now.
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- Start with the number $2$
- Multiply by each of the fourteen fractions until you find one for which the product is an integer
- Starting with this new integer, continue multiplying through the fractions until another integer is produced. (If this process reaches fraction $N=\frac{55}{1}$, the integer’s product with N is guaranteed to be another integer as N has a denominator of $1$; the process continues with this new integer being multiplied by fraction A)
- Continue multiplying through the set to create more integers
- When the integer is a power of $2$, its exponent will be a prime number.
- Listen to this Numberphile interview with Conway on how he invented the Game of Life
- Play the Game of Life
- Aperiodical posts about John Conway
- Conway’s publications on Scholia
- Conway and knots: ‘I proved this when I was at high school in England’
- Graduate Student Solves Decades-Old Conway Knot Problem, in Quanta
John Conway and his fruitful fractions
Following on from the series of ‘Pascal’s Triangle and its Secrets‘ posts, guest author David Benjamin shares another delightful piece of mathematics – this time relating to prime numbers.
At the time of writing the largest known prime number has $24862048$ digits. The number of digits does not reflect the true size of this prime but if we were to type it out at Times New Roman font size 12, it would reach approximately $51.5$ km, or about $32$ miles. Astonishing!
Patrick Laroche from Ocala, Florida discovered this Mersenne prime on December 7, 2018. I was surprised to discover that it’s exponent $82589933$ is the length of the hypotenuse of a primitive Pythagorean triple where $82589933^{2} = 30120165^{2} + 76901708^{2}$ as indeed are 8 of the exponents of those currently ranked from 1 to 10.
The Greek mathematician Euclid of Alexandria ($\sim$325 BC-265 BC) was arguably the first to prove that there are an infinite number of primes – and since then, people have been searching for new ones. Some do it for kudos, for the prize money, to test the power of computers and the need to find more of the large primes used to help protect the massive amount of data which is being moved around the internet.
Mersenne primes, named after the French monk Marin Mersenne, are of the form $2^{p} -1$, where the exponent $p$ is also prime. Mersenne primes are easier to test for primality, which is one reason we find so many large ones (all but one of the top ten known primes are Mersenne). When Mersenne primes are converted to binary they become a string of $1$s, which makes them suitable for computer algorithms and an excellent starting point for any search.
Since generally testing numbers for primality is slow, some have tried to find methods to produce primes using a formula. Euler’s quadratic polynomial $n^2+n+41$ produces this set of $40$ primes for $n = 0$ to $39$. When $n=40$, the polynomial produces the square number $1681$. Other prime-generating polynomials are listed in this Wolfram Mathworld entry.
The French mathematician Lejeune Dirichlet proved that the linear polynomial $a+nb$ will produce an infinite set of primes if $a$ and $b$ are coprime for $n=0,1,2,3,4,…$. Then again, it also produces an infinite number of composite numbers! However, this gem: $224584605939537911 + 1813569659748930n$ produces 27 consecutive primes for $n=0$ to $n=26$ – and of course, all the primes are in arithmetic progression.
14 fruitful fractions
The primes are unpredictable, and become less common as they get larger. Consequently there is no formula that will generate all the prime numbers. However, there is a finite sequence of fractions, that – given an infinite amount of time – would generate all the primes, and in sequential order.
They are the fruitful fractions, created by the brilliant Liverpool-born mathematician, John Horton Conway (1937–2020) who, until his untimely death from complications related to COVID-19, was the John von Neumann Emeritus Professor in Applied and Computational Mathematics at Princeton University, New Jersey, USA.
The fruitful fractions are
$\frac{17}{91}$ | $\frac{78}{85}$ | $\frac{19}{51}$ | $\frac{23}{38}$ | $\frac{29}{33}$ | $\frac{77}{29}$ | $\frac{95}{23}$ | $\frac{77}{19}$ | $\frac{1}{17}$ | $\frac{11}{13}$ | $\frac{13}{11}$ | $\frac{15}{44}$ | $\frac{15}{2}$ | $\frac{55}{1}$ |
A | B | C | D | E | F | G | H | I | J | K | L | M | N |
The first time I encountered this set of fractions was in the wonderful book, The Book of Numbers, by Conway and Guy. I was so intrigued as to how Conway came up with his idea, I emailed him to ask. I was delighted to receive an outline of an explanation and even a second set of fractions, neither of which I can now find – it was 1996 and pre-cloud storage! But no worries… Conway explains everything in this lecture, which also demonstrates his passion for mathematics and his ability to express his ideas in a relaxed and humorous way, even when he searches for an error in his proof on 26 minutes. The lecture also includes an introduction to Conway’s computer language, FRACTRAN, which includes the statement:
‘It should now be obvious to you that you can write a one line fraction program that does almost anything, or one and a half lines if you want to be precise‘.
