When I did my MSc in astrophysics back in '95-'96, I was told that there had been an attempt around the early 20th century to account for the advance of the perihelion of Mercury by altering the Newtonian gravitational equation by changing the exponent of radial distance r to something like 2.0016.
Now obviously this was rapidly debunked by the Einstein theory, but it still strikes me as very odd because in Newtonian theory the exponent has to be exactly 2 and so what could have motivated this complete kludge with no basis in physical or mathematical principle. It is unlike the Nordstrom theory because that, though wrong, is actually mathematically coherent.
My problem is that this was an aside from a lecture 25 years ago and I can't find any details on it. I would like to know more on the reasoning behind this idea and when and by whom this was suggested. If possible I'd like to know if it gained any traction whatsoever.