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Prime Quadruplets and Jump Conditions on Arithmetic Functions

This paper advances the characterization of composite integers satisfying simultaneous jump conditions for Euler's totient and sum-of-divisors functions by proving the conjecture holds for squarefree semiprimes and single prime powers, finding no counterexamples up to $10^{12}$ , and establishing that a proof of the conjecture would imply the infinitude of prime quadruplets.

Original authors: Himaghna Roy Choudhury, Shicheng Wei

Published 2026-06-10

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Himaghna Roy Choudhury, Shicheng Wei

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 are a detective in the world of numbers, looking for a very specific, rare pattern. This paper is about solving a mystery involving two famous "number counters" called Euler's Totient function ( $\phi$ ) and the Sum-of-Divisors function ( $\sigma$ ).

Think of these functions as two different ways to weigh a number:

$\phi(n)$ counts how many smaller numbers are "friendly" with $n$ (they don't share any common factors).
$\sigma(n)$ adds up all the numbers that divide $n$ evenly.

The Mystery: The "Jump" Condition

The authors are investigating a strange phenomenon. Usually, if you take a number $n$ and jump forward by 12 to get $n+12$ , the "weights" ( $\phi$ and $\sigma$ ) change in unpredictable ways.

However, the authors are looking for composite numbers (numbers that aren't prime) where both functions jump up by exactly 12 at the same time.

$\phi(n+12) = \phi(n) + 12$
$\sigma(n+12) = \sigma(n) + 12$

It's like finding a staircase where, no matter which step you are on, taking 12 steps forward always adds exactly 12 units of height to two different measuring tapes simultaneously.

The Big Guess: The "Prime Quadruplet" Theory

Mathematicians R. Stephan and Jud McCranie previously guessed that every time this "double jump" happens, the number $n$ is secretly built from a special family of four prime numbers called a Prime Quadruplet.

A Prime Quadruplet is a tight-knit group of four primes that look like this:
$(p, p+2, p+6, p+8)$
(Example: 5, 7, 11, 13)

The guess (Conjecture 1.1) says: If you find a number $n$ that satisfies the double jump, it must be the product of the first and last numbers of such a family.
In math terms: $n = p \times (p+8)$ .

What This Paper Proves

The authors didn't prove the guess is true for every number in the universe (that's still an open mystery), but they did two very important things to narrow it down:

1. The "Two-Prime" Case is Solved
They proved that if the number $n$ is a "semiprime" (a number made of exactly two different prime numbers, like $65 = 5 \times 13$ ), then the guess is 100% correct.

The Logic: They showed that if $n$ and $n+12$ are both made of just two primes, the math forces those primes to arrange themselves into that specific "Prime Quadruplet" pattern. There is no other way for the numbers to line up.

2. The "Single Prime Power" Case is Ruled Out
They also proved that the solution cannot be a number made of just one prime repeated many times (like $2^5$ or $3^4$ ).

The Logic: They ran the numbers and showed that if $n$ is a power of a single prime, the "jump" of 12 is mathematically impossible. The functions simply won't line up that way.

The Computer Search

Since they couldn't prove it for every possible complex number, they called in the heavy artillery: a computer.

They wrote a program to check every single composite number up to 1 trillion ( $10^{12}$ ).
The Result: They found 166 numbers that fit the "double jump" rule.
The Discovery: Every single one of those 166 numbers was exactly what the guess predicted: a product of a prime quadruplet ( $p \times (p+8)$ ).
They also confirmed that all these numbers leave a remainder of 65 when divided by 72 (fixing a typo in an old database).

The Big Picture Connection

The paper ends with a fascinating "what if."
If this guess is true (that all solutions come from prime quadruplets) AND if there are infinitely many solutions to this "double jump" problem, then it would prove a massive, unsolved problem in mathematics: that there are infinitely many Prime Quadruplets.

Currently, we don't know if there are infinitely many of these prime families. This paper shows that solving the "jump" mystery is essentially the same as solving the "infinite prime family" mystery.

Summary

The Puzzle: Find numbers where two specific math functions both increase by exactly 12 when you add 12 to the number.
The Theory: These numbers are always built from a specific pattern of four primes.
The Proof: The authors proved this theory is true for numbers made of two primes and proved it's impossible for numbers made of one prime repeated.
The Evidence: A computer checked up to 1 trillion numbers and found zero exceptions.
The Stakes: Proving this completely would confirm that there are infinitely many of these special prime families.

