@exite-dasliva It is well-known that there are many solutions in integers to x
2+
y
2 = z
2
, for instance (3, 4, 5),(5, 12, 13). The Babylonians were
aware of the solution (4961, 6480, 8161) as early as around 1500
B.C. Around 1637, Pierre de Fermat wrote a note in the margin
of his copy of Diophantus’ Arithmetica stating that x
n+y
n = z
n
has no solutions in positive integers if n > 2. We will denote
this statement for n (F LT)n. He claimed to have a remarkable
proof. There is some doubt about this for various reasons. First,
this remark was published without his consent, in fact by his
son after his death. Second, in his later correspondence, Fermat
discusses the cases n = 3, 4 with no reference to this purported
proof. It seems likely then that this was an off-the-cuff comment
that Fermat simply omitted to erase. Of course (F LT)n implies
(F LT)αn, for α any positive integer, and so it suffices to prove
(F LT)4 and (F LT)For each prime number > 2.