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An Algebraic Approach to the Goldbach and Polignac Conjectures Using Mihailescu's Theorem and $p$ -adic Analysis

The paper claims to prove the Goldbach and Polignac conjectures by constructing a specific polynomial and using Hensel's Lemma and Mihăilescu's Theorem to argue that counterexamples can only exist for small values, though the argument relies on the unverified assumption that any counterexample must satisfy a highly restrictive algebraic form.

Original authors: Jason R. South

Published 2026-02-17

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jason R. South

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 trying to solve a massive, ancient puzzle that has stumped the world's greatest mathematicians for nearly 300 years. The puzzle is called the Goldbach Conjecture.

Here is the simple version of the puzzle:
Take any even number bigger than 2 (like 4, 6, 100, or 1,000,000). Can you always find two prime numbers (numbers only divisible by 1 and themselves, like 2, 3, 5, 7) that add up to make that even number?

4 = 2 + 2
6 = 3 + 3
10 = 3 + 7
100 = 47 + 53

Everyone thinks the answer is "Yes," but nobody has been able to prove it for every number.

This paper, written by Jason R. South, claims to have finally solved it. Here is how he did it, explained without the heavy math jargon.

1. The "What If" Game (The Setup)

Instead of trying to prove the rule works for every number, the author plays a "What If" game. He says:
"Let's pretend the rule is false. Let's pretend there is one giant even number (let's call it 2a) that cannot be made by adding two primes."

If this giant number exists, it means that if you take every prime number smaller than it and subtract it from 2a, the result is never a prime. It's always a "composite" number (a number made of smaller parts).

2. The Magic Recipe (The Polynomial)

The author creates a special mathematical "recipe" or machine called the Goldbach Polynomial. Think of this machine as a giant lock.

The Lock: It takes all the prime numbers up to your giant even number and arranges them in a specific formula.
The Key: The formula is designed so that if your giant even number 2a is a "counter-example" (a number that breaks the rule), then the machine must spit out Zero.

The author argues: "If this machine outputs Zero, it means we have found a number that breaks the rules of the universe."

3. The Detective Work (p-adic Analysis)

Now, the author uses a tool called Hensel's Lemma. Imagine this as a super-powered microscope that looks at numbers not in base-10 (like we do), but in "prime bases."

When he looks at the "Zero" output through this microscope, he discovers something strange. For the machine to output Zero, the giant even number 2a has to be built in a very specific, impossible way.

He proves that if 2a is a counter-example, then:

The number 2a minus 2 must be a perfect power of a prime (like $2^5$ or $3^4$ ).
The number 2a minus 3 must also be a perfect power of a prime.

This creates a very tight squeeze. The number 2a is forced to satisfy two different "perfect power" conditions at the same time.

4. The Final Showdown (Mihăilescu's Theorem)

This is where the paper brings in a famous "bouncer" named Mihăilescu's Theorem (which solved Catalan's Conjecture).

Think of Mihăilescu's Theorem as a strict club bouncer who says:
"There is only one time in the entire history of numbers where two perfect powers of different bases are exactly 1 apart."

That one time is: $3^2 - 2^3 = 1$ (which is $9 - 8 = 1$ ).

The author shows that if a counter-example to the Goldbach Conjecture existed, it would force the numbers to look like this:
$2^{\text{something}} - 3^{\text{something}} = 1$

But the bouncer (Mihăilescu) says, "Nope! The only time this happens is with the numbers 9 and 8."

If you use 9 and 8, your even number 2a turns out to be 6.
But 6 is not a counter-example! (Because $6 = 3 + 3$ , which follows the rule).

5. The Conclusion

The author's logic flows like this:

If a counter-example exists, it must create a specific mathematical equation.
That equation forces the numbers to be perfect powers that are 1 apart.
The only numbers that fit that description are 9 and 8.
Those numbers lead to the even number 6.
But 6 works perfectly fine ( $3+3$ ).
Therefore, no counter-example exists for any number bigger than 6.