Technical Summary: Prime Quadruplets and Jump Conditions on Arithmetic Functions

Problem Statement
This paper investigates the simultaneous jump conditions for two fundamental arithmetic functions: Euler's totient function $\phi(n)$ and the sum-of-divisors function $\sigma(n)$ . Specifically, the authors analyze composite integers $n$ that satisfy the system of equations:
$\phi(n+12) = \phi(n) + 12 \quad \text{and} \quad \sigma(n+12) = \sigma(n) + 12$
The study is motivated by Conjecture (33) from R. Stephan's 2004 arXiv preprint, which posited a modular constraint on such solutions, and a subsequent conjecture by Jud McCranie suggesting that all such composite solutions are generated by prime quadruplets.

Methodology
The authors employ a combination of analytic number theory proofs and large-scale computational verification:

Analytic Proofs: The paper rigorously derives necessary conditions for solutions by exploiting the multiplicative properties of $\phi$ $ϕ$ and $\sigma$ $σ$ . The authors decompose the problem into specific structural cases for the integer $n$ $n$ :
- Squarefree Semiprimes: Analyzing $n = pq$ where $p$ and $q$ are distinct primes.
- Prime Powers: Analyzing $n = p^k$ for $k \geq 2$ .
- General Composites: Using discriminant analysis and parity arguments to constrain the prime factors of $n$ and $n+12$ .
Computational Search: A complete search was conducted for $n$ in the range $[2, 10^{12}]$ . To handle the scale, the authors utilized a segmented sieve algorithm (avoiding the memory limitations of a flat sieve) implemented in C++ with OpenMP parallelization. The algorithm tracked running values of $\phi$ and $\sigma$ for blocks of integers, utilizing precomputed small primes up to $\sqrt{N} \approx 10^6$ .

Key Contributions and Results

Characterization of Semiprime Solutions: The primary theoretical result (Theorem 2.2) proves that if $n$ and $n+12$ are both squarefree semiprimes, the jump conditions hold if and only if $n = p(p+8)$ where $(p, p+2, p+6, p+8)$ is a prime quadruplet with $p \geq 5$ . The proof demonstrates that the conditions $\phi(n+12) - \phi(n) = 12$ and $\sigma(n+12) - \sigma(n) = 12$ force the prime factors of $n$ and $n+12$ to form the specific arithmetic progression of a prime quadruplet.
Non-Existence of Prime Power Solutions: Theorem 3.1 establishes that no solution exists where $n$ is a power of a single prime ( $n=p^k, k \geq 2$ ). The proof utilizes bounds on $\phi(N)$ relative to $N$ to show contradictions for both even ( $p=2$ ) and odd ( $p \geq 3$ ) prime bases.
Correction of Modular Residue: The paper corrects a typographical error in Stephan's original conjecture regarding the residue modulo 72. While the original text suggested $n \equiv 64 \pmod{72}$ , the authors prove (Corollary 2.4) that if the prime quadruplet characterization holds, then $n \equiv 65 \pmod{72}$ . This is derived using the Chinese Remainder Theorem based on the properties of prime quadruplets.
Computational Verification: The search up to $10^{12}$ $1 0^{12}$ yielded exactly 166 solutions. Every solution found satisfied the following:
- $n \equiv 65 \pmod{72}$ .
- $n$ is a squarefree semiprime of the form $p(p+8)$ .
- The factors correspond to a prime quadruplet.
- No counterexamples to Conjecture 1.1 were found in this range.

Significance and Claims
The paper does not claim to prove Conjecture 1.1 in its entirety (i.e., for all composite integers), but rather provides a complete characterization for the specific and significant subclass of squarefree semiprimes and proves the non-existence of solutions for prime powers.

The authors highlight a profound theoretical implication: if Conjecture 1.1 is proven true and it can be shown that there are infinitely many solutions to these jump conditions, it would logically imply the existence of infinitely many prime quadruplets. This would constitute a proof of the Hardy–Littlewood prime $k$ -tuple conjecture for $k=4$ , a major open problem in number theory.

The work bridges the gap between the behavior of arithmetic functions over shifted intervals and the distribution of prime tuples, offering strong empirical and partial theoretical evidence that the "jump" of 12 in both $\phi$ and $\sigma$ is intrinsically linked to the structure of prime quadruplets.