Let a0a_0 be a decimal digit and A\mathcal{A} the set of integers whose base-10 expansion avoids a0a_0, a sparse set with #A(X)≍Xlog⁡9/log⁡10\#\mathcal{A}(X)\asymp X^{\log 9/\log 10} elements up to XX. Maynard (Inventiones Mathematicae, 2019) proved that A\mathcal{A} contains infinitely many primes — indeed #{p∈A(X)}≍#A(X)/log⁡X\#\{p\in\mathcal{A}(X)\}\asymp\#\mathcal{A}(X)/\log X — but stated two precise questions that remain unsolved. First, prove the conjectural asymptotic formula #{p∈A(X)}=(κA+o(1))#A(X)/log⁡X\#\{p\in\mathcal{A}(X)\}=(\kappa_{\mathcal{A}}+o(1))\#\mathcal{A}(X)/\log X with the explicit constant κA\kappa_{\mathcal{A}} predicted by the coprimality heuristic; this asymptotic is known for one excluded digit in every base q≥12q\ge 12, but in base 10 only the order of magnitude is known, the obstruction being insufficient Type II (bilinear-sum) information. Second, prove that there are infinitely many primes whose decimal expansion avoids any two prescribed digits, where the digit set shrinks to ≍Xlog⁡8/log⁡10\asymp X^{\log 8/\log 10} elements and current methods fail entirely. The later literature — Maynard's companion large-base paper and subsequent work on Vinogradov-type theorems, distribution in arithmetic progressions, and polynomial values in large bases — concerns large bases or auxiliary representations; both base-10 questions remain open.

Background

The decimal (base-10) expansion of an integer nn is its representation n=∑i≥0ni10in=\sum_{i\ge0}n_i10^i with digits ni∈{0,…,9}n_i\in\{0,\dots,9\}. Fix a digit a0a_0 and let A\mathcal{A} denote the set of positive integers whose expansion contains no digit equal to a0a_0. Since each of the roughly log⁡10X\log_{10}X digit positions of an integer below XX takes one of 9 allowed values, #A(X)≍Xlog⁡9/log⁡10=X0.954…\#\mathcal{A}(X)\asymp X^{\log 9/\log 10}=X^{0.954\ldots}, where f≍gf\asymp g means c1f≤g≤c2gc_1f\le g\le c_2g for absolute constants c1,c2>0c_1,c_2>0; thus A\mathcal{A} is a sparse (zero-density) subset of the integers. Very few sparse sets without multiplicative structure are known to contain infinitely many primes, and the natural heuristic — that being prime is statistically independent of digital conditions once the last digit is restricted to be coprime to 10 — predicts about #A(X)/log⁡X\#\mathcal{A}(X)/\log X primes in A(X)\mathcal{A}(X).

The arithmetic of digit-restricted sets has a long history. Dartyge and Mauduit (2000) showed that A\mathcal{A} contains infinitely many integers with at most two prime factors, exploiting that A\mathcal{A} is well-distributed in arithmetic progressions. Mauduit and Rivat (2010) proved that the sum of digits of primes is equidistributed modulo every mm, and Bourgain (2015) proved that one can prescribe a positive proportion of the binary digits and still find primes. Maynard (2019) established the order-of-magnitude result: for every a0∈{0,…,9}a_0\in\{0,\dots,9\}, #{p∈A(X)}≍#A(X)log⁡X,\#\{p\in\mathcal{A}(X)\}\asymp\frac{\#\mathcal{A}(X)}{\log X}, so in particular there are infinitely many primes missing any prescribed decimal digit.

Maynard's proof applies the Hardy–Littlewood circle method — which counts solutions of an additive equation by integrating the product of the Fourier transforms of the two sides, splitting frequencies near rationals with small denominator (major arcs, handled by distribution in arithmetic progressions) from the rest (minor arcs, handled by cancellation) — to the binary problem pairing primes against A\mathcal{A}. The minor arcs are controlled by Harman's sieve, a sieve method that rewrites the von Mangoldt function Λ(n)\Lambda(n) (the logarithm-weighted indicator of prime powers) as a combination of Type I sums, in which the digit-restricted variable is multiplied by a long smooth variable, and Type II bilinear sums ∑a∼A∑b∼Bαaβb\sum_{a\sim A}\sum_{b\sim B}\alpha_a\beta_b over two variables of comparable size. The key input is better-than-square-root cancellation in the L1L^1 norm of the Fourier transform of the digit-restricted set, obtained by decorrelating the Diophantine conditions under which the primes' Fourier transform is large from the digital conditions under which the digit set's Fourier transform is large; the proofs use the geometry of numbers, the large sieve (a quadratic inequality controlling how many integers can lie in prescribed residue classes), and moment estimates for a Markov process governing digit carries in products.

