It's All About the Primes

Understanding the Goldbach conjecture requires familiarity with prime numbers. Before we plunge into one of the oldest and most daunting questions in mathematics, let’s explore why primes are significant.

A prime number is defined as a natural number greater than 1 that can only be divided evenly by 1 and itself. The initial prime numbers include 2, 3, 5, 7, 11, etc. In number theory, we focus on natural numbers, which are the positive integers: 1, 2, 3, 4, 5, and so forth.

In mathematics and nature, a common approach is to analyze the fundamental building blocks that comprise various objects. The fundamental theorem of arithmetic asserts that every natural number greater than 1 can be uniquely expressed as a product of primes. In essence, each natural number has a distinct prime factorization—though this is understood up to the order of the factors.

For instance, the number 6 can be expressed as 2 × 3, and 28 as 2 × 2 × 7. Thus, a thorough understanding of primes can yield substantial insights into natural numbers. This is akin to how physicists study elemental particles and forces, chemists examine atomic interactions, and biologists explore cellular structures.

So, our exploration of primes is essential as they form the foundational elements of natural numbers.

An Innocent-Looking Inquiry

On June 7, 1742, the German mathematician Christian Goldbach penned a letter to the legendary Leonhard Euler. This correspondence, seemingly innocuous, would introduce one of mathematics' greatest mysteries. In this letter, Goldbach proposed the following conjecture:

Any integer expressible as the sum of two primes can also be represented as the sum of an arbitrary number of primes until all components are reduced to units.

It's important to note that at the time, the number 1 was considered prime, so "units" referred to "ones." Goldbach also suggested a second conjecture in the margin of his letter:

Every integer greater than 2 can be expressed as the sum of three primes.

Euler responded on June 30, 1742, reminding Goldbach of a previous discussion where he mentioned that the first conjecture would follow from the assertion:

Every positive even integer can be expressed as the sum of two primes.

This assertion is equivalent to Goldbach's marginal note. Historically, significant ideas often reside in the margins—just ask Fermat!

The conjecture, now known as the Goldbach conjecture, can be articulated in contemporary terms as follows:

The Goldbach Conjecture:

Every even integer greater than 2 can be represented as the sum of two primes.

Let’s verify this with a few examples:

• 4 = 2 + 2
• 6 = 3 + 3
• 8 = 3 + 5
• 10 = 3 + 7 = 5 + 5

Observe that some numbers can be expressed in multiple ways as sums of two primes. The conjecture does not restrict this, making it permissible.

This conjecture has inspired numerous mathematicians and led to the development of various tools to investigate it. Yet, for nearly 300 years, it has eluded the world's most brilliant minds, remaining unsolved.

Euler himself asserted:

> "That every even integer is a sum of two primes, I regard as a completely certain theorem, although I cannot prove it."

What Does the Conjecture Really Imply?

Mathematical statements can often be viewed from multiple perspectives, sometimes revealing clearer interpretations. This concept is known as equivalence. Two statements, A and B, are considered equivalent if A implies B and B implies A.

For example, let S be a subset of real numbers. The statements:

• A: "You can divide the number 1 by any number in S"
• B: "0 is not in S"

are equivalent. If A holds true, then 0 cannot be in S, as division by zero is undefined. Conversely, if B is valid, then division of 1 by any number in S is feasible. Thus, A is also true.

This serves as a simple illustration of equivalence, though proving equivalences in real scenarios can be much more complex.

The Geometry of the Goldbach Conjecture

To grasp the essence of the conjecture, we need to consider the nature of even numbers, which are divisible by 2. The sum of two numbers is even if they are both even or both odd. We can represent this geometrically by recognizing that the equation p + q = 2n translates to (p + q)/2 = n, meaning the average of p and q equals n.

Geometrically, this suggests that there exists a circle with center n, intersecting the number line at p and q. Thus, p and q are symmetrically positioned around n. To summarize, if p, q, and n are natural numbers satisfying p + q = 2n, then p and q are symmetrically distributed around n.

