Let B_n=f(x)=∑_i=0^nvarepsilon_i x^i: varepsilon_i∈0,1, varepsilon_n=1. The Erdős–Mauduit–Sárközy conjecture asserts that there exists a constant c>0 such that #f∈B_n: gcd(f,f')=1 in Z[x]sim c,2^n qquad(n→∞). Equivalently, a positive limiting proportion of integer polynomials with binary coefficients are squarefree.

Erdős–Mauduit–Sárközy conjecture

Let $\mathcal{B}_n=\left\{f(x)=\sum_{i=0}^{n}\varepsilon_i x^i:\ \varepsilon_i\in\{0,1\},\ \varepsilon_n=1\right\}$ . The Erdős–Mauduit–Sárközy conjecture asserts that there exists a constant $c>0$ such that

\#\left\{f\in\mathcal{B}_n:\ \gcd(f,f')=1\text{ in }\mathbb{Z}[x]\right\}\sim c\,2^n\qquad(n\to\infty).

Equivalently, a positive limiting proportion of integer polynomials with binary coefficients are squarefree.

Progress summary

Partially solved

A new paper settles a related missing-digit problem in a function-field setting, but the original integer problem remains open.

The Erdős–Mauduit–Sárközy conjecture concerns squarefreeness for integer polynomials whose coefficients omit digits. The latest result addresses a corresponding function-field version rather than the original conjecture.

August 2026 function-field analogue

An arXiv paper resolves the analogue for binary coefficients and proves a more general estimate for random coefficients. This is genuine progress on the analogue, but it does not settle the original integer-polynomial conjecture.

Current status (as of August 2026): the function-field analogue is resolved for binary coefficients, while the original Erdős–Mauduit–Sárközy conjecture remains open.

Sources

arxiv.org

Sources & referencesView supporting material

Primary source

arXiv

Additional references

Squarefree polynomials with missing digits — arXiv

finite-fields number-theory

2 days ago