In the margins of his copy of arithmetica, Fermat proposed that the equation xⁿ + yⁿ = zⁿ had no positive integer solutions of x, y and z for n > 2, and claimed to have discovered a proof for this. Yet, to the dismay of mathematicians for centuries to come, he proceeded to write ‘I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain’.

After his death, the written proof was not found anywhere among his belongings and, while the individual case n = 4 was proven by Fermat himself and proofs for n = 3, 5 and 7 were also found over the next two centuries, a general proof for n > 2, as Fermat claimed to have found, remained undiscovered. More puzzling was the fact that he had made many such propositions over his lifetime, similarly without proof; while every single one of these claims was eventually proven and verified, only a proof for xⁿ + yⁿ = zⁿ remained undiscovered, and hence became known as ‘Fermat’s last theorem’.

Following the advent of technology in the 20^th century, computers were able to prove that the theorem held for thousands more specific cases of n, but a general proof remained elusive. A general proof would have to come about mathematically.

The final proof was a culmination of many theories, starting in 1955 with the Taniyama-Shimura conjecture, later known as modularity theorem. This conjecture proposes a hypothetical link between elliptical curves and modular forms, specifically that modular forms are alternative representations of elliptical curves. While seemingly unrelated to Fermat’s last theorem, it would turn out to be the crux of the proof, although this would not become apparent until three decades later. For the moment, although much evidence had been gathered in support of it and its consequences had been explored in many papers, a proof of the Taniyama-Shimura conjecture was considered untouchable with current techniques.

The next breakthrough would occur 20 years later, in 1975. Yves Hellegouarch realised that if Fermat’s last theorem was incorrect, this would mean there existed a set of solutions that could be used to form an elliptic curve. By using hypothetical solutions a, b and c, he formed a corresponding elliptic curve of the form y² = x(x – aⁿ)(x + bⁿ) and seven years later, Gerhard Frey used this curve – later named a Frey curve – to link Fermat’s last theorem and the Taniyama-Shimura conjecture. If it were proven that the Frey curve could never be modular, this would contradict the Taniyama-Shimura conjecture, resulting in the conclusion that such a curve could not exist and proving Fermat’s last theorem correct by contradiction.

In 1985 Jean-Pierre Serre attempted to prove that the Frey curve could never be modular, but the proof was incomplete, and the missing part was named the epsilon conjecture. The full proof was discovered in 1986 by Kenneth Ribet and concerns Galois representations of elliptic curves, of the form ρ(E, p). The later renamed ‘Ribet’s theorem’ demonstrated that a curve cannot be modular if its Galois representation exhibits certain specific properties, ultimately proving that the Frey curve could never be modular. While all the components for a complete proof of Fermat’s last theorem seemed to be in place, even after the attention garnered by Frey’s work, the Taniyama-Shimura conjecture still appeared impossible to prove. Ribet himself stated that he was ‘one of the vast majority of people who believed [it] was completely inaccessible’.

But not everyone was so easily dissuaded. When Andrew Wiles, a mathematician experienced in elliptic curves, heard that the epsilon conjecture had been solved, it reignited his childhood interest in Fermat and he spent the next six years attempting to prove the Taniyama-Shimura conjecture for semi-stable elliptic curves. This would only be a partial proof of the conjecture, but enough to account for the category of curves that the Frey curve was classified into. To do this, Wiles used something called modularity lifting: using proof by induction he demonstrated that if the Galois representation, ρ(E, p), of a semi-stable elliptic curve, E, is modular for a prime p > 2, then E itself is modular. Using this new technique, Wiles was able to prove that all semi-stable elliptic curves are modular, ending up with the following (simplified) proof:

Assume that Fermat’s last theorem is incorrect and that the equation xⁿ + yⁿ = zⁿ has at least one set of positive integer solutions x, y, and z for n > 2.  We can use this equation to set up a semi-stable elliptic curve. Using Ribet’s theorem, it can also be shown that this curve is never modular. However, according to modularity theorem, a curve cannot be elliptic and lack a modular form. This is a contradiction and leads to the conclusion that Fermat’s last theorem is in fact correct.

With this proof, Fermat’s last theorem seemed to have been solved. However, upon peer review, it was found that his proof of modularity lifting contained an error that resulted in a gap in his logical framework. While the rest of it was still valid and a great advancement towards a complete proof, Wiles refused to publish, opting instead to work alone at first, and later with his former student Richard Taylor, to resolve the flaw. After almost a year of work, Wiles was able to correct this using Iwasawa theory, and he published his final proof: “Modular elliptic curves and Fermat’s Last Theorem” as well as “Ring theoretic properties of certain Hecke algebras” which he used to justify his corrected working. This new proof was analysed and accepted by the wider mathematical community; Fermat’s last theorem had a generalised proof, after more than 350 years.

This proof represented the triumph of innovation and tenacity against the seemingly impossible, and each of the underlying proofs and techniques developed to solve it were all major advancements in their own right: in particular, Wiles’ proof of the Taniyama-Shimura conjecture for semi-stable curves paved the way for the complete proof, published in 2001 which was a significant breakthrough in number theory. It demonstrated a link between modular forms and elliptical curves, two key topics within number theory, meaning that analytical techniques could be shared between the two, allowing for further insights into each. It also proved a special case of Birch and Swinnerton-Dyer conjecture, another significant but currently unproven problem within mathematics. Wiles’ modularity lifting theorem was another major step for mathematics and has continued to be widely used and developed, as well as furthering the progress of the Langlands program, which seeks to connect Galois groups and automorphic functions.