Using the fractions to find prime numbers
Here’s how the fractions are used to generate primes.
The 19 steps needed to produce the first prime number are:
$2 \overset{ \times M}{\rightarrow} 15 \overset{ \times N}\rightarrow 825\overset{ \times E} \rightarrow 725 \overset{ \times F}\rightarrow 1925\overset{ \times K} \rightarrow 2275 \overset{ \times A}\rightarrow 425 \overset{ \times B}\rightarrow 390 \overset{ \times J}\rightarrow 330 \overset{ \times E}\rightarrow 290 \overset{ \times F}\rightarrow 770 \overset{ \times K}\rightarrow 910\overset{ \times A} \rightarrow 170\overset{ \times B} \rightarrow 156\overset{ \times J} \rightarrow 132\overset{ \times E} \rightarrow 116 \overset{ \times F}\rightarrow 308\overset{ \times K} \rightarrow 364\overset{ \times A} \rightarrow 68 \overset{ \times I}\rightarrow 4 \equiv2^{2}$
The number of steps needed to produce the first 7 primes are shown in the table below:
Prime | 2 | 3 | 5 | 7 | 11 | 13 | 17 |
Steps | 19 | 69 | 281 | 710 | 2375 | 3893 | 8102 |
And here is the start and end of the sequence of fractions used to produce the next prime number from $2^{2}$:
$4 \overset{ \times M}{\rightarrow} 30 \overset{ \times M}\rightarrow 225\overset{ \times N} \rightarrow 12375 \overset{ \times E}\rightarrow 10875 \rightarrow \cdots \rightarrow 232 \overset{ \times F}{\rightarrow} 616 \overset{ \times K}\rightarrow 728\overset{ \times A} \rightarrow 136 \overset{ \times I}\rightarrow 8\equiv2^{3}$
The steps needed for the first 34 primes are given as OEIS A007547 and the first 8102 products in the B-list for A007542.
The successive primes are produced almost like magic – but the number of multiplications needed to produce each new prime becomes larger and larger, and so the method, though wonderfully inventive, is not at all efficient.
Edit: Since this article was first published, the exponent $82589933$ of the Laroche prime has been accepted as the next term in the sequence http://oeis.org/A112634
Further Reading on John Conway
Now I’m calculating with constructive reals!
A while ago I made myself a calculator. I don’t know if anyone else uses it, but for the particular way I like doing calculations, it’s been really good. You’d think that if a calculator does anything, it should perform calculations correctly. But all calculators get things wrong sometimes! This is the story of how I made my calculator a bit more correct, using constructive real arithmetic.
One thing you need to think about when making a calculator is precision. How precise do the answers need to be? Is it OK to do rounding? If you do round, then it’s possible that errors accumulate as you compose operations.
I’ve always wanted to make a calculator that gives exactly correct answers. This isn’t strictly possible: there are more real numbers than a finite number of bits of memory can represent, or a digital display can show, no matter how you encode them. But I’m not going to use every real number, so I’ll be happy with just being correct on the numbers I’m likely to encounter.
British Science Week mathematicians poster competition
I wrote a mathematics-themed competition for British Science Week, which is a UK-wide event lasting ten days taking place this month.
The competition calls for individuals or groups to research the life and/or work of a mathematician and produce a poster to share their findings. The six mathematicians available to choose from are:
MathsJam Leuven recap, February 2022
It’s been a while since we’ve seen a MathsJam recap, but having restarted the MathsJam in Leuven after a hiatus, Dieter was too excited not to share what they’d been up to.
The first (re)edition of the Maths Jam in Leuven (Historic university city in Belgium) was a tiny success. I brought a couple of physical copies of the single page worksheet called the MathsJam Meta Shout with $\sim$10 problems from different sub areas of maths which I received earlier from Katie who coordinates MathsJams internationally.
The problems on the sheet were ranging from simple (?) Fold-and-cut fun, Tangrams (geometry), to some number theory, a touch of Linear Algebra and an easy arithmetic problem, solvable with 12-yo-level calculations (i.e. arrange all numbers from 1-15 so that each adjacent numbers sum to a square of a whole number under 16). Nice to see how broad the difficulty space is on the worksheet. Creative problem solving for (nearly) all ages!
