Theorem Vs Conjecture: A Look Into Fermat's And Goldbach's Work

In the vast landscape of mathematics, two terms, "conjecture" and "theorem," hold distinct positions, embodying the intriguing journey from speculation to validation. Conjectures, often born from the depths of mathematical intuition and insight, represent bold hypotheses awaiting confirmation. On the other hand, the esteemed realm of theorems signifies rigorously proven statements, fortified by irrefutable evidence and logical deduction. Basically, conjecture is an unproven theorem. To unravel the essence of these terms, we embark on a journey exploring the stark disparity between conjectures and theorems, delving into the celebrated tales of Fermat's Last Theorem and the Goldbach Conjecture. These mathematical enigmas, shrouded in mystery and complexity, serve as compelling illustrations of the intricate dynamics between conjectural speculation and the ultimate triumph of proven truth in the realm of mathematical discovery.

Fermat's Last Theorem

In the realm of number theory, Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation $a^n + b^n = c^n$ for any integer value of $n>2$. The cases $n=1$ and $n=2$ have been known since antiquity to have infinitely many solutions.

$$1^1 + 2^1 = 3^1$$

$$3^2 + 4^2 = 5^2$$

Pierre de Fermat initially proposed this theorem around 1637 in the margin of a copy of Arithmetica. He wrote,

“It is impossible for a cube to be a sum of two cubes, a fourth power to be a sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly remarkable proof [of this theorem], but this margin is too small to contain it.”

This statement perplexed mathematicians for centuries, as no one could confirm or refute Fermat's last theorem. This led to skepticism about Fermat's supposed proof, causing it to be referred to as a conjecture rather than a theorem.

Fermat's Proof

$$x^2 + y^2 = z^2$$

Fermat's Proof for $n=4$ utilizes the Pythagoras Theorem.

We can also write the Pythagoras Theorem as

$$x^2 = z^2 - y^2 = (z+y)(z-y)$$

Assumption: x, y and z are all co-prime and greater than 0.

Co-prime numbers are pairs of numbers that do not have any common factor other than 1.

As Pythagorean triples always consist of all even numbers or two odd numbers and an even number, one of x or y must be even and one must be odd, or both of them must be even.

Let $a=3, 4, 5$, the most known and smallest Pythagorean triplets.

$2\cdot a=6, 8, 10$

Notice that $a$ consists of two odd numbers and one even number while $2a$ consists of all even numbers.

Assumption: x is even and y is odd

Since x is even, it is divisible by 2.

$$\left(\frac{x}{2}\right)^2=\left(\frac{z-y}{2}\right)\left(\frac{z+y}{2}\right)$$

Since the product $\frac{x}{2}$ is a square number, $\frac{z-y}{2}$ and $\frac{z+y}{2}$ must also be square numbers.

This rule is proven by:

Suppose that one of the squares is $x^2$ and the other is $y^2$.

Their product,
$(x^2)(y^2)$
will be equal to $(xy)^2$, which is also a perfect square.

By the same reason, the numbers that are multiplied together to give a perfect square number as the product must also be square numbers.

Let $\frac{z-y}{2}= a^2$ and $\frac{z+y}{2}= b^2$.

This gives us the solution:

$$z - y = 2a^2...1'$$

$$z + y = 2b^2...2'$$

$$1'+2': 2z = 2a^2 + 2b^2$$

$$z = a^2 + b^2...3'$$

Sub $3'$ into $1'$:

$$a^2 + b^2 - y = 2a^2$$

$$y = b^2 - a^2...4'$$

$$x^2 + y^2 = z^2$$

Sub $3'$ and $4'$:

$$x^2 + (b^2 - a^2)^2 = (a^2 + b^2)^2$$

$$x = 2ab$$

$x = 2ab$, $y = b^2 - a^2$, $z = a^2 + b^2$, $x^2 + y^2 = z^2$

One of a or b must be even and one must be odd.

