Music Of The Primes: Searching To Solve The Greatest Mystery In Mathematics, Macus du Sautoy, c. 2003 -- IN PROGRESS

Note: for personal use only; for taking notes on this book as I go through it a second (or third time). Much of the information below: direct quotes from the book. 

Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics, Marcus du Sautoy, professor mathematics at University of Oxford, c. 2003.

Chapter 1: Who Wants To Be a Millionaire?

Chapter 2: The Atoms of Arithmetic

Chapter 3: Riemann’s Imaginary Mathematical Looking-Glass

Chapter 4:  The Riemann Hypothesis: From Random Primes to Orderly Zeroes

Chapter 5:  The Mathematical Relay Race: Realising Riemann’s Revolution

Chapter 6:  Ramanujan, The Mathematical Mystic

Chapter 7: Mathematical Exodus: From Gottingen to Princeton

Chapter 8:  Machines of the Mind

Chapter 9: The Computer Age: From the Mind to the Desktop

Chapter 10: Cracking Numbers and Codes

Chapter 11: From Orderly Zeroes to Quantum Chaos

Chapter 12: The Missing Piece of the Jigsaw

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Chapter 1

Who Wants To Be A Millionaire

Hilbert’s 8^th of 23 problems: the Riemann’s hypothesis

Professor Enrico Bombieri, IAS, Princeton

Alain Connes: Institute des Haute Etudes Scientifiques, Paris

Connes: developed a new language for geometry to help explain quantum mechanics

Riemann’s hypothesis: seeks to understand the most fundamental objects of mathematics – prime numbers.

Their importance comes from their power to build all other numbers.

Prime numbers remain the most mysterious objects studied by mathematicians.

Prime numbers seem to be random; one cannot predict whether the next number will be a prime number or not.

Modular mathematics, p. 8 – 9; first mentioned.

Riemann’s hypothesis; not a conjecture. Too many other results depend on Riemann’s hypothesis – p. 10. Because so many other results depend on a formula that predicts a prime number, it is called a hypothesis, and not a conjecture (if I am saying that correctly?)

Bombieri absolutely “believes” Riemann was correct; there is a way to predict the next prime number or to determine whether a number is prime.

RSA, 1970’s: Ron Rivest, Adi Shamir, Leonard Adelman. Today, there are no less than one million prime numbers with 100 digits used to protect electronic accounts.

Of Hilbert’s 23 original problems, #8, the problem of prime numbers , was the only one to make it into the 21^st century unvanquished.

There is no Nobel Prize for mathematicians. So, there is the Fields Medal, restricted to mathematicians under the age of 40. The Fields Medals are awarded every four years. The first one was awarded in Oslo in 1936. The purse: 15,000 Canadian dollars.

Chapter 2

The Atoms Of Arithmetic

Gauss’s guess: Prime Number Conjecture

Prime numbers are considered the atoms of arithmetic because all other numbers can be made from prime numbers.

Prime numbers have their “periodic table” just like the chemicals have their periodic table.

1801: 24 y/o German Carl Friedrich Gauss, a mathematician, becomes an instant celebrity when he predicts exactly when/where a small “planet” called Ceres would come out from behind the sun.

It was a hobby to track Ceres, but Gauss ‘ passion was for looking for patterns in the world of numbers.

At age 19, with compass and straight edge, constructed a 19-sided polygon.  No one prior to him had been able to show how to construct another regular polygon with prime number of sides using such simple equipment.

That convinced him to go into mathematic. He wrote that in his journal which he kept for 18 years, and has only recently been released by the family to the public.

Clock calculator: one of his greatest early inventions.

Clock calculators are now called modular arithmetic. One can use any number of hours on the clock face. See pages 20 – 21.

Internet security: uses clock faces with more numbers than there are atoms in the observable universe.

1801: his first book. The father of “number theory.”  Gauss always called “number theory” the “Queen of mathematics.”

Ishango Bone: dated to 6500 BC; has the prime numbers between 10 and 20. No explanation. To me this is very spooky.

Chinese: even numbers, female attributes; odd numbers, male attributes; odd numbers that are NOT prime (such as 15), given effeminate attributes.  Odd numbers that were also prime, were very, very macho numbers.

Greeks considered prime numbers the atoms of arithmetic, as noted above.

Primes: numbers::quarks:atoms.

Eratosthenes: librarian at the library at Alexandria; third century BC, discovered a way to identify all primes between 1 and 100: the sieve of Eratosthenes.

As a boy, Gauss was given a book with the first several thousand prime numbers; he was unable to find a pattern.

