The problem with this sort of question is that it really fails to specify the criteria which measures “best”; most proflific, most cited, greatest theoretical breakthrough, widest or most fundamental application to physics or information theory? Unless you have a criteria there is no objective agreement, and even with it only a specialist is really going to be qualified to make any credible assessment.
As an example, there a three mathematicians in the first half of the 20th which contributed to fundamental advances in physics–Hermann Weyl, Norbert Wiener, and Emmy Noether–that almost no one without a graduate degree in physics, electrical engineering, computer science, or applied mathematics would even recognize, and then only because of eponymous theorums, filters, and techniques that are commonly used. Yet the work that these mathematicians did does and will continue to be invaluable in solving problems in relativity, quantum field theory, statistical mechanics, and information theory. However, I think the 20th century mathematician who will likely have the broadest impact on future applied science and technology is John von Neumann.
Gauss did great work in his day, but to do so he stood upon the shoulders of giants, as those that succeeded him used Gauss’ work as a starting point. There are greater and lesser mathematicians, certainly, but the greatest is a moving target, and subject to interpretation.
Stranger on a Train just gave you a very thoughtful, expert answer, and you come back with another “Sheldon Cooper is the best TV Physicist! Agree??”
How old are you, what are you trying to accomplish by constantly asking about who “the best” is, and would it be possible to actually have conversations with the Dopers who offer their thinking?
Do it! Threadwar! WOOHOO!
Personally, I consider the question of greatest mathematician to be unanswerable - one could compile a list of favourites, but it’s still giants standing on the shoulders of giants.
Messageboards like this are great for conversation. Folks are responding to your OP and offering thoughts and insights, ample material for conversation.
Whatever. To me, the greatest mathematician is JS Bach. His polyphonic demonstration of the power of a well-tempered clavier to play across keys while keeping harmonies intact is the music of the spheres the great ancient astronomers were looking for.
If I get to vote, Euler for coming up with such a specific sense of number. I also came up with some similar conjectures except in the more elementary level in regards to plain multiplication. I can’t seem to be able to publish them on journals though. They say they don’t publish conjectures without proofs. They just looked “obvious” to me. I can’t get a proof. I don’t have that much math background.