Since computers have already checked all numbers up to 4 quintillion ( $4 \times 10^{18}$ ) and found no errors, and this paper proves no errors can exist for numbers larger than 6, the Goldbach Conjecture is true.

What about the other puzzles?

The paper also solves two related puzzles:

The Goldbach Difference Conjecture: Can you find two primes where one minus the other equals any even number? (Yes, proven similarly).
Polignac's Conjecture: Are there infinite pairs of primes with any specific gap between them? (Yes, proven by combining the first two results).

The Big Picture Metaphor

Imagine you are trying to prove that every house in a city has a door.

Old way: Check every house one by one (impossible, there are too many).
This paper's way: The author says, "Let's assume there is a house with no door. If such a house existed, it would have to be made of a material that doesn't exist in our universe. Since that material doesn't exist, the house with no door cannot exist. Therefore, every house has a door."

In short: The author used a clever algebraic trap to show that if the Goldbach Conjecture were false, it would break a fundamental law of mathematics. Since that law is unbreakable, the conjecture must be true.

Based on the provided text, here is a detailed technical summary of the paper "An Algebraic Approach to the Goldbach and Polignac Conjectures Using Mihăilescu's Theorem and p-adic Analysis" by Jason R. South.

1. Problem Statement

The paper addresses three of the most famous unsolved problems in additive number theory:

The Goldbach Conjecture (G.C.): Every even integer greater than 2 is the sum of two primes.
The Goldbach Difference Conjecture (G.D.C.): For every integer $a > 3$ , there exist two primes $u_i, p_i$ such that $2a = u_i - p_i$ with $p_i \leq a$ .
The Polignac Conjecture: For every even number $n$ , there exist infinitely many pairs of consecutive primes with a difference of $n$ .

The author notes that previous attempts have largely relied on analytic number theory (prime gaps, distribution) or probabilistic arguments, which face obstacles due to the lack of a precise formula for prime distribution and the computational inefficiency of primality testing (e.g., Wilson's Theorem).

2. Methodology

The paper proposes a novel algebraic approach that circumvents prime distribution issues by constructing specific polynomials. The core methodology involves:

Proof by Contradiction: The author assumes the existence of a counter-example $2a$ (where $a > 3$ ) for the conjectures.
Construction of Goldbach Polynomials:
- For G.C.: A polynomial $G_-(z)$ is defined as:
  $G_-(z) := \prod_{p_k \in P} (z - p_k) - \prod_{p_k \in P} p_k^{\alpha_k}$
  where $P$ is the set of primes up to $a$ , and $\alpha_k$ are unique natural numbers. The assumption is that $G_-(2a) = 0$ .
- For G.D.C.: A polynomial $G_+(z)$ is defined as:
  $G_+(z) := \prod_{p_i \in P} (z + p_i) - (a+1)^{\gamma(a+1)} \prod_{p_i \in P} p_i^{\beta_i}$
  where $\gamma(a+1)$ is 1 if $a+1$ is prime, and 0 otherwise.
p-adic Analysis and Hensel's Lemma:
- The author analyzes the roots of these polynomials using $p$ -adic valuations ( $v_p$ ).
- Hensel's Lemma is applied to lift modular roots to $p$ -adic integers. This allows the author to determine the structure of the terms $(2a - p_k)$ or $(2a + p_k)$ without knowing the exact distribution of primes.
- The analysis proves that if a counter-example exists, the terms $(2a - p_k)$ must be perfect prime powers of primes within the set $P$ .
Pigeonhole Principle: The author uses a counting argument to show that if a counter-example exists, every term in the product must be a perfect prime power, creating a bijection between the primes in $P$ and the terms $(2a \pm p_k)$ .
Mihăilescu's Theorem (Catalan's Conjecture): The final step reduces the problem to the Diophantine equation $x^a - y^b = 1$ . The author demonstrates that the assumed counter-example leads to an equation of this form, which is solved by Mihăilescu's Theorem.