Patterns:

·      Triangular numbers, p. 24

·      Fibonacci numbers, p. 25 (13^th century court mathematician); golden ration: 1.61803…

Patterns in nature:

·      Cicadas; two species; one with 17-year-life cycle; one with a 13-year-life cycle.

Conjecture, hypothesis, theorem: p. 29.  A mathematical guess earns the name “theorem” only after a proof is provided.

Hypothesis: “wishful thinking” until a proof is provided.

Theorems need proofs. Goldbach’s Conjecture is true for all numbers up to 4 x 10^14 but is still a conjecture because there is no proof.

Is mathematics an act of discovery or an act of creation? Age-old question, p. 33.

GH Hardy, in his book, A Mathematician’s Apology: “mathematics is not a contemplative but a creative subject.” Graham Greene ranked that book with Henry James’ notebooks as the best account of what it is like to be a creative artist.

The proofs will not be universal in nature; aliens would not be able to understand our proofs.

In proofs, Greeks often started backwards: asking us to prove there were rogue numbers. In this case, “N” was often referred to as the minimal criminal.

Euclid: more than any other Greek mathematician, is regarded as the father of the art of the proof. Part of the research group that Ptolemy I put together in Alexandria; Euclid wrote his book The Elements around 300 BC.

In The Elements, Euclid set down the axioms of geometry. He then used these axioms to write more than 500 theorems of geometry.

The middle part of The Elements: deals with number theory; many regard this section as the first truly brilliant piece of mathematical reasoning. E.g., in Proposition 20: there are an infinite number of prime numbers.

Euclid proposed an infinite number of primes. So, the problem was how to predict the next one.

Twin primes: p. 39 – N & N+2 in which both numbers are prime. The minimal criminals are: 1,000,037 and 1,000,039. Wow. [It cannot be N & N+1 because one of those two numbers has to be an even number.]

Fermat thought he had a formula but did not (2^N + 1). It works until you get to the fifth Fermat number which has 10 digits. It is divisible by 641.

Gauss proved that one could draw any polygon with a prime number of sides using only a straight edge and a compass.

French Monk Marin Mersenne: musician; friend of Gauss; later on Mersenne became the clearing house for mathematicians to report their works.

Mersenne: first to develop a coherent theory of harmonics

Mersenne: worked on primes also; actually did a better job than Fermat.

Merseene asserted correctly that 2^257 – 1 was prime: this number has 77 digits. Did the monk have access to some mystical arithmetic formula that told him why this number, beyond any human computational abilities, was prime?

Mersenne’s numbers: 2^n – 1.  Many are prime, but not all.

Up to this point, mathematicians not interest in proofs, as the Greeks were.

Then, Swiss mathematician Leonhard Euler, b. 1707, developed an interest in proofs.

[Gauss had been born about 1775.]

Euler joins Swiss friends from Basel in Moscow (Peter the Great, Catherine the Great).

Catherine interested in Euler’s work on hydraulics, ship construction, and ballistics.

The seven-bridge problem. Euler proved it impossible to walk all bridge only once; often cited as beginning of topology.

Gauss had his Mersenne; Euler had his Christian Goldbach. Through Goldbach, Euler communicated his conjecture that every even number could be written as a sum of two primes.

Euler: also a proof that certain primes can be written as the sum of two squares.

Euler loved calculating and manipulating mathematical formulas.

Above all else, Euler loved calculating prime numbers.

He produced tables of primes up to 100,000 and a few beyond. He showed that Fermat’s formula for primes, 2^2^n + 1 broke down when n=5.

Uncanny formula: x^2 + x + 41; gives you all primes when x = 0 to 39.

Euler died 1783.

In 1792, Gauss was 15 years old.

Gauss received a math book that at the back had two tables: table of logarithms and table of prime numbers. He saw a connection between the two tables.

[Scottish Baron John Napier: 1614 – conceived a table of logarithms.]

Logarithms important to sailors.

Gauss fascinated with the table of prime numbers.

Asked a new question: how many primes between 1 and 100; between 101 and 1,000; between 1,001 and 10,001, etc. There seemed to be a regular pattern.

Nice discussion of what logarithms are all about. The logarithm on a calculator is for base 10. So, the log of 100 in base 10 is 2. The log of 128 in base 2 is 7. One has to raise 2 to the 7^th power to get 128.

Gauss noted that primes can be counted using logarithms to the base of a special number, called e.  e is as important as π in mathematics; it shows up so often that logs to the base e are called “natural logs.”

The “π(N)” on pages 49 – 50 is a bit confusing. Gauss was using log N but followers started shorthanding it “π(N)” and there are two problems: first, it has nothing to do with π and the arithmetic is lacking to explain how one gets 25 prime numbers between 0 and 100 and 168 primes up to 1,000. – p. 49 – 50.