Two limitations of the base-10 argument are the source of the open questions. First, the available Type II range — the interval of bilinear-sum sizes that can be controlled, roughly N∈[X0.36,X0.425]N\in[X^{0.36},X^{0.425}] — is too narrow to represent Λ\Lambda exactly, so Harman's sieve only minorizes the primes, yielding a lower bound of the correct order but not an asymptotic. The coprimality heuristic predicts, with φ\varphi denoting Euler's totient function, #{p∈A(X)}=(κA+o(1))#A(X)log⁡X,κA=10(φ(10)−1)9φ(10) if (10,a0)=1,κA=109 otherwise,\#\{p\in\mathcal{A}(X)\}=(\kappa_{\mathcal{A}}+o(1))\frac{\#\mathcal{A}(X)}{\log X},\qquad\kappa_{\mathcal{A}}=\frac{10(\varphi(10)-1)}{9\varphi(10)}\ \text{if }(10,a_0)=1,\quad\kappa_{\mathcal{A}}=\frac{10}{9}\ \text{otherwise}, that is, κA=5/6\kappa_{\mathcal{A}}=5/6 when a0∈{1,3,7,9}a_0\in\{1,3,7,9\} and κA=10/9\kappa_{\mathcal{A}}=10/9 otherwise. Maynard can prove this asymptotic for one excluded digit in every base q≥12q\ge 12, and in a companion paper for large bases also with several excluded digits and for polynomial values; only the small bases — base 10 above all — resist. Second, excluding two digits leaves a set of size ≍Xlog⁡8/log⁡10=X0.903…\asymp X^{\log 8/\log 10}=X^{0.903\ldots}, and Maynard states that his methods are not powerful enough to prove that there are even infinitely many primes with two decimal digits absent. Later work has extended the large-base theory — for instance to Vinogradov-type ternary Goldbach theorems with all summands missing a digit (Leng and Sawhney), primes with a missing digit in arithmetic progressions to large moduli (Nath), and primes of the form m2+ℓ2m^2+\ell^2 with ℓ\ell missing three digits (Pratt) — but as of late 2025 the literature still describes the state of the art as: decimal primes may omit any fixed single digit (order of magnitude only), while asymptotic formulas are available only in the large-base regime.

Problem Statement

Two linked questions left open by Maynard's paper, together constituting the base-10 (small-base) frontier of the restricted-digit prime program.

(1) One excluded digit, asymptotic formula. Let a0∈{0,…,9}a_0\in\{0,\dots,9\} and let A(X)={n<X:no digit of n in base 10 equals a0}\mathcal{A}(X)=\{n<X:\text{no digit of }n\text{ in base 10 equals }a_0\}. Prove that, as X→∞X\to\infty, #{p∈A(X)}=(κA+o(1))#A(X)log⁡X,\#\{p\in\mathcal{A}(X)\}=(\kappa_{\mathcal{A}}+o(1))\frac{\#\mathcal{A}(X)}{\log X}, where κA=5/6\kappa_{\mathcal{A}}=5/6 if a0∈{1,3,7,9}a_0\in\{1,3,7,9\} and κA=10/9\kappa_{\mathcal{A}}=10/9 otherwise (equivalently κA=10(φ(10)−1)/(9φ(10))\kappa_{\mathcal{A}}=10(\varphi(10)-1)/(9\varphi(10)) if (10,a0)=1(10,a_0)=1, and 10/910/9 otherwise). This is currently known only within constants in base 10 (Maynard's theorem gives ≍\asymp), although Maynard shows the analogous asymptotic can be obtained for one excluded digit in every base q≥12q\ge 12; he notes it would follow from stronger Type II bilinear information.

(2) Two excluded digits, existence. Let a1≠a2a_1\neq a_2 be two distinct prescribed decimal digits and let A(X)\mathcal{A}(X) be the set of integers below XX containing neither digit. Prove that there are infinitely many primes in A\mathcal{A}; the expected count is of order #A(X)/log⁡X\#\mathcal{A}(X)/\log X with #A(X)≍Xlog⁡8/log⁡10\#\mathcal{A}(X)\asymp X^{\log 8/\log 10}, but even infinitude is beyond current methods. The statement should hold for every choice of the excluded digits (excluding two digits never creates a local obstruction, since at least two of the digits 1,3,7,91,3,7,9 remain available as final digits).

References

  1. James Maynard, Primes with restricted digits, Inventiones Mathematicae 217 (2019), no. 1, 127-218. DOI 10.1007/s00222-019-00865-6, arXiv:1604.01041, https://arxiv.org/abs/1604.01041 (primary source of the open questions recorded here)
  2. James Maynard, Primes and polynomials with restricted digits, International Mathematics Research Notices 2022 (2022), no. 14, 10626-10648. DOI 10.1093/imrn/rnab002, arXiv:1510.07711, https://arxiv.org/abs/1510.07711
  3. James Leng and Mehtaab Sawhney, Vinogradov's theorem for primes with restricted digits, International Mathematics Research Notices 2025 (2025), no. 3. DOI 10.1093/imrn/rnae294, arXiv:2409.06894, https://arxiv.org/abs/2409.06894
  4. Kunjakanan Nath, Primes with a missing digit: distribution in arithmetic progressions and an application in sieve theory, Journal of the London Mathematical Society (2) 109 (2024), no. 1, Paper e12837 (published online 13 November 2023). DOI 10.1112/jlms.12837, arXiv:2108.09212, https://arxiv.org/abs/2108.09212
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