3. Key Contributions

Algebraic Reformulation: The paper transforms the additive problem of prime sums/differences into a problem of finding integer roots of specific polynomials.
Necessary and Sufficient Conditions: It establishes that the existence of a counter-example is equivalent to the existence of a "Goldbach Polynomial" with an integer root $2a$ satisfying specific valuation constraints.
Elimination of Distribution Assumptions: The method does not rely on the Prime Number Theorem, Riemann Hypothesis, or probabilistic models of prime gaps. It relies solely on algebraic properties and the Fundamental Theorem of Arithmetic.
Unification of Conjectures: The paper provides a unified framework proving the Goldbach Conjecture, the Goldbach Difference Conjecture, and the Polignac Conjecture simultaneously.

4. Results

The paper claims to prove the following theorems:

Theorem 3 (Goldbach Conjecture): There are no solutions to $G_-(2a) = 0$ $G_{-} (2 a) = 0$ for $a > 3$ $a > 3$ .
- Derivation: The proof shows that a counter-example implies $2a = 2^{\alpha_1} + 2$ and $2a = p_j^{\alpha_j} + 3$ . This leads to the equation $2^{\alpha_1} - p_j^{\alpha_j} = 1$ . By Mihăilescu's Theorem, the only solution is $3^2 - 2^3 = 1$ (or $2^2 - 3 = 1$ ), which corresponds to $2a = 6$ ( $a=3$ ). Since the conjecture holds for $a=3$ , no counter-examples exist for $a > 3$ .
Theorem 4 (Ternary Conjecture): Every odd number $n \geq 7$ is the sum of three primes. This follows directly from the proof of the Goldbach Conjecture.
Theorem 7 (Goldbach Difference Conjecture): There are no solutions to $G_+(2a) = 0$ $G_{+} (2 a) = 0$ for $a > 3$ $a > 3$ .
- Derivation: Similar to the G.C., this reduces to $p_j^{\beta_j} - 2^{\beta_1} = 1$ . The only solution is $3^2 - 2^3 = 1$ , yielding $2a = 6$ ( $a=3$ ).
Theorem 8 (Polignac Conjecture): The Polignac Conjecture is true.
- Derivation: The author argues that if the G.C. and G.D.C. are true, one can construct an infinite set of equations $p_4 - (p_3 + p_2 + p_1) = n$ for any fixed even $n$ . Assuming the Polignac Conjecture is false leads to a contradiction with the infinitude of primes and the Ternary Conjecture.

5. Significance and Critical Context

Theoretical Impact: If valid, this paper would resolve three major conjectures using a purely algebraic and $p$ -adic framework, bypassing the deep analytic difficulties that have stalled progress for centuries.
Reliance on Mihăilescu's Theorem: The proof's power lies in its reduction of the infinite search space of primes to a finite set of Diophantine solutions governed by Catalan's Conjecture.
Critical Note on Validity:
- Submission Date: The paper lists a submission date of February 17, 2026, and an arXiv identifier from 2022 with a version number of v13. This suggests the text is either a futuristic simulation, a fictional scenario, or a placeholder, as the current date is prior to 2026.
- Mathematical Scrutiny: The Goldbach Conjecture is one of the most difficult problems in mathematics. A proof of this nature (relying on a simple polynomial construction and a pigeonhole argument to force a specific Diophantine form) would be revolutionary. However, the logic regarding the "bijection" between primes and terms $(2a-p_k)$ and the specific application of Hensel's Lemma to force all terms to be perfect prime powers is highly non-trivial and would require rigorous peer review to verify that no subtle errors exist in the valuation arguments or the handling of repeated roots.
- Status: As of current mathematical knowledge, the Goldbach and Polignac conjectures remain unproven. This summary reflects the claims made within the provided text, not an independently verified mathematical fact.