Narayana numbers as concatenations of repdigits

DefinitionConcatenations of two repdigits in a general base#✓fully proved

A natural number \(n\) is a concatenation of two repdigits in base \(b\) if there are digits \(a_1, a_2\) with \(1 \le a_1 \le b - 1\), \(0 \le a_2 \le b - 1\) and \(a_1 \ne a_2\), and block lengths \(l, m \ge 1\), such that \((b - 1) n = a_1 (b^l - 1) b^m + a_2 (b^m - 1)\). The digit conditions force \(b \ge 2\), and the equation says that the base-\(b\) expansion of \(n\) is \(l\) copies of \(a_1\) followed by \(m\) copies of \(a_2\).

Because of the condition \(a_1 \ne a_2\), a repdigit such as \(4 = (11)_3\) is not a concatenation of two repdigits in base \(3\). For \(b = 10\) this is the notion of a concatenation of two decimal repdigits.

public def IsConcatTwoRepdigitsBase (b n : ℕ) : Prop :=
  ∃ a₁ a₂ l m : ℕ, 1 ≤ a₁ ∧ a₁ < b ∧ a₂ < b ∧ 1 ≤ l ∧ 1 ≤ m ∧ a₁ ≠ a₂ ∧
    (b - 1) * n = a₁ * (b ^ l - 1) * b ^ m + a₂ * (b ^ m - 1)

Sources

• Prasanta Kumar Ray and Kisan Bhoi, Narayana Numbers Which Are Concatenations of Two Base b Repdigits, INTEGERS 24 (2024), Article A20, Section 1 (Equation (1)).

Used by 2

Used byExamples of two-repdigit concatenations in bases 2 and 5 · Narayana numbers that concatenate two repdigits in bases 2 to 9

TheoremExamples of two-repdigit concatenations in bases 2 and 5#✓fully proved

The number \(4\) is a concatenation of two repdigits in base \(2\), since \(4 = (100)_2\), and the number \(406\) is a concatenation of two repdigits in base \(5\), since \(406 = (3111)_5\).

public theorem isConcatTwoRepdigitsBase_examples :
    IsConcatTwoRepdigitsBase 2 4 ∧ IsConcatTwoRepdigitsBase 5 406

Sources

• Prasanta Kumar Ray and Kisan Bhoi, Narayana Numbers Which Are Concatenations of Two Base b Repdigits, INTEGERS 24 (2024), Article A20, Theorem 1.

Uses 1

Statement usesConcatenations of two repdigits in a general base

TheoremNarayana numbers that concatenate two repdigits in bases 2 to 9#●ready to prove

Let \((N_n)\) be Narayana's cows sequence and let \(2 \le b \le 9\). Then \(N_n\) is a concatenation of two repdigits in base \(b\) if and only if \((n, b)\) is one of the following \(43\) pairs: \(n = 4\) and \(b = 2\); \(n = 5\) and \(b = 3\); \(n = 6\) and \(b \in \{2, 4\}\); \(n = 7\) and \(b \in \{2, 3, 4, 6\}\); \(n = 8\) and \(b \in \{3, 4, 5, 6, 7, 9\}\); \(n = 9\) and \(b \in \{4, 5, 6, 7, 8, 9\}\); \(n = 10\) and \(b \in \{5, 6, 7, 8, 9\}\); \(n = 11\) and \(b \in \{2, 7, 8, 9\}\); \(n = 12\) and \(b \in \{3, 4, 7, 8, 9\}\); \(n = 13\) and \(b \in \{2, 4, 5, 7, 8\}\); \(n = 14\) and \(b = 6\); \(n = 17\) and \(b \in \{7, 9\}\); \(n = 18\) and \(b = 5\).

The corresponding values and expansions are \(N_4 = 2 = (10)_2\), \(N_5 = 3 = (10)_3\), \(N_6 = 4 = (100)_2 = (10)_4\), \(N_7 = 6 = (110)_2 = (20)_3 = (12)_4 = (10)_6\), \(N_8 = 9 = (100)_3 = (21)_4 = (14)_5 = (13)_6 = (12)_7 = (10)_9\), \(N_9 = 13 = (31)_4 = (23)_5 = (21)_6 = (16)_7 = (15)_8 = (14)_9\), \(N_{10} = 19 = (34)_5 = (31)_6 = (25)_7 = (23)_8 = (21)_9\), \(N_{11} = 28 = (11100)_2 = (40)_7 = (34)_8 = (31)_9\), \(N_{12} = 41 = (1112)_3 = (221)_4 = (56)_7 = (51)_8 = (45)_9\), \(N_{13} = 60 = (111100)_2 = (330)_4 = (220)_5 = (114)_7 = (74)_8\), \(N_{14} = 88 = (224)_6\), \(N_{17} = 277 = (544)_7 = (337)_9\) and \(N_{18} = 406 = (3111)_5\).