On the second to last Tuesday of February 2022 (and hopefully each month from now), we were 3 in total. Which is a good number, I guess, for the Leuven revival anno 2022. Plenty of room to go from there – and a prime number, naturally! :-) Unfortunately we got kicked out of the venue at 21h30 (we started at 20h GMT+1) because we were the only ones left and the venue closes at 22h on Tuesdays (something I didn’t check, nor expected really). But puzzle minded we were, we just overflowed to someone’s home -after a rainy bike intermezzo which refreshened our minds. This didn’t stop us continuing our puzzling until 23h.
One of the attendees (a mathematician by degree) aced all the problems in <3h all the while (attempting) to explain his rationale. It was quite impressive to see! And fun too, because I definitely learned quite some things that night. I was still attempting to fold-and-cut the necessary T-shape with the proper dimensions (3 unit squares on top & 4 from top to bottom) when others had already finished their second tangrams (with some clever area proportion estimates). I forgot to bring my edition of the Set game so we didn’t participate in the online inter-MathsJam set-hunt – being only three we were too eager to just dive into the puzzles first.
After that second to last Tuesday, I tried some of the puzzles I hadn’t completed that night myself and I still haven’t finished them all just yet. (Some of them really make the gears in my brain grind!) I really liked the mix of complexity and variation in type of problems (kudos/thx/merci to all those involved in the making of the Shout).
I’m already eagerly looking forward to the next edition Shout and the next physical meetup by extension (Tuesday 22th of March), and have arranged a new venue for this month in the bar Café Entrepot of the local art center Opek. This seems very fitting for the subtle art of maths and I’ve got the guarantee that they will host us at length, yay! I sure hope to see you there on a second to last Tuesday soon. :-)
Each Edge Peach Pear Plum
At the 2021 UK MathsJam Gathering, I gave a talk on a subject that has bothered me more than is reasonable: the graph-theoretic layout of the narrative of the baby’s book Each Peach Pear Plum, by Janet and Allan Ahlberg.
It’s one of my son’s favourite books to fall asleep to. It was his older sister’s favourite, and mine and my wife’s when we were little. I agree with the quote on the back cover, that it’s “the perfect first book”.
BUT
Probability, statistics and Pascal’s other contributions
This is the final part in the Pascal pentalogy, a series of guest posts by David Benjamin exploring the secrets of Pascal’s Triangle.
Probability and combinations
In Part 1 of this series we stated that Pascal is credited with being the founder of probability theory – but credit also needs to be given to other mathematicians, in particular the Italian polymath Girolamo Cardano.
The connection between probability and the numbers in Pascal’s triangle can be shown by looking at the outcomes when one or more coins are tossed. The table below, from row two, lists the outcomes for one, two and three unbiased coins.
$1$ | ||||||||
$1$ H | $1$ T | |||||||
$1$ HH | $2$ HT, TH | $1$ TT | ||||||
$1$ HHH | $3$ HHT, HTH, THH | $3$ HTT, THT, TTH | $1$ TTT | |||||
$1$ | $4$ | $6$ | $4$ | $1$ |
For four coins there is $1$ outcome for four heads, $4$ outcomes for three heads and one tail, $6$ outcomes for two heads and two tails, $4$ outcomes for one head and three tails and one outcome for $4$ tails.
Row four shows us that when three unbiased coins are tossed, the probability they will land showing two heads and one tail in any order is $\frac{3}{1+3+3+1}=\frac{3}{8}$.
As the sum of the $n^{th}$ row is $2^{n}$, the number of possible outcomes for four coins is $2^4=16$, $32$ for five coins, $64$ for six coins, …
Quincunx
A Quincunx, or Galton Board, is named after the English explorer and anthropologist Francis Galton (1822-1911) – although this name is now less popular, because of Galton’s views on eugenics and racist attitudes.
The board is a triangular array of pegs. Balls are dropped onto the top peg and then bounce their way down to the bottom where they are collected in containers. Each time a ball hits one of the pegs, it bounces either left or right with an equal probability of $\frac{1}{2}$ and the balls collect in the containers to form the classic bell-shaped curve of the normal distribution.
The Quincunx is like Pascal’s triangle with pegs instead of numbers. The number on each peg represents the number of different paths a ball can take to reach that peg. If there are $10$ rows and the last row contains the containers, then the probability of landing in the third container from the right can be calculated by using the formula for the Binomial distribution.
The probability of landing in the third bin from the right is $120\times(\frac{1}{2})^3\times(\frac{1}{2})^7=\frac{15}{128}=0.1171875$, where $120$ is the number of different paths to that bin.
Statistics and permutations
The link between statistics and the triangle can be demonstrated using combinations. Consider these 5 mathematicians Euler, Pascal, Ramanujan, Hilbert and Conway and the possible teams for a three-legged race.
