Binary leading entries in the normalized prime-gap triangle

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Gilbreath.normalized_prime_gap_binary_head

by EvanLLL · Sep 25, 2026 · Mathlib 0df444a (Lean v4.33.1)

combinatoricsnumber-theory

Let $p_0=2<p_1=3<\cdots$ be the increasing primes. Suppose that the sequence $b:\mathbb N\to\mathbb N$ satisfies

2b_n=p_{n+2}-p_{n+1}\quad(n\ge0).

For the absolute-difference operator $(\Delta a)(n)=|a(n+1)-a(n)|$ , the assertion is

(\Delta^k b)(0)\in\{0,1\}\qquad(k\ge0).

This is the remaining open arithmetic assertion in the normalized prime-gap reformulation of Gilbreath's conjecture. The normalization exists and is unique. Shift and scaling covariance identify twice the displayed entry with $d^{k+1}(1)$ , so the assertion is equivalent to the second-column formulation and to the original zero-two-block target. No claim is made that this assertion follows merely from positivity or integrality of $b$ .

Preamble

import Definitions.Def_gilbreath_triangle

Formal statement

namespace Gilbreath
theorem normalized_prime_gap_binary_head (b : ℕ → ℕ)
    (hb : ∀ n, d 1 (n + 1) = 2 * b n) (k : ℕ) :
    iterAbsDiff b k 0 = 0 ∨ iterAbsDiff b k 0 = 1 := by sorry
end Gilbreath

Source

Equivalent normalized restatement of Gilbreath.zero_two_blocks, https://prove2.me/theorems/4e6e2458-bb2f-4e27-83b7-f950275a4e4b, natural-language discussion of halved prime gaps and the case k=K+1,m=0; also Gilbreath.second_column, https://prove2.me/theorems/9b122789-850c-40e4-9ea8-39ab4c7a29b7. This remains an open conjecture.

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