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Note on shifted primes with large prime factors

This paper improves upon Ding's recent quantitative bound for the proportion of shifted primes $p \le x$ with a large prime factor $P^+(p-1) \ge p^c$ by establishing a tighter upper limit of $-\frac{7}{2}\log c$ for the range $e^{-\frac{2}{7}} < c < 1$ .

Original authors: Yuchen Ding, Zhiwei Wang

Published 2026-02-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yuchen Ding, Zhiwei Wang

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

Imagine you have a giant bag of numbers, specifically the prime numbers (numbers like 2, 3, 5, 7, 11 that can only be divided by 1 and themselves). These are the building blocks of mathematics.

Now, take any prime number, say $p$ . If you subtract 1 from it, you get a new number ( $p-1$ ). This new number is usually a "composite" number, meaning it's made of smaller prime factors multiplied together. For example, if $p=13$ , then $p-1=12$ , which is made of $2 \times 2 \times 3$ . The "largest prime factor" here is 3.

The paper by Yuchen Ding and Zhiwei Wang is a detective story about finding primes where this "largest piece" is surprisingly huge.

The Big Question: How Big Can the Pieces Be?

The authors are asking a specific question: If we look at all the prime numbers up to a very large number $x$ , how many of them have a "largest piece" (let's call it $P(p-1)$ ) that is at least a certain fraction of the prime itself?

Let's say we pick a fraction $c$ (like 0.9, meaning 90%). We want to know: How many primes $p$ are there where the biggest piece of $p-1$ is at least 90% of $p$ ?

Mathematicians have been trying to figure out the "density" of these special primes. In other words, if you pick a random prime from a huge list, what are the odds it has this giant piece?

The Previous Detective Work

The Old Map (1935): A famous mathematician named Erdős proved that as your fraction $c$ gets closer and closer to 1 (meaning you demand the piece to be almost the whole number), the number of such primes drops to almost zero. He showed they become incredibly rare.
The Recent Map (2023): A researcher named Ding improved this. He gave a specific formula to estimate how rare they are when $c$ is very close to 1 (between 0.88 and 1). He found an upper limit (a ceiling) on how many of these primes could possibly exist.

The New Discovery: A Sharper Lens

Ding and Wang's paper is about sharpening that lens. They didn't just look at the same area; they found a way to see a wider range and get a tighter, more accurate estimate.

Here is what they did, using simple analogies:

1. The "Sieve" Analogy
Imagine you have a bucket of sand (all the numbers) and you want to find the gold nuggets (the specific primes we are looking for). You use a sieve (a mesh screen) to filter out the dirt.

Old Sieve: Previous methods used a sieve that was good, but it had some "holes" where dirt could slip through, or it wasn't fine enough to catch the smallest grains of gold.
The New Sieve: The authors used a more sophisticated tool called a Linear Sieve. Think of this as a high-tech, adjustable mesh that fits the shape of the gold nuggets much better than the old square mesh. It filters out the "noise" (numbers that don't fit the criteria) much more efficiently.

2. The "Distribution" Problem
When you are counting these special primes, you have to deal with "error terms"—mistakes in your count because the primes aren't perfectly evenly spaced.

The Old Way: Previous mathematicians could only trust their counts up to a certain distance (let's call it the "halfway mark"). Beyond that, the errors got too big to ignore.
The New Way: The authors used a powerful new theorem (related to the work of Bombieri, Friedlander, and Iwaniec) that allowed them to trust their counts much further out—up to a "four-sevenths" mark. This is like being able to see clearly through a foggy window that previously blocked your view.

The Result: A Tighter Ceiling

By combining their better sieve with their ability to see further into the fog, they improved the "ceiling" on how many of these special primes can exist.

The Range: They proved their new, tighter formula works for a wider range of fractions $c$ . Specifically, it works for any $c$ between roughly 0.75 and 1. (Previous work only worked for $c$ between 0.88 and 1).
The Precision: For any number in that range, their new formula gives a lower (better) upper bound. It says, "There are definitely fewer of these special primes than we thought before."

Why Does This Matter? (According to the Paper)

The paper doesn't claim this will immediately fix a computer virus or cure a disease. Instead, it highlights why this math is interesting in the world of pure theory:

The Twin Prime Connection: Finding primes where $p-1$ has a huge factor is mathematically linked to the Twin Prime Conjecture (the idea that there are infinitely many pairs of primes that differ by 2, like 3 and 5, or 11 and 13). If you can understand these "shifted primes" better, you get closer to solving that famous puzzle.
Fermat's Last Theorem: There is a surprising, deep connection between these large prime factors and the first case of Fermat's Last Theorem (a famous problem solved in the 1990s).
Cryptography: The paper mentions that the opposite of what they are studying (primes where the factors are small) is used in cryptography (security codes). While they are studying the "large factor" side, understanding the full landscape of prime factors helps security experts know which numbers are safe to use and which are weak.