There are $10$ different teams of $3$:
EPR EPH EPC ERH ERC EHC PRH PRC PHC RHC
The formula to calculate the number of combinations is $_n{C}_r =\frac{n!}{r!(n-r)!}$ where $n$ represents the total we are choosing from, $r$ the number in the team and
\[ n!=n\times(n-1)\times(n-2)\times(n-3)\times…\times1\]
In our example $n=5$, $r=3$ and $\frac{5!}{3!(5-3)!}=\frac{120}{6\times2}=10$
$_n{C}_r$ can be used to calculate the rows of Pascal’s triangle as shown below for row $6$, where in the calculation of $_5{C}_0$, $0!=1$
$_5{C}_0$ | $_5{C}_1$ | $_5{C}_2$ | $_5{C}_3$ | $_5{C}_4$ | $_5{C}_5$ |
$1$ | $5$ | $10$ | $10$ | $5$ | $1$ |
The animation film Of Dice and Men by John Weldon is a lovely way to introduce students to probability and statistics.
Pascal the polymath: mathematics, inventor, science and religion
Pascal’s father was a tax collector and in 1642 Blaise invented a mechanical calculator to assist his father. It was called the Pascaline and had a wheel with eight movable parts for dialing. Each part corresponded to a particular digit in a number. Numbers could be added by turning the wheels located along the bottom of the machine. Subtraction was carried out by exploiting a method called nines’ complement representation, the use of which allows subtraction to be reduced to addition. Each digit in the answer was displayed in a separate window. The workings of the Pascaline are demonstrated here.
The Musée des Arts et Métiers in Paris has one of the original Pascalines. The invention was not a commercial success – it was very expensive and often only purchased as a novelty rather than for use. Essentially, it was an adding machine. Subtraction was turned into a form of addition, as was multiplication. Division was done by repeated subtraction. Nines’ complement representation is still used in modern digital computers by a similar technique called ones’ complement which is used to represent negative numbers and hence perform subtraction in the same way as addition. Pascal did not discover this method but his calculator is the earliest known device to employ it. He continued to make improvements to his design until 1652.
Conic sections – normally just called conics – are obtained when a mathematical cone is sliced by a plane. Depending on the angle of the slice, the intersections create a circle, an ellipse, a parabola and a hyperbola. Conics have many applications including the wheel of course, ophthalmic, parabolic mirrors and reflectors, telescopes, searchlights and projectile motion.
Pascal wrote a short treatise, Essai pour les coniques (Essay on Conics) when only 16. In it he included what is known as Pascal’s Theorem which states that if a hexagon is inscribed in a conic section then the three intersection points of opposite sides lie on a straight line – the Pascal line. The theorem [also referred to as Pascal’s Hexagrammum Mysticum Theorem] was his first important mathematical discovery and a breakthrough in the field of projective geometry.
In 1647 Pascal expanded on the work of the Italian physicist Evangelista Torricelli, the inventor of the barometer by writing Experiences nouvelles touchant le vide (New experiments with the vacuum) in which Pascal gave detailed rules to describe to what degree various liquids could be supported by air pressure. In 1971 the SI unit for pressure [equal to one newton per square metre] was named the pascal.
Also in 1647 he discovered Pascal’s Law of hydrostatics allowing for the development of the hydraulic press. Pascal himself used the principle to invent the syringe.
Pascal wrote an extremely influential theological work which was unfinished at the time of his death. It was posthumously called Pensées (Thoughts) and contained a detailed and coherent examination and defence of the Christian faith.
In 1655 Pascal was trying to invent a perpetual motion machine, a machine that continues to operate without drawing energy from an external source. The laws of physics now say this is impossible. Naturally he failed but he ended up inventing a basic roulette wheel, now upgraded and used in casinos as a game of chance.
The Swiss computer scientist Niklaus Emil Wirth, born in 1934, named one of his programming languages Pascal in honour of Blaise. Wirth along with Helmut Weber also designed the programming language named after another mathematician, Euler. [Recommended read: Euler: The Master of Us All ]
Pascal died in extreme pain at the young age of 39. He had a malignant growth in his stomach which had spread to his brain. Like many others, such as Évariste Galois and Franz Schubert, we are left wondering what else Pascal could have achieved had he lived longer. His work with Fermat into the calculus of probabilities helped the German mathematician Gottfried Leibniz [1646-1716] develop the infinitesimal calculus. Pascal is buried in the Saint-Étienne-du-Mont church in Paris and his death mask is held at the J. Paul Getty museum in Los Angeles, California.