Although the theorem now has a proof, one mystery remains. Wiles’ proof is dependent on modularity theorem, which would not be conjecture for centuries after Fermat’s death. This raises the obvious question: what was Fermat’s proof? There are still people who believe Fermat had found such a cunning and innovative proof that it still eludes us. Yet, given the absence of any records from Fermat himself and the considerable amount of work and collaboration it took for the rest of the world to find a proof, most mathematicians nowadays believe that either his proof was flawed, or that he never found a one to begin with.

Has anyone else woken up in the morning and just felt dread at the thought of getting out of bed? Like the world is covered in a thick grey fog and it is too difficult to do anything? When this happens, is it raining or grey outside? Is the room you are in painted in grey or have a cold grey light coming through the windows? This heavy dampening or even depressing feeling may in part be because of the colour and the psychological effects it has on humans.

Colour is essential to us. We can see a huge variety of shades, up to 10 million, all which hold some significance due to our mental associations with those colours. We can link colours to many things such as memories, items, or even concepts like war. This is because when the brain views colour in the memory being made, the chances of stimuli being transferred to memory are increased. Allowing our brain to compare and recall information more effectively. This aids our memory when it comes to remembering scenarios. For example, if someone sees a colour similar to the walls in their childhood bedroom, it may help them picture their room in more detail, and they may even think of a specific occasion when the walls were important. On the other hand, if someone has a traumatic memory, and they or someone else was wearing a certain colour, then that colour may remind them and give them anxiety.

The effect of colours on our mood is so influential that it is called Colour Psychology. Because of its influence, it is used by marketing teams, who go through rigorous processes to choose colours for products and advertisements so that they will sell best. Colour communicates things on a less individual scale as well. For example, green is occasionally used by petrol companies in marketing as it represents nature and life. The company wants to present itself as good for the environment despite the fact that it is selling oil. Red is often used on stop signs and traffic lights as it has associations to danger such as the bright warning colour that resembles many poisonous fruits or blood. People even use it to demonstrate things about themselves. Such as black or white for mourning, or yellow for vibrancy and joy.

In Europe and North America in the 1930s, grey came to symbolise industrialisation and war. Its previous connotations with industry have now come to mean modern, and grey is often used in minimalist architecture and housing décor. It has become so popular because it is a neutral shade that fits well with most colours. Therefore, it is easy to rent and sell properties with grey walls or fixed furniture. However, it is never seen as a happy, uplifting, or ‘fun’ colour; and often has connotations with indifference, sadness, and frigidity, giving it an isolating or depressing feeling if one is surrounded by too much grey.  Generally, it brings down the energy of a room and mutes the other colours in the area. In this way it is used in prisons, intentionally or unintentionally, to convey a sense of control, lack of hope, and dampness, which may reduce rebellion or plans to escape. 

Unfortunately, our world, especially urban environments like London, are becoming more and more grey. According to The Guardian, ‘Elephant’s Breath – described as an “uplifting” mid-grey, with a hint of magenta – has been called a paint colour of the decade in the UK, ranking among Farrow & Ball’s top 10 shades for the past 12 years’ and ‘Sherwin Williams’ top 50 colours, meanwhile, span from beige to dark grey but mostly split the difference with a rich spectrum of greige.’ Whist the top three new car colours in the UK (2022) are Grey – 25.7%, Black – 20.1%, and White – 16.7%.

Increases in the colour grey has been noted in other areas of life as well. In October 2020 an analysis done by the Science Museum group studied the colours in more than 7000 photographs of everyday objects from around 1800 to present day. They grouped them into 21 categories based on use and counted the colour of each pixel in all the photos. The study found that 40% of all the objects from 2020 were black or grey, compared to a very low 8% in 1800. Although the objects studied were not selected randomly, as they were part of a collection, The results still show a significant increase in the use of grey in everyday objects. The increase of grey objects in modern society also has to do with the increase of objects in general. Yes, grey objects have increased since the 1800s, but so have coloured objects, as we produce so much more than we did then. Furthermore, in urban environments, such as London, grey is everywhere! From concrete, to the constantly cloudy sky, to the Thames, and even buildings, grey is a constant in our background environment without us even realising it.

Luckily, there are some positive effects from grey as well! It is a neutral colour and when used in moderation can create feelings of calm and safety. It can promote concentration, as it relaxes the mind in a non-distracting way allowing one to fully focus on their work. When mixed with other colours, it can create different tones and shades of grey, thus making warm or cold environments without changing its neutral calming base. Grey is a chameleon colour of sorts if used properly. Ultimately, the colour grey is not a bad or negative colour, nor is it positive. It is completely neutral. However, the excess of grey in our environment can be a problem. Its use as an easy-to-style base has made grey products easier to sell, with manufacturers increasing their production and therefore their prevalence in our lives. The excess of grey surrounding us then elicits negative emotions such as isolation and dullness. If you feel like your environment is too grey and affecting your mood, you could try wearing bright articles of clothing and accessories or consider adding some brighter décor in your home. Grey should not be allowed to subconsciously negatively influence our emotions and adversely affect our lives.

Throughout history, Crimea has been under threat by Russian forces since as early as the Russian empire. Although many are aware of the geopolitical battles Crimea has faced, there is often little acknowledgement of the untold Christianization forced upon Crimean citizens.