Fermat’s Theorem:

(aka Fermat's Last Theorem) states that no 3 positive integers a, b & c satisfy the equation:

for any integer value of n greater than 2.  The cases n = 1 and n = 2 have been known to have infinitely many solutions since antiquity; first conjectured by Fermat (1637) in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin.  The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century; among the most notable theorems in the history of mathematics & until 1994 it was in the Guinness Book of World Records as the "most difficult mathematical problem", one of the reasons being that it has the largest number of unsuccessful proofs.