みんな大好きフィボナッチ数列

[Header Image Credit] The image is taken from Pinterest, and the text is from the book "The Da Vinci Code" by Dan Brown.

Spirals in nature

Does the following spiral look familiar?

It's the Lucas spiral, which is far less famous than the Fibonacci spiral.
Lucas numbers are defined in a manner very similar to Fibonacci numbers:

A more important concept is the Lucas sequence, which is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891).
The proper noun is pronounced in French, as the title of Japanese version of Wikipedia suggests: リュカ, NOT ルーカス. I checked YouTube videos and found that most of Americans pronounce as ルーカス, but let's not get too hung up on pronunciation.
The definition of the Lucas sequence is as follows:

The characteristic equation of the recurrence relation for Lucas sequences Uₙ(P,Q) and Vₙ(P,Q) is:
    x² − Px + Q = 0.
Let α and β be the roots of the characteristic equation x² − Px + Q = 0.
Let D be the discriminant : D = P² − 4Q. If D ≠ 0, α and β are distinct, and
    Uₙ = (αⁿ − βⁿ) / (α − β), 
    Vₙ = αⁿ + βⁿ.

By the definition, Fibonacci numbers are Uₙ(1, −1), and Lucas numbers are Vₙ(1, −1). Their formulae are:

Simply put, we should not confuse the Golden Ratio with the Fibonacci sequence:

821

今日8月21日は MMDD という十進表現にすると 821 である。YYYYMMDD と表現すると 20250821 が素数になるのは8月5日の記事に書いた通り：

この821という数(数字とは敢えて言わない)は一寸面白い：

• {821, 823, 827, 829} is the largest prime quadruplet of 3 digits.

• 821 = 14² + 5⁴ (Friedlander-Iwaniec prime).

• 821 is 56th irregular prime, and 827 is 57th irregular prime.

Bernoulli numbers ベルヌーイ数 & 関孝和

The definition of irregularity of a prime number is fundamental but not easy:
An odd prime number p is defined to be regular if it does not divide the class number 類数 of the p-th cyclotomic field 円分体 Q(ζₚ) where ζₚ is a primitive p-th root of unity. The class number of the cyclotomic field Q(ζₚ) is the number of ideals of the ring of integers Z(ζₚ) up to equivalence, in other words the order of the Ideal class group イデアル類群.
Therefore an odd prime number p is called irregular 非正則 when p divides the class number of the p-th cyclotomic field Q(ζₚ).
The first such example takes place at p = 37.
There is an equivalent definition, thanks to Kummer's criterion.
Ernst Kummer showed in 1850 that an equivalent criterion for regularity is that p does not divide the numerator of any of the Bernoulli numbers Bₖ for k = 2, 4, 6, ..., p−3.
There are 3 irregular primes of 2 digits, namely 37, 59 & 67, which we can prove using this criterion and the following facts:

In the 19th century the concept of regular primes played a crucial, albeit limited, role in the progress towards proving Fermat's Last Theorem.
この奇素数に対する正則性という概念は19世紀にフェルマー予想の部分的な証明において極めて重要な役割を果たしている。

The importance of Bernoulli numbers cannot be explained in a short sentence, but let me just mention that Seki Takakazu 関孝和 is known to have discovered "Bernoulli numbers" independently from Jacob Bernoulli as mentioned in Wikipedia. Here's another article by Discuss Japan:

Whilst Seki Takakazu and his school made significant contributions to 和算Japanese mathematics, they did not explicitly study the Fibonacci sequence. However, those mathematicians in Edo period studied 連分数 continued fractions, which leads to the Golden Ratio 黄金比 φ.

A caveat here. The Golden Ratio φ generates the Fibonacci sequence, but is NOT equivalent to the Fibonacci sequence, as mentioned earlier.

N.B. Let me take the liberty to clarify that I don't have the faintest interest in an empty propaganda if not a nonsense by those One Delta Ten Tango with the key phrase "Japanese First" which is effectively the same as 禍 MAGA.

Fibonacci numbers フィボナッチ数

では 821 はフィボナッチ数列とどう関係するのか？
これには二つの質問が考えられる：

• 821番目のフィボナッチ数は？ What about the 821st Fibonacci number?

• 821 はフィボナッチ数列の素因数に現れるか？ Are there any Fibonacci numbers which have 821 as a factor?

① It is denoted F₈₂₁, and is a composite:

② And 821 appears in F₂₀₅:

For any prime p, the rank of apparition, denoted Z(p), is the smallest index n>0 such that p divides Fₙ. Since the word "apparition" is so evocative, one might recall L'Apparition by Gustave Moreau.

素数は出現するのである。

Using this term, the rank of apparition Z(821) is 205. Then 821 should appear in ​F₄₁₀, F₆₁₅, F₈₂₀,... As a matter of fact:

これから次の一般的な問題が生じる：素数 p が フィボナッチ数の素因数になる条件は何か？
A question naturally arises: Under what condition a prime number p becomes a factor of a Fibonacci number?

