Arithmetic Properties of Integers with Restricted Digits

The study of integers whose digital expansions omit or constrain certain symbols has deep roots in analytic number theory and combinatorics. At its core lies the analysis of how sets defined by forbidden digit patterns—whether excluding a single numeral in base-10 or barring blocks of symbols in an arbitrary radix—behave under classical arithmetic operations. Central themes include the infinitude and density of primes within these sets, uniformity of distribution in residue classes, growth rates of counting functions, and the fine structure of error terms in summatory functions. Techniques draw on the Hardy–Littlewood circle method, sieve theory, large sieve inequalities and tools from the geometry of numbers. Beyond pure theory, these investigations inform pseudorandom-number generation, coding theory and the complexity analysis of digital algorithms. Concrete examples range from proving that infinitely many primes lack a given digit in their decimal form to establishing equidistribution of digit sums along polynomial sequences over finite fields. The interplay between digital and arithmetic constraints continues to reveal unexpected regularities and bridges to additive combinatorics, ergodic theory and theoretical computer science.

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Arithmetic Properties of Integers with Restricted Digits publication trend

The graph below shows the total number of articles in arithmetic properties of integers with restricted digits across all publications each year (not limited to Nature Index journals).

Technical terms

Digital expansion: The representation of an integer as a sequence of digits in a fixed base.

Restricted digit set: A subset of digits forbidden from appearing in the digital expansion of an integer.

Sieve method: A combinatorial technique to estimate the size of sets of integers under arithmetic constraints, often by successively “sieving out” multiples of primes.

Sum-of-digits function: A mapping that assigns to each integer the sum of its digits in a specified base.

Equidistribution: The property that a sequence of numbers is uniformly spread across residue classes or subintervals, within a given modulus or interval.

Palindromic integer: An integer whose sequence of digits reads the same forwards and backwards in a given base.

References

1. Primes with restricted digits. Inventiones Mathematicae (2019).
2. Primes and Polynomials With Restricted Digits. International Mathematics Research Notices (2021).
3. Incomplete exponential sums over finite fields and their applications to new inversive pseudorandom number generators. Acta Arithmetica (2000).
4. Normality of the Thue–Morse function for finite fields along polynomial values. Research in Number Theory (2022).
5. Infinitude of Palindromic Almost-Prime Numbers. International Mathematics Research Notices (2024).