a step beyond Euclid and Fermat, 5

For the sake of reference, you may like to open the previous note, step 4, in a separate window.

Let

Par(n)  :=  { k : 0 ≤ k ≤ n and k = n mod 2}

be the set of all non-negative numbers k ≤ n, which have the same parity as n; e.g.

• Par(0) = {0}
• Par(1) = {1}
• Par(2} = {0 2}
• Par(3) = {1 3}
• Par(4) = {0 2 4}

etc.

It follows from the initial values of the sequences (g(n) and (A(n), and from the last equality of "step 4" that:

• A(n)  =  ∏_{k ∈ Par(n)} g(k)     ∀ n=1 2...
  

The first three terms, g(0) g(1) g(2), of sequence (g(n)), are pairwise relatively prime.

If every two of the terms g(0) ... g(n) (with different indices) are relatively prime then obviously A(n-1) and A(n) are relatively prime too. Since, by the very definition, g(n+1) is the difference of A(n) and A(n-1), g(n+1) is relatively prime with A(n-1) and with A(n), which means that g(n+1) is relatively prime to every previous term g(0) ... g(n). This holds for every n=2 3 ... Thus every two different terms of sequence (g(n)) are relatively prime.

a step beyond Euclid and Fermat, 4

I've introduced the Euclid sequence e(n), and Fermat introduced his Fermat sequence f(n). Now let me introduce sequence g(n), which again has pairwise relatively prime terms. In another post I will show that sequence g is essentially different from sequences e and f and from all sequences euf. It is also satisfying that the terms g(n) are asymptotically smaller than e(n) and f(n), that they are of the order of the square root of e(n) or of f(n).

First, let me introduce an auxiliary sequence A(n):

• A(1) := 3
• A(2) := 10
• A(n+1) := A(n-1) ⋅ (A(n) - A(n-1))     ∀ n= 2 3 ...

so that A(3) = 3*(10-3) = 21, A(4) = 10*(21-10) = 110, etc.

It is not too hard to see that

A(n) = n mod 2     ∀ n=1 2 ...

Now let

• g(0) := 2
• g(1) := 3
• g(2) := 5
• g(n+1) := A(n) - A(n-1)     ∀ n=2 3 ...

so that g(3) = 7, g(4) = 11, g(5) = 89, etc. Also:

• A(n+1) := A(n-1) ⋅ g(n+1)     ∀ n= 2 3 ...

It is a fascinating topic to me but I am tired now. I'll continue later.

neologism, 1

poe tr y

        poetry you're a difficult lover
        i know you prefer it outdoors
        you desire cold shores and tall mountains
        sun burning  rain camouflage  and soft snow

        you like fireplay randomly cracking
        ornaments moving on the wall
        puffed pillows under your convex buttocks
        never worried about closing your doors






wh,
© 1991

==================

neologism:

• fireplay

In "advancing into the sunset" (see post kenning, 3) there was another neologism:

• selfboat

a step beyond Euclid and Fermat, 3

Let's make our first step beyond Euclid and Fermat. Their sequences are so similar that they beg to be placed under a roof of a common generalization. Let a b be two integers. Let's define:

• euf_{a b}(0)  :=  a+b;
• euf_{a b}(n+1)  :=  (euf_{a b}(n) - b) ⋅ euf_{a b}(n) + b
  

for every n=0 1...

Then

• euf_{a b}(n+1)  =  b + ∏_k=0...n euf_{a b}(k)
  

for every n=0 1...

Of course, if  B := euf_{a b}(k) - a  for certain non-negative integer k then:

• euf_{a B}(n)  =  euf_{a b}(k+n)

for every n=0 1...

For special values of integers a b, the sequence (euf_{a b}(n) : n=0 1...) becomes the Euclid sequence, when (a b) := (1 1), or the Fermat sequence, when (a b) := (2 3); when b=1 then we suppress  b  by writing

euf_a(n)  :=  euf_{a 1}(n)
Thus:

• e(n) = euf₁(n) — the Euclid sequence;
• f(n) = euf₂(n) — the Fermat sequence.

On the other hand, when a:=0, we obtain the constant sequence of values b.

The following properties of an Euclid-Fermat sequence euf_{a b} are equivalent:

• integers a b are relatively prime, i.e. gcd(a b) = 1;
• euf_{a b}(k) and euf_{a b}(n) are relatively prime whenever k and n are different.

