Fibonacci numbers and a metric on coprime pairs

In this paper, we introduce a metric on the set of pairs of coprime natural numbers. We explicitly construct a quasi-isometric embedding from the set of natural numbers into this metric space via Fibonacci numbers.

💡 Research Summary

The paper introduces a natural metric on the set X of unordered pairs of coprime natural numbers and shows that the sequence of consecutive Fibonacci pairs provides a (1, 1)‑quasi‑isometric embedding of the integers into this metric space.

Construction of the metric.
For a pair n = {n₁,n₂}∈X, every natural number m can be expressed as a linear combination m = n₁x + n₂y with integers x,y because {n₁,n₂} generates ℤ. The authors define

 qₙ(m) = min {|x| + |y| : m = n₁x + n₂y}.

Thus qₙ(m) is the smallest ℓ¹‑norm of a representation of m by the generators of n. For two pairs m,n∈X they set

 dₐ(m,n) = logₐ (max{qₙ(m), q_m(n)}) (a > 1).

Lemma 2.1 proves the sub‑multiplicative inequality qₙ(k) ≤ q_m(k)·qₙ(m), which yields the triangle inequality for dₐ. Non‑degeneracy follows from the observation that qₙ(m)=1 iff m⊂n, i.e. m is one of the two generators of n. Consequently dₐ is a genuine metric. The metric is essentially the restriction of Tsuboi’s metric (originally defined on certain groups) to the abelian group ℤ, and different choices of the base a>1 give quasi‑isometric metrics differing only by a constant multiplicative factor (logₐ b).

Embedding via Fibonacci pairs.
Let Fₙ denote the n‑th Fibonacci number and φ = (1+√5)/2 the golden ratio. Since consecutive Fibonacci numbers are coprime, the pairs

 fₙ = {Fₙ, Fₙ₊₁} ∈ X

are well defined. The main theorem (Theorem 1.1) states that the map

 ℕ → X, n ↦ fₙ

is a (1, 1)‑quasi‑isometric embedding when the metric d = d_φ is used.

The proof reduces to estimating q_{fₙ}(F_m) and q_{fₙ}(F_m+1). Using Honsberger’s identity

 F_{m}=F_{m‑n‑1}Fₙ + F_{m‑n}Fₙ₊₁,

the authors exhibit an explicit representation (x,y) = (F_{m‑n‑1}, F_{m‑n}) that yields the upper bound

 q_{fₙ}(F_m) ≤ |F_{m‑n‑1}| + |F_{m‑n}|.

For the lower bound they note that any representation F_m = xFₙ + yFₙ₊₁ implies

 F_m ≤ (|x|+|y|)F_{n+1},

hence q_{fₙ}(F_m) ≥ F_m/F_{n+1}.

Standard exponential estimates for Fibonacci numbers, namely

 φ^{n‑2} ≤ Fₙ ≤ φ^{n‑1},

translate these bounds into

 m‑n‑1 ≤ log_φ q_{fₙ}(F_m) ≤ m‑n + 1,

and the same holds for F_m+1. Consequently

 |m‑n|‑1 ≤ d(f_m,f_n) ≤ |m‑n| + 1,

which is precisely the definition of a (1, 1)‑quasi‑isometric embedding.

Generalizations.

1. k‑Fibonacci numbers.
   For any integer k≥1, the k‑Fibonacci sequence is defined by

 F_{k,0}=0, F_{k,1}=1, F_{k,n+2}=kF_{k,n+1}+F_{k,n}.

The associated “k‑metallic ratio” φ_k = (k+√(k²+4))/2 plays the role of φ. Lemma 4.2 gives the analogue of Honsberger’s identity, and Lemma 4.3 supplies exponential bounds for F_{k,n}. Repeating the same argument shows that the map

 n ↦ {F_{k,n}, F_{k,n+1}}

is a (1, 1)‑quasi‑isometric embedding of ℕ into (X, d_{φ_k}).

2. ℓ‑coprime tuples.
   For ℓ≥2 the authors consider the set

 X_ℓ = { {n₁,…,n_ℓ}⊂ℕ | gcd(n₁,…,n_ℓ)=1 }.

The function q_n(m) is extended to the ℓ‑dimensional ℓ¹‑norm, and the metric

 d_ℓ(m,n) = log_φ (max{q_n(m), q_m(n)})

is shown to be a metric. By taking the ℓ‑tuple of consecutive Fibonacci numbers

 f_ℓ,n = {F_n, F_{n+1},…, F_{n+ℓ‑1}},

the same bounding technique yields

 |m‑n|‑1 ≤ d_ℓ(f_ℓ,m, f_ℓ,n) ≤ |m‑n| + 1,

so the map n ↦ f_ℓ,n is again a (1, 1)‑quasi‑isometric embedding.

Significance and outlook.
The work bridges elementary number theory (properties of coprime pairs and Fibonacci identities) with large‑scale geometric concepts (quasi‑isometry). The metric is remarkably simple—based on the minimal ℓ¹‑norm of a linear representation—yet it captures enough structure to reflect the exponential growth of the Fibonacci sequence. The fact that the embedding constants are optimal (K=1, L=1) underscores the tightness of the construction. The extensions to k‑Fibonacci and higher‑arity coprime tuples suggest a broad framework: any linear recurrence with exponential growth and a suitable coprimality property can generate a quasi‑isometric copy of ℤ inside a combinatorial metric space built from coprime generators. Future research may explore other recurrences (e.g., Lucas, Tribonacci), connections with continued fractions, or applications to the geometry of groups where similar “generator‑norm” metrics appear.

Overall, the paper provides a clean, self‑contained demonstration that the natural arithmetic of Fibonacci numbers yields a precise quasi‑isometric model of the integer line inside a metric space defined purely by coprimality and linear combinations.