Let's substitute some values into the equations derived to validate the proof.

$$a=2, b=1: 3^2 + 4^2 = 5^2$$

$$a=7, b=2: 45^2 + 28^2 = 53^2$$

By using the equations derived, we can now prove Fermat's Last Theorem when $n=4$.

$$x^4 + y^4 = z^4$$

$$(x^2)^2 + (y^2)^2 = (z^2)^2$$

As usual, one of x or y must be even and one must be odd.

Substitute in the solutions for the Pythagoras equation:

$$x^2 = 2ab, y^2 = b^2 - a^2$$

Assume $b$ is odd in this case. Since a and b are co-primes, b must also be a co-prime to 2a.

Let $c$ and $d$ be positive real numbers.

$$a = 2c^2$$

Since x is a square number, 2a and b must also be square numbers.

$$x^2 = 2a \cdot b$$

$$2\cdot2c^2 = 2^2\cdot c^2$$

$$b=d^2$$

$$y^2 = d^4 - 4c^2$$

We can rewrite this equation as:

$$4c^4 + y^2 = d^4$$

This is again in the form of Pythagoras equation where $(2c^2)^2 + y^2 = (d^2)^2$

$$2c^2 = 2ef, d^2 = e^2 + f^2, y = e^2 - f^2$$

Since $2c^2 = 2ef$ where $e$ and $f$ are co-primes, we can then write $e = g^2$ and $f = h^2$ since a square number is always a product of two square numbers.

By utilizing the values of $e$ and $f$, we can now rewrite the equation $d^2 = e^2 + f^2$ into $d^2 = g^4 + h^4$.

Now, let's compare the equation $d^2 = g^4 + h^4$ with $x^4 + y^4 = z^4$. There is an obvious difference in the degree of power of d and z.

However, since $(x^2)^2 + (y^2)^2 = (z^2)^2$ indicates that $z^4 = (z^2)^2$, where $z^4$ is just a square of a square number. We can then rewrite the equations into $x^4 + y^4 = z^2$ and $(x^2)^2 + (y^2)^2 = z^2$.

We can manipulate the equations in such way because a square of a square number can always be written as a number squared.

$$(3^2)^2 = 9^2$$

Well, if I inform you that this concludes Fermat's proof, you might be perplexed. What exactly have we proven? Well, we have shown that if we have a solution for the equation $x^4 + y^4 = z^2$, then we can find a solution to the same equation $d^2 = g^4 + h^4$, which may sound completely useless. However, the solution has become smaller.

Therefore, what Fermat demonstrated is that for any positive integer solution of $x^4 + y^4 = z^2$, we can find a solution with smaller integers. This process can be repeated, generating an infinite sequence of progressively smaller positive integers. According to the proof by infinite descent, this is impossible. Hence, we can conclude that $x^4 + y^4 = z^2$ has no solution when $x, y, z > 0$.

A proof by infinite descent is a specific type of proof by contradiction employed to demonstrate that a statement cannot be true for any number. This is achieved by establishing that if the statement were valid for a certain number, then it would also hold true for a smaller number, creating an infinite descent and ultimately resulting in a contradiction.

Wiles's Proof of Fermat's Last Theorem

By 1993, with the help of computers, mathematicians confirmed that Fermat's Last Theorem holds for all prime numbers $n < 4,000,000$. During this period, they also uncovered that proving a special case of a result from algebraic geometry and number theory known as the Shimura-Taniyama-Weil conjecture (now known as the Modularity theorem) would be equivalent to proving Fermat’s last theorem.

The Shimura-Taniyama-Weil conjecture states that elliptic curves over the field of rational numbers are related to modular forms.

$$a^n + b^n = c^n$$

Andrew Wiles has proven Fermat's Last Theorem by the method of contradiction. Wiles began with assuming Fermat's Last Theorem to be false.

