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.