We may rewrite the above simple recursive formula, which expresses euf_{a b}(n+1) in terms of euf_{a b}(n) as follows:

• euf_{a b}(n+1)  =  (euf_{a b}(n) - b/2)² + b - b²/4

This formula allows to study the rate of increase (or decrease) of an Euclid-Fermat sequence.

a step beyond Euclid and Fermat, 2

Fermat has defined his numbers:

• f(n) := 2²ⁿ + 1 for every n=0 1 ...

hoping that all of them are prime. Following Euler, we will see in another entry to this blog that this is not so. At this time let's just compare the Euclid numbers e(n) with Fermat numbers f(n). Surprisingly, they are quite similar in more than one way:

• f(0) = 3
• f(n+1) = (f(n) - 2) ⋅ f(n) + 2
• f(n+1) = 2 + ∏_k=0...n f(k)
  

for every n=0 1...

Polya has used the last formula to partially vindicate Fermat's hope—as in the case of the Euclidean sequence, also every two different Fermat numbers are relatively prime (hence once again we see that there are infinitely many different prime numbers). It follows that:

• p(n) ≤ f(n-1)     for every n=1 2...
  

---

The iterative formula for  f(n+1)  can be rewritten equivalently as follows:

• f(n+1) = (f(n) - 1)² + 1

for every n=0 1...

a step beyond Euclid and Fermat, 1

The eternally elegant Euclid's proof of infinitude of the set of prime numbers can be rephrased as follows: let

• e(0)  :=  2
  
• e(n+1)  :=  (e(n) - 1) ⋅ e(n) + 1

Then  (e(n) : n=1 2...)  is an increasing sequence of natural numbers such that each of its two different terms are relatively prime.

Indeed,

• e(n+1)  :=  1 + ∏_k=1...n e(k)

for every  n = 0 1 ...

Let  (p(k) : k=0 1 ...)  be the increasing sequence of all prime numbers. We see that

• p(n)  ≤  e(n)   for every n=0 1...

In fact:

• p(0) = e(0) = 2
• p(1) = e(1) = 3

but

• p(2) = 5 < 7 = e(2)

and

• p(n)  <  e(n)   for every n=2 3...

kennings, 3

  advancing into the sunset

    the two dimensional surface
    and the selfboat of my body

    from another dimension
    your warm presence
    seems
           in this cool ocean
    more real
          than my own

    with every breath of the salty water
    i taste roses
      of your wavy breasts

    the ocean passes by (the sun is red)

    i wear horizon
        'round my head

wh ©
1989/1990
dec/dec

---

There is 1 diagonal and 2 non-diagonal kennings in the above poem:

diagonal:

• roses of breasts — nipples and their aureoles;

non-diagonal:

• the selfboat of my body — this kenning is an extraction. And "selfboat" is a neologism.
• breath of the salty water

The last phrase has a potential to become a diagonal kenning. In the given poem it describes the breath of the lyrical subject rather than of the ocean. The latter interpretation is possible as a secondary one. Anyway, that latter interpretation allows for a diagonal kenning of fog or wind at the sea. Thus I will include this last kenning in both categories of kennings.

---

Let me compile the kennings which have appeared in the 3 poems posted so far.

Diagonal kennings:

• breath of the salty water — sea wind or sea fog;
• God's toys — church buildings;
• roses of breasts -- nipples and their aureoles;
  
• stars' backyard — mountains or a town up in the mountains.

Non-diagonal kennings:

• breath of the salty water
  

• grass lawn

• the code of behavior

• the meadow of slumber

• the selfboat of my body -- an extraction
  

• the sheets of music

• the sky of May

Neologisms:

• selfboat -- the body of a swimmer or simply a swimmer.

kennings, 2

This time I'll present a 2-part poem which features non-diagonal kennings.

==


a concert outdoors

          the code of behavior
          --------------------


  the trees and bushes
   steal sun and shadow
    from one another
     and
  step on your toes  trip you  trip you
   till you fall
    into the narrow patch
     of the meadow of slumber
  ... you HEAR
   the sheets
    attached to branches and shoots
     by clothes-pins
  the sheets of music ...

  silence
   wakes you up
    the guys under the tree
     clean instruments
  alright  you clap


____________________________________________



         Small Girls
      Have Great Future



  the sky of May
  punishes the audience

  a small girl
  knows none of that

  the orchestra breathes
  under the oak

  the girl follows her yellow and black
  pacifier   the propeller must be rotating
                      too fast to see

  she sounds
  like a small bee

  20 years from now
  she'll marry
  a devoted bear

wh ©
1991-05-05



======================================

I am much more interested in diagonal kenning than in the non-diagonal ones. Since people tend to confuse them, I have decided to list both kinds, and to separate them. Most of the time the distinction is very clear, the obscure cases are relatively rare.