Assumption: There exists non-zero integer solutions $a$, $b$ and $c$ for $n > 2$

Assume that Fermat's Last Theorem is false. This implies the existence of a set of numbers $(a, b, c, n)$ that satisfies Fermat's equation, allowing us to construct a Frey curve, which is elliptic. We proceed with the assumption that a solution has been found and a corresponding curve has been created.

A corresponding Frey curve considering the two conditions above can be drawn with the equation $y^2 = x(x - a^n)(x + b^n)$, which is never modular. If we can prove that all such elliptic curves will be modular (meaning that they match a modular form), it leads to a contradiction, disproving our initial assumption.

A function is said to be modular if it follows the following expression:

$$f(\frac{{az + b}}{{cz + d}}) = (cz + d)^k f(z)$$

Here, $f(z)$ is a holomorphic function where every point in the neighborhood of a point in the field is analytic (infinitely differentiable).

$$\begin{pmatrix}a&b\ c&d\end{pmatrix}\in SL_2\left(Z\right)$$

The matrix of $a$, $b$, $c$ and $d$ belongs to the set of two by two matrices with integers with determinant 1. $Z$ lies on the upper half of the Argand plane which means that the imaginary part of $Z$ is greater than 0 ($Im(z) > 0$) and k is a positive integer.

Comparing elliptic curves directly with modular forms is a challenging task. However, elliptic curves can be effectively represented within Galois theory. Wiles recognized that by focusing on the representations of elliptic curves rather than the curves themselves, the process of counting and aligning them with modular forms would become significantly more manageable. Subsequently, the primary objective of the proof shifts towards establishing:

(1) if the geometric Galois representation of a semistable elliptic curve is modular, so is the curve itself

(2) all geometric Galois representations of semistable elliptic curves exhibit modularity.

Together, these propositions enable us to work with representations of curves rather than dealing directly with elliptic curves.

Wiles described this realization as a "key breakthrough".

Wiles' initial objective was to establish a proof concerning these representations: if a semistable elliptic curve $E$ possesses a Galois representation $ρ(E, p)$ that is modular, then the elliptic curve $E$ must also exhibit modularity.

Technically, this involves demonstrating that if the Galois representation $ρ(E, p)$ is a modular form, then all the associated Galois representations $ρ(E, p∞)$ for every power of $p$ are also modular forms. This is referred to as the "modular lifting problem," and Wiles approached it using deformations. Any elliptic curve, or its representation, can be categorized as either reducible or irreducible.

Wiles's initial approach includes counting and matching using proof by induction and a class number formula ("CNF"). In this approach, once the hypothesis is established for one elliptic curve, it extends to be proven for all subsequent elliptic curves.

It was in this area that Wiles encountered challenges, initially with horizontal Iwasawa theory and subsequently with his extension of Kolyvagin–Flach. His work in extending Kolyvagin–Flach were primarily focused on enhancing its strength to prove the complete CNF he intended to use. Eventually, it became evident that neither of these approaches alone could generate a CNF capable of encompassing all types of semistable elliptic curves. The final component of his proof in 1995 involved realizing that success could be achieved by fortifying Iwasawa theory with the techniques from Kolyvagin–Flach.

At this point, the proof has shown a key point about Galois representations. If the geometric Galois representation $ρ(E, p)$ of a semistable elliptic curve E is both irreducible and modular for some prime number $p > 2$, then E is modular.

Crucially, this outcome not only indicates that modular irreducible representations lead to modular curves but also allows us to demonstrate the modularity of a representation by employing any prime number $> 2$ that is most convenient for our purposes, as proving it for one prime $> 2$ automatically establishes it for all primes $> 2$.

At this point, our proof started with:

1) Assuming Fermat's Last Theorem, $a^n + b^n = c^n$, to be false

2) Constructing a semistable elliptic curve using the number $a$, $b$, $c$ and $n$, which is never modular.

3) Try to prove that all such elliptic curves will be modular, then we have our contradiction and proved our assumption wrong.