In this framework, the Goldbach conjecture posits:

For every whole number n ≥ 2, there exists a circle in the plane with center n and radius r such that 0 ≤ r ≤ n-2. Either n is prime with r = 0, or the circle intersects the number line at two prime numbers.

This perspective may not provide a clearer understanding, but it offers a geometric intuition regarding the underlying symmetry between whole numbers and primes.

Goldbach circles in the plane.

While the circles themselves are not essential, they offer a useful visual representation of the symmetric relationships between numbers.

In the subsequent section, we will explore this concept further and prove an additional equivalence.

The Semiprime Equivalence

In number theory, problems are often categorized as either additive or multiplicative. For instance, the ability to prime factorize a natural number greater than 1 is a multiplicative challenge, while both the twin prime conjecture and the Goldbach conjecture are more additive in nature.

What if we could reinterpret the Goldbach conjecture in a more multiplicative framework?

Recall that a semiprime is defined as a natural number that is the product of exactly two prime numbers. The initial few semiprimes include 4, 6, 9, 10, etc. While semiprimes are not as widely discussed as primes, they are "close" to being prime, making them worthy of examination.

I propose that the following statement is equivalent to the Goldbach conjecture:

Statement 1:

For all n ≥ 2, there exists a whole number m such that 0 ≤ m ≤ n-2 and n² - m² is a semiprime.

Let’s prove this proposition:

Proposition:

Statement 1 is equivalent to the Goldbach conjecture.

Proof:

Assuming the Goldbach conjecture is true, for a given whole number n ≥ 2, we have 2n = p + q for some primes p and q. Without loss of generality, assume p ≤ q. Based on our earlier discussion, there exists an m such that 0 ≤ m ≤ n-2, hence p = n - m and q = n + m.

This leads to:

n² - m² = (n - m)(n + m) = p ⋅ q,

indicating that n² - m² is a semiprime.

Conversely, if we assume Statement 1 is true for a number 2n with n ≥ 2, we need to demonstrate that 2n can be expressed as the sum of two primes.

By assumption, we can find an m such that 0 ≤ m ≤ n-2, and n² - m² is a semiprime. Since n² - m² = (n - m)(n + m), both n - m and n + m must be primes, thus:

2n = (n - m) + (n + m),

showing that 2n is indeed a sum of two primes.

Q.E.D.

This indicates that proving Statement 1 would simultaneously validate the Goldbach conjecture, and vice versa.

Visualizing the Conjecture

What does this look like visually? Whole numbers can be represented as boxes in 1, 2, or 3 dimensions, constructed from smaller cubes. For instance, the number 6 can be arranged as a 1 × 6 cube in one dimension or as a 2 × 3 rectangle in two dimensions.

Consider a square constructed from these smaller cubes. The perspective we've discussed regarding the Goldbach conjecture suggests that regardless of the size of your square, you can remove a smaller square (or choose not to) such that the resulting shape can only be reconstructed into a box in one or two dimensions, but not in three.

Looking from above, you would see a structure resembling this:

Here we see 9² - 4² from above. The pink cubes cannot form a box in three dimensions but can only create a 5 × 13 box in two dimensions or a 1 × 65 box in one dimension.

The Importance of Curiosity and Abstractions

The study of prime numbers holds great significance, as they are the building blocks of all other numbers—a philosophy that has persisted for over 2000 years. However, what the ancient Greeks could not foresee is that 2300 years later, knowledge of prime numbers would be instrumental in cybersecurity and online transactions. Euclid was brilliant, but he could not have imagined the internet!

This illustrates that while certain areas of pure mathematics may not seem directly applicable to our daily lives, they could spark developments that transform human existence centuries later.

Curiosity remains the most crucial gift in the realm of science.

The first video, "Goldbach Conjecture - Numberphile," delves into the fascinating aspects of this conjecture and its historical significance.

The second video, "The Goldbach Conjecture," further explores its implications and challenges in modern mathematics.