Non-diagonal kennings:

• the code of behavior

• the meadow of slumber

• the sheets of music

• the sky of May

For the sake of convenience, let me collect the kennings from the previously posted poem ("6000 feet"). There were two, and both were diagonal:

Diagonal kennings:

• stars' backyard -- mountains or a town up in the mountains;
• God's toys -- church buildings.

Actually, there was also one non-diagonal:

• grass lawn

One can see already from the above 5 examples of non-diagonal kennings that they represent several different semantical constructions, and they also have various relation to the common language--e.g. the expression "grass lawn" is simply a part of the common English, and "the sheets of music" is almost like that too. However, there is a dramatic difference between the usage of the two in the respective poems.

kennings, 1

Let me (try to) start a presentation of my poems and my kennings featured in my poems.


6000 feet

       6000 feet up plus two
       of my own
       in the stars' backyard
       surrounded by God's toys
       catholic lutheran methodist pentecostal...
       and grass lawns which grow
       cigarette butts
       will I join
       the midget trees
       in the midget town?



wh, ©
1996-jan/feb

Diagonal kennings:

stars' backyard -- mountains or a town up in the mountains;
God's toys -- church buildings.

projects?

Let me think aloud. I am considering writing here, in an ad hoc, irregular way, about some of my projects or projects of projects. Despite the chaotic, unsystematic entries, I still may collect here some materials, something like fractional data bases. This may complement and help me to develop special sites for the respective projects (more ☻) systematically.

This plan doesn't feel like a blog, it may even go against the idea and spirit of a blog, but this blog, or whatever it is going to be, is first of all for me. Also, such a who knows what may be just as interesting/boring as a regular blog.

In particular, I am thinking about starting here a data base of what I call diagonal kennings. I used to call them "skaldic kennings", and I would prefer such a nice name if it were not for the misunderstandings caused by such a name--I would use it formally, according to my own definition, while others would quote some poems, and would claim that I am wrong ☺. If I were (and perhaps I will) to have here such a data base of diagonal kennings, then I would also post here the poems which feature the respective kennings.

meeting a great man (part 2)

Professor gave me his notes to read. He was making a steady progress in a nice, tasty, delicate style, introducing neat notions on his way. On the other hand, the whole task was infinitely simpler than what he has done in the past; the problem was well within the framework of elementary computer science, it could have been an exercise for bright high school kids, or for undergraduates, or for the beginning graduate students; furthermore, Professor didn't know, and the company didn't recognize that I have already solved that problem for them; more than that, with their programmer, the two of us have programmed my solution; furthermore--with my brother, we programmed the inverse problem, which was significantly harder (but still not such a big deal). The company had the working code from us, the two of us and their programmer had tested our program.

Professor knows and has achieved in the past a zillion times more than me; the result, for which he is famous, will be a jewel of the history of mathematics forever; but these days a single man, even when he's as outstanding as him, knows only a fraction of what science and technology has to offer.

Also, Professor was not exactly young at the time. He is older than me, while I was at the time not just old, and not even ancient, but already archaeological.

meeting a great man

The last time I met a great mathematician, I'll call him Professor, was five and a half years ago, under weird circumstances. We both had consulted (independently) for a company owned and run by a crazy guy, whose obsession was to be famous for a scientific and technological break through. Unfortunately, the owner's idea was obviously totally wrong, much more obviously wrong than Perpetuum Mobile.

It was also the last time I made some money (actually, good money, which is another story). Professor made good money too but I am not sure if he got everything which they owed him (the company was a bit less stable than Microsoft or Google).

I'd seen Professor only for two days. It was enough to observe his modest but dignified, diplomatic conduct, and quiet, systematic way of working. I said "diplomatic" because Professor would never say anything critical about the company's technological trust. He just would say nothing except for calling it "NN's technology", with just a slight smile. I was telling them, all the time, that their effort does not make sense. Professor acted in a much more mature way. They didn't ask him about their "technology", hence he didn't volunteer his opinion. He was only concerned about his own task, which he had treated seriously. He came to the office in the morning, then he worked the full day. Day after day (I don't know for how long).

***

It's nearly 4am. I'll continue later.

three musketeers

It is almost 2am now. My brother and I will take our father to his doctor before noon.

Let me start and hope for the best

I have started and abandoned a small bunch of blogs at different servers already. Will I stick to this one? I hope so (again ☺) but it has to be seen.