An SO(2,2) Iterated Function System Part 4 - Integer spinors

 The great merchant, Hypatia of Alexandria, was dead. In her will, she left her vast fortune not to a person, but to a puzzle. Her wealth was locked in a magnificent chest with not one, but three complex locks.

Her will stipulated: "My fortune shall go to the first of my former students who can provide the number of coins in this chest. To aid you, I can say this: the number is more than 100 but less than 200. If you attempt to count the coins by twos, threes, fours, up to tens, there will always be a specific number of coins left over."

A young scholar, Euclid, was the first to arrive. He listened to the executor read the conditions and immediately asked for the remainders.

"The remainders are as follows," the executor said, unrolling a scroll.

Divisible by 2: Remainder 1

Divisible by 3: Remainder 2

Divisible by 4: Remainder 3

Divisible by 5: Remainder 4

Divisible by 6: Remainder 5

Divisible by 7: Remainder 0

Divisible by 8: Remainder 1

Divisible by 9: Remainder 2

Divisible by 10: Remainder 3

Euclid's face fell. "This is chaos! The conditions are not coprime. Four, six, eight, nine, and ten are all composite! This will take me a week of calculations!"

Just then, his rival, the quick-witted Sun Tzu, arrived. He glanced at the list and chuckled. "You are thinking like a laborer, Euclid, not a mathematician. You are looking at the locks; I am looking at the key."

"And what key is that?" Euclid scoffed.

"The key," said Sun Tzu, "is to see what the number almost is, not what it is. Look at the list. What do you see?"

I will skip the rest of the story. Stories from the past are fine, but we have a story of the present,  and it needs your full attention. Here it is.

In this note we continue the discussion started in An SO(2,2) Iterated Function System Part 3 - Weyl spinors and null vectors, but now replacing the field of real numbers R by the ring of integers Z. We will show that the following Proposition holds:

Proposition 1. Let A =  A = {{a,b},{c,d}} be a non-zero matrix with integer entries, A ∈ Z^2⨉2, with det(A) = ad - bc = 0. Then there exist vectors v,w ∈ Z² such that

A = vw^T.                (1)

The factorization is unique up to a sign: if A = vw^T, then A = (-v)(-w^T) is the only other (trivial) variation.

The proof of this Proposition uses the concept of the Greatest Common Divisor (gcd). The standard definition of gcd is:

Definition. For any two integers a,b, at least one of which is nonzero,  we denote by gcd(a,b) their greatest common divisor, that is the greatest integer d such that there exist integers e,f such that a = de, b = df. In particular, if (a) is nonzero, we have  gcd(0,a) = gcd(a,0) = |a|.

But that definition is tailored for integers, while the concept of gcd is more general, and it has to do with the Unique Factorization Property. So here is a deeper and more formal definition (for non-zero integers)

In other words gcd(a,b) is the product  it's the product of all common primes, each raised to the power of the smallest exponent found in either number. The signs of the original integers do not affect the result.

Before proving the Proposition, we first prove the following Lemma

Lemma 1. Let a,b,c,d be integers such that gcd(a,b)=1 and ad = bc. Then there is an integer k such that

c = ka,

d = kb.                (1)

Proof. First note that if a = 0 and b = 0, then gcd(a,b) is undefined, therefore this case is implicitly excluded by the assumption that gcd(a,b) = 1. Suppose a = 0 and b ≠ 0. Then gcd(0,b) = |b|, and, since gcd(a,b) = 1, it follows that b = ± 1. From ad = bc it follows that c = 0. Take k =  ±d. Then d = kb and c = ka. Similarly if b = 0 and a ≠ 0. Let us now assume that both a ≠ 0 and b ≠ 0. By assumption  gcd(a,b) = 1 and ad = bc. Think of the unique prime decomposition of (b) on the right hand side of ad = bc. None of these primes entering the composition of b is in (a), since gcd(ab)=1. Therefore they must be all contained in d. Therefore there is an integer k such that d = kb. But then, from ad = bc, it follows kab = bc, and, since b ≠ 0, we have c = ka. QED                                                                                      

We can now return to proving Proposition 1.

Proof (of Proposition 1).  Since A is a nonzero matrix, at least one of it rows is nonzero. Suppose it is the first row: (a,b). For the second row the proof goes in a complete analogy.