4) However, it is very difficult to deal with the elliptic curves directly. Therefore, we represent them with the Galois representations and try to draw comparison from there; if the geometric Galois representation of a semistable elliptic curve is modular, so is the curve itself.

5) If the geometric Galois representation $ρ(E, p)$ of a semistable elliptic curve $E$ is irreducible and modular (for some prime number $p > 2$), $E$ is modular.

We will classify all semistable elliptic curves according to the reducibility of their Galois representations and apply the lifting theorem to the outcomes. We can choose any prime number that is most convenient. Since 3 is the smallest prime number greater than 2, and there has been prior research on representations of elliptic curves using $ρ(E, 3)$, selecting 3 as our prime number is a helpful starting point.

Wiles discovered that it was more straightforward to demonstrate the modularity of the representation by selecting a prime $p = 3$ in cases where the representation $ρ(E, 3)$ is irreducible. However, the proof for cases where $ρ(E, 3)$ is reducible was easier to prove by choosing $p = 5$. Hence, the proof diverges into two paths at this juncture.

If the Galois representation $ρ(E, 3)$ is irreducible, it has been established since around 1980 that its Galois representation is always modular, but if the representation is both irreducible and modular, then $E$ itself is modular.

So, let's explore the scenario when $ρ(E, 3)$ is reducible. Wiles discovered that when the representation of an elliptic curve using $p = 3$ is reducible, it was more convenient to work with $p = 5$ and apply his power lifting theorem. This allowed him to establish that $ρ(E, 5)$ will also be modular, instead of attempting to directly prove that $ρ(E, 3)$ itself is modular.

If $ρ(E, 3)$ and $ρ(E, 5)$ are both reducible, Wiles directly demonstrated that $ρ(E, 5)$ must possess modularity.

Finally, if $ρ(E, 3)$ is reducible and $ρ(E, 5)$ is irreducible. In such instances, Wiles demonstrated that it is always possible to identify an additional semistable elliptic curve, denoted as $F$, wherein the representation $ρ(F, 3)$ is irreducible, and the representations $ρ(E, 5)$ and $ρ(F, 5)$ share an isomorphic relationship, indicating identical structures.

In this case, $ρ(F, 3)$ is irreducible, which implies $F$ is modular. As $F$ is modular, $ρ(F, 5)$ must also be modular. Since $ρ(E, 5)$ and $ρ(F, 5)$ have identical structures, and $ρ(F, 5)$ is modular, Wiles concluded that $ρ(E, 5)$ must also be modular.

Hence, in cases where $ρ(E, 3)$ is found to be reducible, it has been proven that $ρ(E, 5)$ will always exhibit modularity. Subsequently, as per the modularity lifting theorem, the modularity of $ρ(E, 5)$ implies that $E$ is also modular.

We have successfully proven that irrespective of whether $ρ(E, 3)$ is irreducible or reducible, any semistable elliptic curve $E$ will always exhibit modularity. This proves the validity of the Taniyama–Shimura–Weil conjecture for semistable elliptic curves; and since there is no room for contradiction, it also proves that the elliptic curves proposed by Frey cannot exist. Consequently, the non-existence of solutions to Fermat's equation is proven, validating Fermat's Last Theorem.

From here, we have our proof by contradiction. We have proven that if Fermat's Last Theorem is false, we could create a semistable elliptic curve that cannot be modular according to Ribet's Theorem but also must be modular at the same time according to Wiles' previous part of proving.

Ribet's theorem shows that if the Galois representation associated with an elliptic curve has certain properties, then that curve cannot be modular

As it cannot be both, the only answer is that no such curve exists. Therefore, our assumption is false and Fermat's Last Theorem $a^n + b^n = c^n$ is true for any $n > 2$.

This concludes the entire proof for Fermat's Last Theorem. Personally, I am captivated by the theorem's complexity and the various proofs it entails. Exploring the specific case of the exponent $n = 4$ introduced me to innovative applications of the well-known Pythagorean equation, extending beyond traditional triangle length solutions. The exploration of concepts like Pythagorean triples and the product of square numbers added an extra layer of fascination to the overall journey.

