"In agony, Maedhros threw himself and his Silmaril into a fiery pit..."
From lotr wikia
"The Arkenstone also known as the "Heart of the Mountain" of Thrain was a wondrous gem sought by Thorin Oakenshield in J. R. R. Tolkien's The Hobbit. It was discovered beneath the Lonely Mountain (Erebor)..."
It isn't mentioned what fiery pit, but the Arkenstone was found deep in the mountain which most likely could have a fiery origin.
Are they the same stone?
If so, why wasn't this leveraged in the Lord of the Rings.
There's a few things I think I need to add to support this question.
- The Arkenstone is larger. The explanation for the dimmer light and the cutting of the stone is the fact that more weaker gemstone has formed around the Silmaril.
- The Arkenstone emits its own light. Only Silmarils were known to be able to do this. The reason is because life essence was placed into the stone and it held light from the two trees. A gem formed in the Earth wouldn't have essence.
- Things can travel large distances through the core of the Earth.
Ultimately, I think this is a weak connection that Tolkien may have thought to add, but either changed his mind or didn't think was important enough to cross-check and ensure equivalency. In other words, it was a forgotten implication.