Let g = gcd(a,b). Since at least one of a,b is nonzero, g = gcd(a,b) is well defined. Thus g is a positive integer, and there exist integers a',b' such that

a = ga',
b= gb',                (2)

and gcd(a',b') = 1. Since ad = bc, and g is positive, from ad = bc it follows that a'd = b'c. We can now apply Lemma 1 to deduce that there exist an integer k such that

c = ka',
d = k b'.            (3)

From (2) and (3) it follows that A is of the form:

A = {{ga',gb'},{ka',kb'}}. Setting v = (g,k)^T, w = (a',b') we have A = vw^T. QED

It follows also from the proof that the decomposition is unique if we require the two components of (w) to be coprime (i.e. if their gcd is 1).

Exercise 1. Find the decomposition of the matrix  x^, where x = (3,14,6,13) as mentioned  in the Example Part 2.

To be continued...

The magic of numbers 3,4,5

 Pikabu the cat knows how to count 0,1,2. Dr. Marek Wolf, the author of "Plato was right", in a response to my previous post "The magic of numbers 0,1,2" (featuring Pikabu ze cat), kindly informed me in a private communication, that there exists empirical evidence that hens (and other birds) can count to three. And indeed, on Researchgate we can easily locate the appropriate paper on this subject. The paper "Any bird and chicken can count to three " by Alexander Yurkin provides us with the following observation:

"It is known that any bird and chicken can count to three (if the chicken incubates 4 eggs or more, and someone  takes an egg, then she sits quietly, if the chicken incubates 3 eggs and someone takes one egg, then she begins to scream, about the fact that her eggs are stolen)."

Newborn chicks can count, at least up to three,
a new study suggests

The paper by Yurkin was inspired, in turn,  by another paper, by Pradeep Mutalik (or Hradeep Mitalik?) in Quanta Magazine, "How Randomness Can Arise From Determinism". Quanta Magazine goes into the Quantum Future of Quantum Waves. We are not there .... yet. So, let us, for a while, remain at the level of cats and birds, and integer numbers.

Reality is all entangled. Everything connects to everything in an acausal synchronistic web of connecting arrows. Every day, while in the hyperbaric chamber, at the pressure of 2.0 atmospheres and oxygen level 90%, I am either meditating or listening to an audio book. Recently I am listening to Dan Brown's novel "Origin".

And so yesterday I listened, in particular, to a fragment containing this piece:

"The Ancient of Days, Langdon thought, squinting through the darkness at Blake’s famous 1794 watercolor etching.(...)
The piece was so futuristic in style that, centuries later, the renowned physicist and atheist Stephen Hawking had selected it as the jacket art for his book God Created the Integers."

2006 edition cover.
2007 edition cover is, for some reason, totally unimpressive.

The paper "How Randomness Can Arise From Determinism", mentioned above, features the Pascal triangle

Puzzle 3: Predicting Individual Behavior

It starts with 1,11,then 121. There are no zeros shown in this triangle. In my previous post, "The magic of numbers 0,1,2", there were lot of zeros. Let us have a look at these 0,1,2 again:

Our attention concentrates on triangular patterns. It is so easy to miss the main point: 

In each row a squared plus b squared gives c squared.

The numbers are big, so you would need a 24 digit calculator (or Wolfram Alpha) to check this statement.  For instance

21x21 + 220x220 = 221x221

etc. 

Each row is a Pythagorean triple. 

How many are there of such triples? How to generate them? 

Everybody knows the famous 3,4,5 and 9+16=25. This is the "Mother Triple". 

But then what would be the "genealogical tree" of these big 0,1,2 triples in the image above? And what does it all have to do with "The Origin" and "Quantum Future"?

To be continued

P.S. 31-08-23 Actually studying:

Andrzej Trautman, Pythagorean Spinors and Penrose Twistors

Plato was right

Today's guest post is by Marek Wolf

When in 1997 I visited Isaac Newton Institute in Cambridge I bought a book "Men of Mathematics" written by E.T. Bell, edition from 1937 [1]. In the five story bookshop there were a few shelves with classics: "Mathematical Principles of Natural Philosophy" by Isaac Newton, "On the Origin of Species" by Charles Darwin, "Principles of Quantum Mechanics" by Paul A.M. Dirac from 1958 and the above-mentioned book by E.T. Bell.

