All numbers of the form F(N)=2^(2)^(N)+1 (N power of 2 powers of 2 plus 1) where N is a whole number are known as Fermat numbers. F(0)=2^(2)^(0)+1=3, F(1)=2^(2)^(1)+1=5, F(2)=2^(2)^(2)+1=17, F(3)=2^(2)^(3)+1=257, F(4)=2^(2)^(4)+1=65537, and F(5)=2^(2)^(5)+1=4294967297. While F(0), F(1), F(2), F(3), and F(4) are known prime numbers similar to the ancient discovery that there can only be five Platonic solids: tetrahedron, cube, octahedron, icosahedrons, and dodecahedron; F(5), F(6), F(7), F(8 ), F(9), F(10), F(11), F(12), F(13), F(14), F(15), F(16), F(17), F(18 ), F(19), F(20), F(21), F(22), F(23), F(24), F(25), F(26), F(27), F(29), F(30), F(31), and F(32) are known composite numbers, that is numbers which can be divided by other numbers in addition to 1 and itself. Many larger values (N>32) are also believed to be composites. Although there are no proofs, there are heuristic arguments which suggest that only a finite number of them are primes, but of course the discoverer of Fermat numbers was Pierre de Fermat himself. He accomplished this feat around the middle 1600s. He was the same Fermat who said he had a proof for his last theorem (the one that Andrew Wiles finally proved between 1994 and 1995 using elliptic curves, modular forms and ideal numbers) but the margin of the book he was reading could not provide the space to write it down (in fact it was a copy of one of the six surviving books by Diophantus called the thirteen books of the Arithmetica). However, regarding Fermat numbers, he admitted outright that he could not provide any plausible proofs although he asserted that N power of 2 powers of 2 plus 1 represents a series of prime numbers. Later on, he was the first to doubt its correctness. Approximately 370 years to this day, using a sieve of Diophantus his doubt can now be lifted one way or the other.