Bernstein 1994 noniterative 2adic

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conjecture_p1: Bernstein's non-iterative conjecture that every positive integer is the image under the 2-adic permutation Phi of a third of an integer, which his Theorems 2 and 3 show equivalent to the 3N+1 conjecture.

corollary_1: Bernstein's corollary that the 2-adic permutation Phi sends rational 2-adic integers to rational 2-adic integers, one half of the Periodicity Conjecture of Lagarias's survey.

corollary_2: Müller's theorem, reproved by Bernstein from his expansion of Phi, that the 2-adic permutation Phi conjugating H to the Collatz map is nowhere differentiable.

theorem_1: Bernstein's identity that the 2-adic permutation Phi carries the simple map H, which halves even inputs and subtracts one from odd inputs, to the Collatz map C, so that C of Phi of Q equals Phi of H of Q.

theorem_2: Bernstein's theorem that a 2-adic integer N some iterate of the Collatz map sends to one lies in the image under Phi of the thirds of integers, so the 3N+1 conjecture implies his non-iterative conjecture.

theorem_3: Bernstein's theorem that a positive integer N lying in the image under Phi of the thirds of integers has some Collatz iterate equal to one, so his non-iterative conjecture implies the 3N+1 conjecture.

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Daniel J. Bernstein, A non-iterative 2-adic statement of the 3N+1 conjecture, Proc. Amer. Math. Soc. 121 (1994) 405-408 (author-hosted copy, cr.yp.to/papers/231.pdf; 4 pp.).

Defines the permutation Phi of the 2-adic integers by paired expansions Q = 2^{d_0} + 2^{d_1} + ... and N = (-1/3)2^{d_0} + (-1/9)2^{d_1} + ..., over increasing sequences 0 <= d_0 < d_1 < ..., and proves (Theorems 2 and 3, p. 2 of the copy read) that the 3N+1 conjecture is equivalent to the non-iterative statement that Z^+ is contained in Phi((1/3)Z). The bridge is Theorem 1, that Phi conjugates H (halve if even, subtract one if odd) to the Collatz map C (halve if even, 3N+1 if odd). The applications on p. 3 give Corollary 1, that Phi maps rational 2-adic integers to rationals (half of the Periodicity Conjecture of Lagarias's 1985 survey), and Corollary 2, Müller's theorem that Phi is nowhere differentiable; the paper also derives from the expansions that Phi is a 2-adic homeomorphism and recovers Theorems L and B of that survey, and notes that the expansion extends to the AN+B problem for odd A and B. Pages cited are those of the typescript, numbered 1--4. Relevance: Gives an equivalent non-iterative 2-adic restatement of the 3N+1 conjecture, i.e. a reformulation of problem 1135.

Source: PDF. The copy read for this card is the author's typescript, which prints the header "Proceedings of the American Mathematical Society 121 (1994), 405–408." and no copyright or license line (pp. 1 and 4 read on the page images on 2026-10-02), from the author's papers page (https://cr.yp.to/papers.html, read 2026-10-02), which states no copyright, license or terms line; the term is unstated.

Bears on. #1135: Theorems 2 and 3 show that a positive integer reaches 1 under the Collatz map C exactly when it lies in Phi((1/3)Z), so the Conjecture (p. 1) is equivalent to the 3N+1 conjecture; the problem page records that a C-orbit reaches 1 exactly when the orbit of the problem's shortcut map f does. The paper proves the equivalence only, not either conjecture.

Result pages.

• Conjecture (p. 1): the definition of Phi and the statement that Z^+ is contained in Phi((1/3)Z).
• Theorem 1 (p. 2): C(Phi(Q)) = Phi(H(Q)).
• Theorem 2 (p. 2): if C^k(N) = 1 then N lies in Phi((1/3)Z).
• Theorem 3 (p. 2): a positive integer in Phi((1/3)Z) has some iterate C^k(N) = 1.
• Corollary 1 (p. 3): Phi maps rational 2-adic integers to rational 2-adic integers.
• Corollary 2 (p. 3): Phi is nowhere differentiable (Müller).

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