Chapter nine of Bell's book  is devoted to Leonhard Euler (1707--1783). On page 147, citing de Morgan, Bell tells an anecdote, about Euler who was a deeply religious Calvinist. At the same time there lived Denis Diderot, who, as a progressive philosopher,  was atheist. The two men  met one day and Diderot asked Euler if he has a proof that God exists. Euler wrote some formula and said: "Hence God exists. Sir, please reply". Diderot embarrassed quickly walked away.

We do not know what the formula was, but here, I will present two formulas in the same spirit, which I often expose  students to during the first calculus lecture. The first is following: 

 Add two first odd numbers: 1+3 = 4, we added two numbers and the result is 4=22. Next we add five: 1+3+5=9, the summations of three first odd numbers gives 9=32. Next we add 7 and obtain 16=42, again the square of a number of added terms. It continues to be true up to to infinity. In formula we can write it as

The simplest proof is by induction.

The second example is following:

 We add cubes of consecutive natural numbers. Thus we have

1 + 23 = 1 + 8 = 9= 32 = (1+2)2 ,

1 + 23 + 33 = 36 = 62 = (1 + 2 + 3)2 ,

1 + 23 + 33 +  43 = 100 = 102 =  (1+2+3+4)2 : the sum of cubes of the consecutive integer numbers is a square of the sum of these numbers.

And again so on to infinity.

Isn't it miraculous and mysterious? It seems to hide some deep secret. As an equation we write

This identity is sometimes called Nicomachus’s theorem, after Nicomachus of Gerasa (c. 60 – c. 120 CE). About four centuries ago, Johannes Faulhaber (1580--1635) developed formulas for the power sums. He published many papers on this topic, initially he found particular sums up to exponent 7 in 1614. Subsequently Faulhaber gave a formula expressing the sum of the k-th powers of the first n positive integers. In no other case is a sum of powers of consecutive natural numbers equal to square of another sum. In [2] Sheila M. Edmonds proved that no higher power of n sums would be expressed as the square of the sum of powers of 1, 2, ...,, n. More precisely she proved that if

then p=3 and q=1.

These formulas (1) and (2) are the absolute and eternal truth. It means that they were true even before mankind appeared on the Earth. They were true also before the Big Bang (if you believe in it). It means that there is a world containing such mathematical truths independent of our real Universe. This idea was first put forward by the Greek philosopher Plato (428/427 or 424/423 – 348/347 BC). This system of beliefs in philosophy is called "Platonism". 

 What is the reality of mathematical theorems? Has the theorem existed anywhere until the moment it is formulated by a mathematician, or is it only the appearance of it in the head of a human that creates the reality it concerns?

Many mathematicians consider themselves Platonists.

Today Euler could argue using the axiom of choice. It is a kind of mathematical joke: choice axiom is equivalent to the existence of God. The choice axiom is a statement from set theory and was formulated in 1904 by Ernst Zermelo. The axiom of choice says that given any collection of non--empty sets (even if the collection is uncountable), it is possible to take one element from each set and construct a new set from these elements. Assuming the axiom of choice,   Zermelo proved that each set can be well-ordered. It means that in each set there exists the first element with respect to some ordering relation. In particular a set of all causes can be well ordered.  Hence there exists the first cause. 

 According to Saint Thomas Aquinas the first cause of everything was God. In the opposite direction it is simpler: God is an omnipotent being and thus can choose from each set by one element and the axiom of choice follows. In 1963 Paul Cohen (1934 -- 2007) proved that the axiom of choice is independent of other axioms of the set theory. It means it is possible to have consistent mathematics with or without the choice axiom. Therefore it "means" it is possible to have a world with God and a world without God.

German mathematician Don Zagier in his habilitation lecture in 1977 said "... upon looking at prime numbers one has the feeling of being in the presence of one of the inexplicable secrets of creation." [3]


[1] E.T. Bell, "Men of Mathematics. The lives and achievements of great mathematician from Zeno to Poincare ", A touchstone book, 1937, New York, London, Sydney

[2] Sheila M. Edmonds, Sums of Powers of the Natural Numbers, Mathematical Gazette, Vol. 41, No. 337 (Oct., 1957), pp. 187-188

[3] D.Zagier, The first 50 million prime number,  Math. Intellig. no. 0 (1977), p.8, left column

Marek Wolf