Cryptarithmically unique terms in integer sequences

TL;DR

This paper investigates the distribution of cryptarithmically unique terms within various integer sequences, providing estimates of their density and exploring their properties in primes and perfect squares.

Contribution

It introduces the concept of cryptarithmically unique terms in sequences and estimates their density, revealing new insights into their distribution among primes and squares.

Findings

Nearly half of perfect squares below 10^12 are cryptarithmically unique.

Cryptarithmically unique primes are extremely rare, with density dropping below 10^-300 for large primes.

The density of such primes recovers and approaches unity for very large primes.

Abstract

A cryptarithm (or alphametic) is a mathematical puzzle in which numbers are represented with words in such a way that identical letters stand for equal digits and distinct letters for unequal digits. An alphametic puzzle is usually given in the form of an equation that needs to be solved, such as SEND + MORE = MONEY. Alternatively, here we will consider cryptarithms constrained not by an equation but by a particular subsequence of natural numbers, for example perfect squares or primes. Such a cryptarithm has a unique solution if there is exactly one term in the sequence that has the corresponding pattern of digits. We will call such terms cryptarithmically unique. Here we estimate the density of such terms in an arbitrary sequence for which the overall density of terms among integers is known. In particular, among all perfect squares below 10^12, slightly less than one half are cryptarithmically unique, their density increasing toward larger numbers. Cryptarithmically unique prime numbers, however, are initially very scarce. Combinatorial estimates suggest that their density should drop below 10^-300 for decimal lengths of approximately 1829 digits, but then it recovers and is asymptotic to unity for very large primes. Finally, we introduce and discuss primonumerophobic digit patterns that no prime number happens to have.