 Fermat tried to interest his contemporaries in the properties of whole numbers. He believed they might yield discoveries at least as important as those of geometry. Wallace's reaction was typical. I look upon problems of this nature to have more in them of labor than either of use or difficulty. About a century later, Christian Goldbach rediscovered Fermat and tried to interest his colleagues, well, technically his underlings. On December 1st, 1729, Goldbach wrote to Euler, Do you know of Fermat's observation that all numbers of the form 2 to the 2 to the n minus 1 plus 1, such as 3, 5, 17, and so on, are prime? Euler responded, Probably nothing can be discovered from this observation of Fermat. But that changed shortly afterward, and by June of 1730, Euler had become interested in Fermat's work. Ironically, for the person who would make Fermat synonymous with the foundation of number theory, Euler's first publication refuted the Fermat conjecture communicated by Goldbach. In notes of a result, Fermat and others observed about primes, presented to the St. Petersburg Academy on September 26, 1732, Euler noted, If n is 2n plus 1, then a to the n plus 1 has a factor of a plus 1. And if n is some number times 2n plus 1, then a to the n has a factor of a to the p plus 1. Euler offered no proof, but both can be proven using ordinary algebra. We won't go into the details, but note that this relies on the factor theorem, that if f of x is a polynomial with f of a equals 0, then x minus a is a factor of f of x. And as an example that will easily generalize into a proof, we'll show why 2 to the power 37 plus 1 has a factor of 2 plus 1. If f of x is x to the 37 plus 1, then f of negative 1 is 0. So f of x has a factor of x plus 1. And so we can write x to the 37 plus 1 as x plus 1 times something else. And consequently, 2 to the 37 plus 1 is 2 plus 1 times something else. And so there's our factor of 2 plus 1. Similarly, consider something like 2 to the 40th plus 1. And again, let's consider the polynomial x to the 40th plus 1. And we can rewrite this. And taking y equals x to the 8th, this could be rewritten as, which has a factor of y plus 1. And so it's y plus 1 times something. And restoring x to the 8th equals y gives us. And so consequently, 2 to the 40th plus 1 has a factor of 2 to the 8th plus 1. Now, this means that certain powers plus 1 are definitely composite. Conversely, it follows that a to the n plus 1 can only be prime if a is even and n is a power of 2. And this leads to Fermat's claim that numbers of the form 2 to the 2 to the n plus 1 are always prime. This seems to be true if we take a look at the first five of those numbers we find. But Euler noted that the sixth number actually has a factor of 641. One important feature of this paper of Euler is that it followed Fermat's practice of presenting results without really explaining where they came from. And in particular, in this case, Euler didn't explain how he found this factor of 641, but merely hinted that he had a more general method. Fortunately, Euler eventually changed his mind and began publishing number theory papers as ordinary mathematics papers. And we'll see how he obtained this factor of 641.