public theorem_wanted narayanaCowsSequence_isConcatTwoRepdigitsBase_iff {n b : ℕ} (hb : 2 ≤ b)
    (hb9 : b ≤ 9) :
    IsConcatTwoRepdigitsBase b (narayanaCowsSequence n) ↔
      (n, b) ∈ ({(4, 2), (5, 3), (6, 2), (6, 4), (7, 2), (7, 3), (7, 4), (7, 6), (8, 3),
        (8, 4), (8, 5), (8, 6), (8, 7), (8, 9), (9, 4), (9, 5), (9, 6), (9, 7), (9, 8), (9, 9),
        (10, 5), (10, 6), (10, 7), (10, 8), (10, 9), (11, 2), (11, 7), (11, 8), (11, 9),
        (12, 3), (12, 4), (12, 7), (12, 8), (12, 9), (13, 2), (13, 4), (13, 5), (13, 7),
        (13, 8), (14, 6), (17, 7), (17, 9), (18, 5)} : Finset (ℕ × ℕ))

Sources

• Prasanta Kumar Ray and Kisan Bhoi, Narayana Numbers Which Are Concatenations of Two Base b Repdigits, INTEGERS 24 (2024), Article A20, Theorem 1.

Uses 2

Statement usesConcatenations of two repdigits in a general base · Narayana's cows sequence

TheoremGuzmán Sánchez–Luca lemma: \(L < 2^r H (\log H)^r\)#●ready to state

Lemma 3 (Guzmán Sánchez and Luca): for real r ≥ 1 and H with H > (4r²)^r and H > L / (log L)^r, we have L < 2^r H (log H)^r. The hypothesis H > 0 is implied by H > (4r²)^r and is dropped.

Sources

• Prasanta Kumar Ray and Kisan Bhoi, Narayana Numbers Which Are Concatenations of Two Base b Repdigits, INTEGERS 24 (2024), Article A20, Lemma 3.

TheoremLogarithmic bounds for Narayana numbers that concatenate two repdigits#●ready to prove

Let \((N_n)\) be Narayana's cows sequence and let \(\alpha\) be a real number with \(\alpha^3 = \alpha^2 + 1\) (so \(\alpha \approx 1.46557\) is the unique real root of \(x^3 - x^2 - 1\)). Let \(b \ge 2\), \(n \ge 1\) and \(l, m \ge 1\) be natural numbers, and let \(a_1, a_2\) be digits with \(1 \le a_1 \le b - 1\), \(0 \le a_2 \le b - 1\) and \(a_1 \ne a_2\). If \((b - 1) N_n = a_1 (b^l - 1) b^m + a_2 (b^m - 1)\), that is, the base-\(b\) expansion of \(N_n\) is \(l\) copies of \(a_1\) followed by \(m\) copies of \(a_2\), then \((l + m - 1) \log b + \log \alpha < n \log \alpha < (l + m) \log b + 1\).

public theorem_wanted narayanaCowsSequence_concat_log_bounds {α : ℝ} (hα : α ^ 3 = α ^ 2 + 1)
    {n b a₁ a₂ l m : ℕ} (hb : 2 ≤ b) (hn : 1 ≤ n) (ha₁ : 1 ≤ a₁) (ha₁b : a₁ < b)
    (ha₂b : a₂ < b) (hl : 1 ≤ l) (hm : 1 ≤ m) (hne : a₁ ≠ a₂)
    (heq : (b - 1) * narayanaCowsSequence n = a₁ * (b ^ l - 1) * b ^ m + a₂ * (b ^ m - 1)) :
    ((l + m - 1 : ℕ) : ℝ) * Real.log b + Real.log α < n * Real.log α ∧
      n * Real.log α < ((l + m : ℕ) : ℝ) * Real.log b + 1

Sources

• Prasanta Kumar Ray and Kisan Bhoi, Narayana Numbers Which Are Concatenations of Two Base b Repdigits, INTEGERS 24 (2024), Article A20, Lemma 1.

Uses 1

Statement usesNarayana's cows sequence

2 fully proved2 ready to prove1 ready to state

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