It's been known since before Pythagoras that if the lengths of the sides of a right-angled triangle are a, b and c, with c the longest, then a, b and c satisfy the equation $a^2+b^2=c^2.$

 This is a simple example of a Diophantine equation.
That equation has many solutions in which a, b and c which are integers. For example:

• a=5, b=12 and c=13
• a=7, b=24 and c=25
• a=8, b=15 and c=17
Fermat made the hypothesis that $a^n+b^n=c^n$ has no integer solutions when n > 2. In 1637 he wrote, in his copy of Claude-Gaspar Bachet's translation of the famous Arithmetica of Diophantus, "I have a truly marvellous proof of this proposition which this margin is too narrow to contain." (Original Latin: "Cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet." )

He omitted to write it down anywhere else.

For over 350 years many mathematicians, despite much effort, failed to produce a correct proof until the British mathematician Andrew Wiles published a proof in 1995. He was later knighted for his effort.

It is now generally believed that Fermat's proof had a flaw, which he discovered later. There is a flawed proof in which the flaw is to use something which at the time was generally thought to be true, but was later shown to be false. This is, of course, sheer speculation, but it is mildly satisfying.