psifenixposted about the "one-element field F1", which patently doesn't exist and also is possibly the key to proving the Riemann Hypothesis. Did I mention that it completely, absolutely, definitely by definition doesn't exist by definition (since even in a ring, 0≠1, and must therefore have at least two elements) and still is under intense and active study? Did I mention I love mathematicians?
I remember having heard about it in John Baez's writings at some point. There is lots more about it on this blog and on this blog