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$d$ -Translated Unit Sensitive Primes

This paper introduces the concept of $d$ -translated unit sensitive primes, constructs an arithmetic progression containing arbitrarily long sequences of such primes that are also Brier numbers, and identifies the first known prime satisfying this property for all nonnegative integers $d$ .

Original authors: Thomas Luckner, R. James Philpott

Published 2026-07-29

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Thomas Luckner, R. James Philpott

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Great Number Hunt: When Primes Hide in Plain Sight

Imagine the world of mathematics as a vast, infinite library where every book is a whole number. In this library, the most famous and mysterious characters are the prime numbers. These are the "atoms" of arithmetic: numbers like 2, 3, 5, and 7 that can only be divided evenly by 1 and themselves. They are the building blocks of everything else, yet they seem to appear in the number line with no predictable pattern, like fireflies blinking in the dark. For centuries, mathematicians have tried to find rules that govern where these fireflies hide.

One of the biggest questions in this library is about "composite" numbers—the messy, divisible numbers like 4, 6, or 100. Sometimes, mathematicians want to prove that a certain type of number is always composite, no matter how you tweak it. To do this, they use a clever trick called a covering system. Think of this like a security blanket made of different-sized nets. If you throw a net over every possible number, you can prove that no matter which number you pick, it will always get caught in at least one net. If every net is designed to catch only composite numbers, then you've proven that your specific type of number can never be a prime. This paper dives deep into this game of cat-and-mouse, using these nets to hunt for very special, very stubborn primes.

The Paper's Big Discovery: The Ultimate "Sensitive" Primes

This paper introduces a new, super-charged version of a number property called being "unit sensitive." To understand this, imagine you have a number, like 97. If you change just the very last digit (the "unit" digit) to any other number from 0 to 9, you get a new list of numbers: 90, 91, 92, and so on. If every single one of those new numbers is composite (not prime), then 97 is "unit sensitive." It's like a number that is so fragile that the slightest touch to its tail turns it into a non-prime.

The authors, Thomas Luckner and R. James Philpott, take this idea and stretch it out. They ask: What if you don't just change the last digit, but first add a bunch of zeros to the end of the number? This is called being "d-translated unit sensitive."

If $d=0$ , you just change the last digit (the original "unit sensitive" idea).
If $d=1$ , you add one zero (making 97 into 970) and then change the last digit (970, 971, 972...).
If $d=2$ , you add two zeros (9700) and change the last digit.

The paper proves something amazing: There are infinitely many prime numbers that are "sensitive" to this trick for any number of zeros you add. No matter how many zeros you tack on, if you change the final digit, the result is always a composite number. The authors call these [0, ∞)-translated unit sensitive primes.

How They Built the "Unbreakable" Number

To find these elusive primes, the authors didn't just guess; they built a mathematical machine using a method pioneered by the famous mathematician Paul Erdős. They constructed a specific arithmetic progression. Think of this as a train track where the numbers are spaced out perfectly: $Am + B$.

$m$ is a counter that goes 0, 1, 2, 3...
$A$ and $B$ are two giant, carefully chosen numbers.

The authors designed $A$ and $B$ so that every number on this track satisfies four strict rules (or "criteria"):

The Zero Rule: If you add zeros and change the last digit, the number gets caught in a "net" (a prime divisor) that makes it composite.
The Tail Rule: If you just change the last digit (without adding zeros), it also gets caught in a net.
The Sierpinski Rule: If you multiply the number by powers of 2 and add 1, it's always composite.
The Riesel Rule: If you multiply the number by powers of 2 and subtract 1, it's always composite.

By using a complex web of congruences (mathematical "nets" based on remainders), they ensured that for every number on this track, the modified versions (those with added zeros or changed digits) are always composite. Crucially, they chose $A$ and $B$ to be relatively prime, which guarantees by Dirichlet's Theorem that the track itself contains infinitely many primes. The construction ensures that while the neighbors of these primes (created by the tricks above) are composite, the primes themselves remain prime.

The Results: A Prime That Does It All

The paper presents two main theorems:

Theorem 1: There are infinitely many primes that are [0, ∞)-translated unit sensitive. The authors found a specific starting prime, 41459060189171787548442999328384678040412832671445258454633 (which is their $B$ value), that is the first known example of such a prime.
Theorem 2: They went even further. They found an arithmetic progression where the numbers are not only [0, ∞)-translated unit sensitive but are also Brier numbers. A Brier number is a rare beast that is both a Sierpinski number and a Riesel number (satisfying rules 3 and 4 above).

The paper proves that within this specific train track, there are infinitely many primes that are all of these things at once. In fact, they show that for any number $k$ , you can find $k$ consecutive primes on this track that are all [0, ∞)-translated unit sensitive and Brier numbers.

What They Don't Claim (and What They Rule Out)

It is important to note what this paper does not say.

It does not claim that all primes are sensitive. In fact, the paper shows that the first 106 primes are not [1, 4381]-translated unit sensitive. Most primes are "tough" and can survive a few changes to their digits.
It does not suggest that finding these numbers is easy. The authors note that while a computer can check if a number is sensitive for a small range of zeros, checking for all possible zeros (the infinite case) is impossible for a computer to brute-force. That is why they had to build this mathematical "net" proof instead of just running a simulation.
It does not claim that these numbers have a practical use in the real world (like cryptography or engineering). The paper is purely a theoretical exploration of number patterns.