While the entirety of Andrew Wiles's proof is undoubtedly complex and can be challenging to fully understand, the underlying mathematical concepts are undeniably fascinating. The proof highlights the interconnections of various mathematical ideas and theorems, allowing mathematicians to explore their creativity and concepts to gradually advance toward their ultimate objectives.

The application of familiar concepts, such as proof by contradiction, vividly demonstrates the connection between foundational mathematical principles and more advanced concepts. The deduction of the Galois representations adds a layer of beauty to the proof, showcasing the elegance of discovering a more straightforward method that ultimately resolves the theorem. It highlights the evolution of mathematical techniques and the continuous exploration of innovative approaches to problem-solving in the field.

Goldbach's Conjecture

Goldbach's conjecture remains as one of the oldest and most prominent unsolved problems in both number theory and the broader realm of mathematics. The conjecture states that every even natural number greater than 2 can be expressed as the sum of two prime numbers. While it has been verified for all integers less than 4×10^18, extensive efforts have been dedicated to proving it conclusively, yet the conjecture continues to elude resolution.

Origin

Originally proposed in its current form as the weak Goldbach conjecture, this mathematical proposition was introduced by the Prussian mathematician Christian Goldbach in a letter dated June 7, 1742, addressed to Leonhard Euler. In essence, it stated that any whole number $> 5$ can be expressed as the sum of three prime numbers.

For example:

$$12 = 2 + 3 +7$$

$$25 = 3 + 5 +17$$

$$49 = 7 + 19 + 23$$

Euler restated this in an equivalent form as what is now called the strong Goldbach conjecture or simply the Goldbach conjecture. Euler stated that every even number $> 2$ is the sum of two primes.

For example:

$$4 = 2 + 2$$

$$10 = 3 + 7$$

$$100 = 53 + 47$$

Attempted Proof

For small values of $n$, the direct proving of the strong Goldbach conjecture and consequently the weak Goldbach conjecture is feasible. For example, in 1938, Nils Pipping confirmed the conjecture up to $n=100000$. The advent of computers facilitated extensive checks for numerous $n$ values; T. Oliveira e Silva conducted a distributed computer search, confirming the conjecture for $n ≤ 4×10^18$ by 2013. Notably, one outcome of this search is that 3325581707333960528 is the smallest number incapable of being expressed as the sum of two primes.

Statistical considerations that center on the probabilistic distribution of prime numbers present informal evidence favoring the conjecture, both in its weak and strong forms, when dealing with sufficiently large integers. As the integer size increases, the number of ways it can be expressed as the sum of two or three other numbers also grows, making it increasingly "likely" that at least one of these representations comprises exclusively of primes.

A very basic version of the heuristic probabilistic argument for the strong Goldbach conjecture can be outlined as follows. According to the prime number theorem, an integer $m$ chosen at random has approximately a $\frac{1}{\ln \left(m\right):}$ chance of being prime. Therefore, for a large even integer $n$ and $m$ as a number between 3 and $\frac{n}{2}$​, the probability of both $m$ and $n - m$ being prime might be expected to be $\frac{1}{\ln \left(m\right)\ln \left(n-m\right)}$. If one follows this heuristic, the anticipated total number of ways to represent a large even integer $n$ as the sum of two odd primes could be roughly expressed as:
$$\sum\limits_{m = 3}^{\frac{n}
{2}} {\frac{1}
{{\ln m}}} \frac{1}
{{\ln \left( {n - m} \right)}} \approx \frac{n}
{{2\left( {\ln n} \right)^2 }}$$

Given that $\ln \left(n\right)<<n$​, since logarithm does not change very fast and is a very slow growing function, this quantity tends towards infinity as $n$ increases, suggesting that every large even integer is likely to have not just one but indeed numerous representations as the sum of two primes.