Summary in One Sentence

Ding and Wang built a better mathematical "net" and a clearer "telescope" to prove that the number of prime numbers with a giant "largest piece" is even smaller and more restricted than we previously knew, specifically for a wider range of sizes.

Technical Summary: Note on Shifted Primes with Large Prime Factors

Problem Statement
This paper investigates the distribution of shifted primes $p$ such that the largest prime factor of $p-1$ , denoted $P(p-1)$ , exceeds a specific threshold $x^c$ . Specifically, the authors study the quantity $T_c(x)$ , defined as the count of primes $p \le x$ satisfying $P(p-1) \ge x^c$ , where $0 < c < 1$ . The primary objective is to establish a non-trivial unconditional upper bound for the ratio $T_c(x)/\pi(x)$ as $x \to \infty$ , particularly for values of $c$ close to 1.

Context and Motivation
The study of $P(p-1)$ is significant in number theory for several reasons. It is linked to the twin prime conjecture (via the condition $P(p+2) > p$ ) and the first case of Fermat's Last Theorem. While conditional results based on the Elliott–Halberstam conjecture suggest an asymptotic formula $T_c(x)/\pi(x) \to 1 - \rho(1/c)$ (where $\rho$ is the Dickman function), unconditional results are more challenging.

Previous work by Erdős (1935) established that $\limsup T_c(x)/\pi(x) \to 0$ as $c \to 1$ . Recently, Ding (2023) provided a quantitative form of this result, showing $\limsup T_c(x)/\pi(x) \le 8(c^{-1}-1)$ for $8/9 < c < 1$ . The authors note that this bound relies on a two-dimensional sieve applied to a polynomial sequence and the classical Bombieri-Vinogradov theorem (level of distribution $x^{1/2-\varepsilon}$ ).

Methodology
The authors improve upon Ding's result by introducing two key methodological advancements:

Linear Sieve on Prime Variables: Instead of applying a two-dimensional sieve to a polynomial in integer variables, the authors employ the linear sieve directly on the sequence of prime variables. They define a sequence $\mathcal{A} = \{ \ell p + 1 : \ell \le x^{1-c}, \ell p \le x, 2|\ell \}$ and sieve out primes within this set.
Enhanced Level of Distribution: To handle the error terms arising from the linear sieve, the authors utilize a theorem of Bombieri-Friedlander-Iwaniec type. Crucially, they apply this with a level of distribution $x^{4/7-\varepsilon}$ , which is superior to the classical $x^{1/2-\varepsilon}$ used in previous unconditional proofs. This allows for a more precise estimation of the error terms when summing over moduli.

The proof relies on:

Iwaniec's Linear Sieve Weights: Using upper and lower bound sieve weights ( $\lambda_d^\pm$ ) to bound the sifted set.
Bombieri-Friedlander-Iwaniec Theorem: Applying a convolution generalization with well-factorable weights to estimate sums involving primes in arithmetic progressions with the improved level of distribution.
Asymptotic Analysis of Multiplicative Functions: Utilizing results by La Bretèche, Pomerance, and Tenenbaum regarding the partial sums of specific multiplicative functions derived from the sieve weights.

Key Results
The main theorem establishes the following unconditional upper bound:

Theorem 1.1: For any $e^{-2/7} < c < 1$ ,
$\limsup_{x \to \infty} \frac{T_c(x)}{\pi(x)} \le -\frac{7}{2} \log c.$

This result extends the valid range of $c$ from $8/9 \approx 0.888$ (in Ding's result) down to $e^{-2/7} \approx 0.751$ . Furthermore, for the overlapping range $8/9 < c < 1$ , the new bound $-\frac{7}{2} \log c$ is strictly smaller than Ding's bound $8(c^{-1}-1)$ , providing a tighter constraint.

Corollary and Implications
A direct consequence of Theorem 1.1 is Corollary 1.1, which states that for any $c > e^{-1/7}$ (approximately $0.867$),
$\limsup_{x \to \infty} \frac{T_c(x)}{\pi(x)} < \frac{1}{2}.$
This improves upon previous unconditional results (such as Ding's $c \approx 0.993$ and the earlier $c \approx 0.993375$ derived from explicit constants in the Brun-Titchmarsh inequality) regarding the threshold at which the density of such primes drops below $1/2$ .

Significance
The paper claims its significance lies in providing a non-trivial upper bound for $T_c(x)$ over a wider range of $c$ and with a sharper constant than previously known unconditional results. The authors note that while the conditional asymptotic density is expected to be $-\log c$ (under the Elliott–Halberstam conjecture), their unconditional bound is $3.5$ times larger ( $-\frac{7}{2} \log c$ ). The work demonstrates that by combining the linear sieve with the Bombieri-Friedlander-Iwaniec theorem at a higher level of distribution, one can effectively refine estimates for the distribution of large prime factors in shifted primes without relying on unproven conjectures.