I'm looking for examples where, after a long time with little progress, a simultaneous mathematical discovery, solution, or breakthrough was made independently by at least two different people/groups. Two examples come to mind:
- Prime Number Theorem. This was proved by Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896.
- Sum of three fourth powers equals a fourth power. In 1986, Noam Elkies proved that there are infinitely many integer solutions to $a^4 + b^4 + c^4 = d^4$. His smallest example was $2682440^4 + 15365639^4 + 18796760^4 = 20615673^4$. Don Zagier reported that he found a solution independently just weeks later.
Can you give other instances?
Best Answer
One of the most entertaining seminar talks I ever attended was by Michael Atiyah on the moment map on a Lie group. As the talk progressed, the people in the front row became more and more agitated, until Raoul Bott finally spoke up, said something like, "Michael, these guys just proved the same theorem last week!", and pointed to Victor Guillemin and Shlomo Sternberg.