Another way to approach this would be let $m$ be a positive integer and $p$ and $q$ be prime numbers.

$$2m = p + q$$

If we write $2m$, which is an even number, as the sum of two prime numbers $p$ and $q$, one of $p$ and $q$ has to be greater or equal to $m$. Let's assume $p\ge m$ and $q\le m$ in this case. If we look at a particular number that is a little bigger than $m$, between $m$ and $2m$. Its chance of being prime is again $\frac{1}{\ln \left(m\right):}$, which is the same for $q$. Therefore, we can write the chance of both $p$ and $q$ being prime as $\frac{:1}{:\ln \left(m\right)^2}$

However, this heuristic argument is somewhat imprecise because it assumes that the chance of $p$ and $q$ are prime to be independent events.

A Goldbach's Comet shows for each even number $m$, the number of ways of writing as the sum of two primes.

As illustrated in Fig 2, the number of ways of writing a number as the sum of two primes grows just as you would expect. There are some variation of course where some number have lots of ways, some have only a few. However, even if we consider the ones with the fewest ways, the number seems to grow pretty steadily. No one has ever discovered a really really big even number that has only one way of writing it as the sum of two primes. This piece of information really shows how the proof presented above was a really good guess, despite of the inaccurate assumption of the independency of the events.

Proving Goldbach's Weak Conjecture

Goldbach's weak conjecture states that every odd number greater than 5 can be expressed as the sum of three primes where a prime may be used more than once in the same sum. This conjecture is called "weak" because if Goldbach's strong conjecture is proven, then this would also be true. In 2013, Harald Helfgott released a proof of Goldbach's weak conjecture.

Proving Strong Goldbach's Conjecture

The strong Goldbach conjecture poses a greater challenge compared to the weak Goldbach conjecture. Utilizing Vinogradov's method, Nikolai Chudakov, Johannes van der Corput, and Theodor Estermann demonstrated that almost all even numbers can be expressed as the sum of two primes.

In 1930, Lev Schnirelmann established that any natural number greater than 1 can be represented as the sum of not more than a computable constant C primes, introducing Schnirelmann density. Schnirelmann's constant was determined to be less than 800,000 by Schnirelmann himself. Subsequent improvements by various authors, including Olivier Ramaré in 1995, revealed that every even number $n\ge 4$ is the sum of at most 6 primes. The most recent advancement comes from Harald Helfgott's proof of the weak Goldbach conjecture, which is mentioned above.

In 1948, employing sieve theory methods, Alfréd Rényi demonstrated that every sufficiently large even number can be expressed as the sum of a prime and an almost prime with at most $K$ factors.

Sieve theory is a set of general techniques in number theory, designed to count, or more realistically to estimate the size of, sifted sets of integers. The prototypical example of a sifted set is the set of prime numbers up to some prescribed limit X.

Chen Jingrun, in 1973, also utilizing sieve theory, established that every sufficiently large even number can be represented as the sum of either two primes or a prime and a semiprime, which is the product of two primes. Ying Chun Cai established the following in 2002:

There exists a natural number N, such that every even integer n larger than N is the sum of a prime less than or equal to n^0.95 and a number with at most two prime factors.

In 1975, Hugh Montgomery and Robert Charles Vaughan demonstrated that "most" even numbers can be expressed as the sum of two primes. Specifically, they established the existence of positive constants $c$ and $C$ such that, for all sufficiently large numbers $N$, every even number less than $N$ is the sum of two primes, with at most $CN^(1 − c)$ exceptions. In 1951, Yuri Linnik established the existence of a constant $K$, such that every sufficiently large even number is the sum of two primes and at most $K$ powers of 2. János Pintz and Imre Ruzsa confirmed in 2020 that $K = 8$ is effective.