これに対する回答の一つが Carmichaelの定理であるが、その手法は既に述べたリュカ数列を用いる一般的なものである。
One theory that address the above mentioned question is known as Carmichael's theorem, which uses the Lucas sequence.

Fibonacci primes フィボナッチ素数

素数であるようなフィボナッチ数は存在するのか否か？
Are there prime numbers that are also Fibonacci numbers ?

YES. It's not difficult to find them in the following sequence:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155,…

Fibonacci primes フィボナッチ素数 are listed in the following sequence:
https://oeis.org/A005478

2, 3, 5, 13, 89, 233, 1597, 28657, 514229, 433494437, 2971215073, 99194853094755497, 1066340417491710595814572169, 19134702400093278081449423917, 475420437734698220747368027166749382927701417016557193662268716376935476241,...

These numbers are indexed as Fₚ such as 89 = F₁₁, 233 = F₁₃ , 1597 = F₁₇, 28657 = F₂₃,... Some more Fibonacci primes are:

Interesting facts on Fibonacci numbers

This ⬆️ is not by accident, but is derived from the definition of the Fibonacci sequence. Another example is

Primes generated by Fibonacci primes F₄₃ & F₄₇

Examples of primes of the form x² + y⁴ (as per Friedlander-Iwaniec) using F₄₃ = 433494437 as a base:

Examples of primes of the form x³ + 2y³ (as per Heath-Brown) using F₄₇ = 2971215073 as a base:

Brilliant number, Chen prime, and Sophie-Germain prime close to F₁₃₁

Arithmetic progression from 137 to F₁₃₇

Two cases of two Fibonacci primes indexed by twin primes

(431, 433) is a twin prime pair (p₈₃, p₈₄).
The 431st & 443rd Fibonacci numbers are both prime numbers:

(569, 571) is a twin prime pair (p₁₀₄, p₁₀₅).
The 569th & 571st Fibonacci numbers are both prime numbers of 119 digits:

Clusters of primes which appear after Fibonacci prime F₅₀₉

F₅₀₉ is a prime number of 107 digits:
10597999265301490732599643671505003412515860435409421932560009680142974347195483140293254396195769876129909

And its next prime is F₅₀₉ + 668:
10597999265301490732599643671505003412515860435409421932560009680142974347195483140293254396195769876130577

The smallest twin prime pair larger than F₅₀₉ is (F₅₀₉ + 5292, F₅₀₉ + 5294):
10597999265301490732599643671505003412515860435409421932560009680142974347195483140293254396195769876135201
10597999265301490732599643671505003412515860435409421932560009680142974347195483140293254396195769876135203

The smallest prime quadruplet larger than F₅₀₉ is {p, p+2, p+6, p+8}
where p = F₅₀₉ + 1599416232 = 10597999265301490732599643671505003412515860435409421932560009680142974347195483140293254396195771475546141

These examples show that a twin prime pair is found much closer to F₅₀₉ than a prime quadruplet and scarcity of prime quadruplets.

Fibonacci semiprimes フィボナッチ半素数

F₃₄ & F₉₄ provide interesting examples of Fibonacci semiprimes:
    F₃₄ = 5702887 = 1597 × 3571 = F₁₇ × L₁₇
    F₉₄ = 19740274219868223167 = 2971215073 × 6643838879 = F₄₇ × L₄₇
where Lₙ stands for n-th Lucas number.
It's very easy to prove that

Here are two more examples showing the above relation:

Fibonacci number Fₙ is semiprime for the following n (9 ≤ n ≤ 1000):
19, 22, 26, 31, 34, 41, 53, 59, 61, 71, 73, 79, 89, 94, 101, 107, 109, 113, 121, 127, 151, 167, 173, 191, 193, 199, 227, 251, 271, 277, 293, 331, 353, 397, 401, 467, 587, 599, 601, 613, 631, 653, 743.
Hopefully the above list is correct…
One example of Fibonacci semiprime is:
F₆₁₃ = 89824816176020664407957288521932489478482660934516789842973 × 640533921256762287061021544191816475281134371932094984116800532167821

Closing

Let me close this article with two prime numbers 9311 (p₁₁₅₂) & 9677 (p₁₁₉₄):

• Both are irregular primes.

• F₉₃₁₁ is a 1946 digit prime and F₉₆₇₇ is a 2023 digit prime.

F₉₆₇₇ is the largest Fibonacci prime with 4-digit index.
After p = 9677, Fₚ (p-th Fibonacci number) is prime for the following:
14431, 25561, 30757, 35999, 37511, 50833, 81839, 104911, 130021, 148091, 201107,...
フィボナッチ素数は更に続くが、フィボナッチ素数が無限に存在するか否かは現時点でも未解決問題である。

🔳

P.S. There's a term called silver ratio 白銀比. In English version of Wikipedia it means 1:1+√2, but Japanese version of Wikipedia describes both 1:1+√2 and 1:√2. They are obviously different. The name silver ratio is by analogy with the golden ratio.
And only recently I learned the term "Yamato ratio" 大和比 for the first time. It represents 1:√2.
1+√2 has a connection with Pell numbers, very similar to Lucas numbers and Fibonacci numbers.