The Takeaway

In the end, this paper is a masterclass in mathematical construction. The authors took a simple idea—changing the last digit of a number—and stretched it to the limit, adding infinite zeros to the mix. By weaving together a complex tapestry of prime number "nets," they proved that there exists an infinite family of primes that are incredibly "sensitive." No matter how you try to tweak them by adding zeros or changing the final digit, the resulting numbers always collapse into composite numbers. And the cherry on top? These fragile primes are also the rare Brier numbers, proving that even in the chaotic world of prime numbers, there are patterns so strong they can be built, brick by brick, into an infinite tower.

Technical Summary: d-Translated Unit Sensitive Primes

Problem Statement
This paper addresses the existence and construction of specific classes of integers defined by their sensitivity to digit manipulation and their properties regarding exponential forms. The authors introduce the concept of $d$ -translated unit sensitive integers. A nonnegative integer $n$ is defined as $d$ -translated unit sensitive if appending $d$ zeros to the right of $n$ and changing the resulting unit digit to any other possible digit (yielding a number distinct from $n$ ) results in a composite number. When $d=0$ , this reduces to the standard definition of a unit sensitive prime.

The paper investigates whether there exist primes that satisfy this condition for any nonnegative integer $d$ (termed $[0, \infty)$ -translated unit sensitive). Furthermore, the authors seek to construct arithmetic progressions containing such primes that simultaneously satisfy the properties of Brier numbers (integers that are both Sierpíński numbers, where $k \cdot 2^n + 1$ is composite for all $n$ , and Riesel numbers, where $k \cdot 2^n - 1$ is composite for all $n$ ).

Methodology
The authors employ the technique of covering systems, a method originally utilized by Paul Erdős to construct arithmetic progressions of integers with specific composite properties. The construction relies on defining a finite set of congruences such that every integer in a specific domain satisfies at least one congruence.

The methodology involves constructing an arithmetic progression of the form $Am + B$ (where $m$ is a nonnegative integer) that satisfies four specific criteria:

Criterion (1): For any $d \ge 1$ and any digit $n \in \{0, \dots, 9\}$ , the number $(Am+B) \cdot 10^d + n$ is divisible by a prime factor of $A$ . This ensures $Am+B$ is $d$ -translated unit sensitive for all $d \ge 1$ .
Criterion (2): For any nonzero integer $x \in \{-3, \dots, 6\}$ , the number $(Am+B) + x$ is divisible by a prime factor of $A$ . This ensures $Am+B$ is unit sensitive (the $d=0$ case), provided $Am+B \equiv 3 \pmod{10}$ .
Criterion (3): For any $k \ge 0$ , the number $(Am+B) \cdot 2^k + 1$ is divisible by a prime factor of $A$ . This ensures the progression consists of Sierpíński numbers.
Criterion (4): For any $k \ge 0$ , the number $(Am+B) \cdot 2^k - 1$ is divisible by a prime factor of $A$ . This ensures the progression consists of Riesel numbers.

To satisfy these criteria, the authors construct a covering system for the exponents $d$ and $k$ . For each congruence class in the covering (e.g., $d \equiv a \pmod b$ ), they associate a unique prime $p$ (or a set of primes) such that $p$ divides $A$ . They then determine the necessary residue class for $B$ modulo $p$ to ensure divisibility. Specifically:

For Criterion (1), $B \equiv -n \cdot 10^{-a} \pmod p$ .
For Criterion (3), $B \equiv -1 \cdot 2^{-a} \pmod p$ .
For Criterion (4), $B \equiv 1 \cdot 2^{-a} \pmod p$ .

The authors utilize the Chinese Remainder Theorem to combine these modular constraints into a single value for $B$ , while $A$ is defined as the product of all unique primes used in the coverings. To ensure the progression contains infinitely many primes, $A$ and $B$ are constructed to be relatively prime, invoking Dirichlet's Theorem on arithmetic progressions. The verification of the coverings was performed using the computer algebra systems Magma and Maple.

Key Contributions and Results
The paper establishes the existence of infinitely many primes with these combined properties through the following theorems and corollaries:

Theorem 1: There exist infinitely many primes of the form $Am + B$ that are $[0, \infty)$ -translated unit sensitive. The authors provide explicit values for $A$ and $B$ . The value $B$ is identified as the first known prime that is $[0, \infty)$ -translated unit sensitive.
Theorem 2: There exist infinitely many primes of the form $Am + B$ that are both $[0, \infty)$ -translated unit sensitive and Brier numbers. The authors provide explicit, large integer values for $A$ and $B$ for this progression. The first known prime satisfying both conditions is identified as $A(11) + B$ .
Corollaries 1–3: The construction yields arithmetic progressions containing infinitely many prime Sierpíński numbers, prime Riesel numbers, and prime Brier numbers, respectively. These progressions are distinct from previously known examples in the literature.
Corollaries 4–5: By applying results from D. Shiu and J. Maynard regarding consecutive primes in arithmetic progressions, the authors prove that for any positive integer $k$ , there exist $k$ consecutive primes that are simultaneously $[0, \infty)$ -translated unit sensitive and Brier numbers.

Significance
The paper claims significance in extending the application of covering systems to a new class of integers defined by digit sensitivity. While previous work by Erdős, Sierpíński, and Riesel focused on the compositeness of numbers in the forms $x - 2^k$ , $k \cdot 2^n + 1$ , and $k \cdot 2^n - 1$ , this work integrates these concepts with the sensitivity of the unit digit after translation (appending zeros).

The authors note that while computational checks can verify unit sensitivity for small $d$ , the condition for $[0, \infty)$ -translated unit sensitivity is not computationally feasible to verify directly for a single integer due to the infinite nature of $d$ . The constructive proof using covering systems provides a rigorous existence proof for such numbers where direct computation fails. The paper also contributes to the catalog of Brier numbers by providing new arithmetic progressions containing them, including the first known prime that is both a Brier number and $[0, \infty)$ -translated unit sensitive.