In conclusion, the attempted proof of the Goldbach Conjecture represents a fascinating journey through the realms of number theory and mathematical exploration. Despite the intricate efforts, both historical and contemporary, the conjecture remains an unsolved puzzle that has eluded conclusive validation. The interplay of various mathematical techniques, from the classical approaches of Hardy and Littlewood to modern advancements like Vinogradov's method, showcases the enduring complexity of this centuries-old problem. As researchers continue to delve into the depths of number theory, the quest for a comprehensive resolution to the Goldbach Conjecture persists, inviting mathematicians to unravel its secrets and contribute to the ever-evolving landscape of mathematical understanding. The pursuit of such enigmatic conjectures not only pushes the boundaries of mathematical knowledge but also highlights the beauty and resilience inherent in the pursuit of truth within the realm of numbers.

Personally, I was genuinely surprised by the enduring mystery surrounding such a seemingly straightforward statement at first. After conducting further research, I discovered that the underlying concepts are more intricate than I initially perceived. A highlight for me in the proof was the courageous guess made based on established facts. Despite its inaccuracy in considering the dependency of the events, the utilization of statistics, particularly probability in pure mathematics, was a novel concept that hadn't occurred to me before. The exploration of prime number theory was genuinely captivating. Additionally, the use of computer power, demonstrated through the plotting of Goldbach's comet and the intensive calculations, underscored the transformative impact of technology not only in our daily lives but also in the realm of mathematics.

Conclusion

In the realm of mathematics, the dichotomy between conjecture and theorem unfolds as a captivating narrative, blending intuition with rigorous proof. Fermat's Last Theorem and the Goldbach Conjecture stand as testaments to the intricate dance between speculation and confirmation, where the brilliance of conjectural insight meets the unyielding scrutiny of mathematical validation.

As we navigate these mathematical odysseys, we witness how conjectures emerge from the minds of visionaries, daring to challenge the unproven. The discoveries made along way is what we should truly cherish. The arduous paths toward theorems reveal the meticulous craftsmanship of mathematicians, employing logic, deduction, and at times, unforeseen techniques. Fermat's Last Theorem, a centuries-old enigma, showcases the persistence required to unravel complexities, while the Goldbach Conjecture beckons us to ponder the probabilistic dance of numbers.

In the final act, the transition from conjecture to theorem reflects not just the triumph of a single idea but the resilience of the mathematical spirit. The stories etched by these mathematical puzzles remind us that, in the pursuit of truth, conjectures may linger in uncertainty, but the journey of proving theorems transforms speculation into enduring mathematical legacy.

Despite the complexity in mathematical knowledge involved in both Fermat's Last Theorem and Goldbach Conjecture, I have acquired several general concepts regarding the subject. Different mathematical skills and knowledge can be utilized to simplify or break a complex question down into pieces as demonstrated in Fermat's Last Theorem. This reminds me of George Polya's four steps in solving a question:

1. Identify the problem
2. Devise a plan
3. Execute the plan
4. Review the solution

I think an additional step can be added in between step 2 and 3 where we should review the plan and check if it is the optimum route. Although different individuals may have different preferences while approaching the same question, it is still crucial for us to be aware of the optimum route and try to incorporate it into our current working.

On the other hand, Goldbach's conjecture illustrates how mathematics is taken for granted. There are rules or facts that we use in our calculation without knowing the origin of it. Mathematics to me, is a language which is not commonly spoken and appreciated. It is truly amazing how such abstract concept has been simplified for us to be utilized in calculations.

The vast scope for creative exploration is truly remarkable, evident in the process of proving the theorem where various concepts were intricately connected to achieve the ultimate result. Similarly, in the pursuit of proving the conjecture, diverse mathematicians employed distinct approaches, all converging toward a common objective. This flexible nature of mathematics is one that I truly appreciate, where diverse paths lead to a shared goal allowing mathematicians to explore their creativity. The outcome is indeed important, but the attempts to acquire solutions to questions where 'no solution' is purported to be the panacea to complexities are not to be overlooked.

$$Journey \ge Destination$$

References and Further Reading

Wiles’s proof of Fermat’s Last Theorem - Wikipedia

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