A new mathematical preprint recasts the Collatz 3n + 1 map as an operator on sequences of Boolean polynomials and derives explicit formulas for several of its carry sequences. The work is a formal algebraic reformulation; it does not provide empirical evidence that every positive integer reaches 1.
Posted on arXiv as arXiv:2608.25791v1 and dated 26 Aug 2026, the manuscript asks how to represent the map in this language and how to express its carry sequences explicitly. Its supplied analysis also identifies two additional carry-sequence calculations as future work.
Binary arithmetic, recast
At the foundation is a Boolean quotient ring R over F2. The construction fixes an integer n and uses indeterminates x_i for 1≤i≤n; R∞ is the set of sequences in R. In the finite-support specialization over F2, those sequences correspond to base-2 expansions of an integer k. Here, “Boolean” means the algebra is built for 0-or-1 values, so binary information can be handled as polynomial coordinates.
Addition is modeled by a carry recursion. For sequences u and v, coordinate i is u(i)+v(i)+c(u,v;i). The initial carry is c(u,v;1)=0, while the next one is c(u,v;i+1)=u(i)v(i)+c(u,v;i)(u(i)+v(i)). In ordinary terms, the rule carries information from one binary position into the next.
The map in a different form
The paper introduces an alternative integer map by adding a power of two to three times the input, written O(k)=2v₂(k)+3k. It builds the matching formal operations as Add, PlusPowerTwo and Shift1, so the algebraic construction can be compared with the stated integer arithmetic.
Under the finite-support specialization, Add(y, PlusPowerTwo(y)) represents k+2v₂(k), Shift1(y) represents 2k, and O(y) represents 3k+2v₂(k). This is the paper’s stated bridge from the polynomial sequences to integer arithmetic, and it is explicitly tied to that specialization.
From recursion to formulas
The central coordinate identity gives O(y)(i)=δ(y;i)+γ(y;i)+y(i)P(y;i−1)+y(i)+y(i−1). Each coordinate therefore combines the two carry sequences with a current term, a prefix product and the preceding term.
For the first carry, Theorem 2 supplies a direct finite sum. At j+1, γ(y;j+1) is summed over k from 0 through ⌊(j−3)/2⌋, with terms P(y;j−2k−2)U(j−2k,j). The recursion is thus replaced, within the defined setting, by an explicit polynomial expression.
A lemma decomposes the second carry as δ(y;j+1)=δ1(y;j)+δ2(y;j). Theorem 3 then gives δ2(y;j+1) as a sum over i=1 through j−1 of y(i)V2(i+1,j), making that component explicit in y and V2.
The remaining component, δ1, is handled through a combinatorial sum over S∈Λ(t). Theorem 4 evaluates that sum as two finite sums involving P, V1 and U, with the conventions P(1,−1)=P(1,0)=U(t+1,t)=1.
A corollary gives δ1(y;j+1) a two-part finite-sum form. Its terms include P(y,1,i)V1(y,i+2,i+3+4j)U(y,i+5+4j,t) and V1(y,1,4i+1)U(4i+3,t). Together with the δ2 result, this makes both pieces of the second carry explicit in the Boolean-polynomial setting.
Lemma 1 states γ(y;j)δ(y;j)=0 for j≥1. The result is an algebraic exclusion identity within the paper’s definitions, not a measured numerical finding.
A formal result, not a final answer
The work stays within the formal setting it defines: sequences in R∞, Boolean polynomial identities and finite-support specializations. Its correspondence to integer arithmetic is stated under the specified specialization rather than as an experimental test.
The Further Work section leaves two analyses unresolved: the carry sequence for PlusPowerTwo(y) and Shift1(y), and the carry sequence for Add(PlusPowerTwo(y), Shift1(y)) and y. For that reason, the preprint’s contribution is best read as a structured algebraic reformulation with explicit identities—not as a solution to the original all-positive-integer convergence question.
Paper data and sources
Original title: A Boolean polynomial operator for the Collatz $3n+1$ problem
Authors: Mario DeFranco
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text