The graph and this concept is considered one of the most miraculous findings in mathematics. And Gauss never mentioned it except in his journals.

Gauss did not have a proof for this concept; and this was the turning point in mathematics for Gauss, from needing proofs. – p. 51. He pressed on even if he couldn’t provide a proof.

Paris mathematician Legendre (25 years old than Gauss) also noted the relationship between logs and prime numbers, but Gauss beat him to it. Legendre, to be fair, had a better “formula” than Gauss.

Controversy developed over who (Legendre or Gauss) was first to see relationship between logs and primes. Gauss did not care a whole lot. Legendre even said he beat Gauss to prediction regarding Ceres.

Riemann’s Hypothesis can be interpreted: given a choice between an ugly world and an aesthetic one, Nature always chooses the latter.  It is a constant source of amazement for most mathematicians that mathematics should be like this, and explains why they so often get wound up about the beauty of their subject.

[Note, the word “probability” is starting to rear it’s (ugly) head. When I think “probability,” I think “quantum mechanics.”]

Gauss refined his formula for determining primes.

By the time of his death, Gauss had primes up to 3,000,000. Jakub Kulik, a professor at U of Prague: prime numbers up to 100,000,000; eight volumes.

Gauss’ intuition up to 10^16 was off by one ten-millionth of one percent; Legendre was off by one-tenth of one percent.

So, Gauss had discovered that Nature was using a weighted coin when tossing to decide when the next prime number would appear.  But the formula would have to wait for a new generation.

Gauss distracted by astronomy (Ceres) and publicly left mathematics.

One of his very few students, Riemann picked up the baton.

Chapter Three

Riemann’s Imaginary Mathematical Looking-Glass

1809: Wilhelm von Humboldt, education minister of Prussia (northern German state). Writes to Goethe (1816) with his plans for education.

·      Gymasiums – new concept

·      return to classical studies

·      to be created across Prussia and neighboring state of Hanover

·      teachers not to be clergy, but from graduates from the new universities and polytechnics that were built during this period

Jewel in this crown: Berlin University.

For the first time: promote research alongside teaching.

Despite his passion for the Ancient World, he pioneered the introduction of new disciplines to sit beside the classical faculties of Law, Medicine, Philosophy, and Theology.

For the first time, the study of mathematics was to form a major part of the curriculum in the Gymnasiums and universities.

Opposite of what Napoleon was doing in France, and the new universities ultimately brought down Napoleon.

Humboldt, Prussia/Hanover, revolutionized mathematics, particularly the study of prime numbers.

Lüneberg: the Gymnasium Johanneum.

Enter, student Bernhard Reimann, around 1842.

Three classicists: Euclid, Archimedes, and Apollonius.

Riemann read Legendre’s book and learned of the strange relationship between logs and prime numbers. The book: 859 large quarto pages; Riemann mastered it in six days!!

To satisfy his father, he went to university town of Göttingen for a religious education/career.

Göttingen: Lower Saxony; Brothers Grimm; Gauss appointed professor of astronomy and directory of the observatory  in 1807; things changed; Göttingen was on its way to becoming famous for its science, not theology.

Gauss becoming old; lecturing only on astronomy. Riemann had outgrown Göttingen.

He moved to University of Berlin, which had benefitted from the French occupation. Brilliant mathetician there: Peter Gustav Lejeune-Dirichlet.

Dirichlet important in Riemann’s study of primes.

Pored over papers by the mathematical revolutionary Augustin-Louis Cauchy.

Some wrote that Cauchy was only mathematician doing pure mathematics.

Riemann secluded himself with Cauchy’s writings; re-emerged: what had captured Cauchy and Riemann’s imagination was the emerging power of imaginary numbers.

Imaginary numbers – a new mathematical vista

The square root of minus one: they hold the key to the 20^th century world of subatomic particles.

On a larger scale, airplanes would not fly without imaginary numbers.

Babylonians and Egyptians: fractions

Greeks (6^th century BC): Pythagoras theorem  not always compatible/solvable with fractions.

For example, a one-unit right triangle has a hypotenuse with length square root of 2.

Square root of 2 cannot be written as a fraction.

Fractions: decimal expansions have repeating patterns.

Square root of 2: no repeating pattern. Ever.

Mathematicians (at the time) felt uncomfortable that one could not write a number with absolute specificity/or absolutely precisely, and so they called such numbers irrational numbers.

Negative numbers were discovered similarly out of attempts to solve such simple equations as x + 3 = 1. Hindus proposed negative numbers in the 7^th century.

European mathematicians not happy with negative numbers until the 17^th century.

But there were still other “crazy” equations that couldn’t be solved with irrational numbers and with negative numbers. For example: x^2 = -1.