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TL;DR: It is a coincidence.

1. Firstly, the power laws don't match for $n\neq 3$ spatial dimensions:

• On one hand, Bertrand's theorem is constricted to a 2D orbit plane (due to angular momentum conservation), and don't depend on the ambient spatial dimension $n$.

• On the other hand, Newton's shell theorem is tied to Gauss' law. Gauss surfaces are hypersurfaces of dimension $n-1$.

2. Secondly, even if we restrict to $n=3$ spatial dimensions, the solutions are different:

• On one hand, Bertrand's theorem only works for a $1/r^2$ force law and Hooke's law but not for non-trivial linear combinations thereof.

• On the other hand, the converse Newton's shell theorem also works for linear combinations thereof.

3. Thirdly, the known proofs of Bertrand's theorem are longer and the requirement of closed orbits leads to a rationality condition that has no counterpart in the converse Newton's shell theorem.

TL;DR: It is a coincidence.

1. Firstly, the power laws don't match for $n\neq 3$ spatial dimensions:

• On one hand, Bertrand's theorem is confined to a 2D orbit plane (due to angular momentum conservation), and doesn't depend on the ambient spatial dimension $n$.

• On the other hand, Newton's shell theorem is tied to Gauss' law. Gauss surfaces are hypersurfaces of dimension $n-1$.

2. Secondly, even if we restrict to $n=3$ spatial dimensions, the solutions are different:

• On one hand, Bertrand's theorem only works for a $1/r^2$ force law and Hooke's law separately but not for non-trivial linear combinations thereof.

• On the other hand, the converse Newton's shell theorem also works for linear combinations thereof.

3. Thirdly, the known proofs of Bertrand's theorem are longer and the requirement of closed orbits leads to a rationality condition, which has no counterpart in the converse Newton's shell theorem.

It seems to be a coincidence.

1. Firstly, the solutions are different:

• On one hand, Bertrand's theorem only works for a $1/r^2$ force law and Hooke's law but not for non-trivial linear combinations thereof.

• On the other hand, the converse Newton's shell theorem also works for linear combinations thereof.

2. Secondly, the known proofs of Bertrand's theorem are longer and the requirement of closed orbits leads to a rationality condition that appears to have no counterpart in the converse Newton's shell theorem.

TL;DR: It is a coincidence.

1. Firstly, the power laws don't match for $n\neq 3$ spatial dimensions:

• On one hand, Bertrand's theorem is constricted to a 2D orbit plane (due to angular momentum conservation), and don't depend on the ambient spatial dimension $n$.

• On the other hand, Newton's shell theorem is tied to Gauss' law. Gauss surfaces are hypersurfaces of dimension $n-1$.

2. Secondly, even if we restrict to $n=3$ spatial dimensions, the solutions are different:

• On one hand, Bertrand's theorem only works for a $1/r^2$ force law and Hooke's law but not for non-trivial linear combinations thereof.

• On the other hand, the converse Newton's shell theorem also works for linear combinations thereof.

3. Thirdly, the known proofs of Bertrand's theorem are longer and the requirement of closed orbits leads to a rationality condition that has no counterpart in the converse Newton's shell theorem.

It seems to be a coincidence.

1. Firstly, the solutions are different:

• On one hand, Bertrand's theorem only works for a $1/r^2$ force law and Hooke's law but not for non-trivial linear combinations thereof.
• On the other hand, the converse Newton's shell theorem also works for a constant force law (besides the two power laws already mentioned above), and moreover it works for any linear combinations of all these three power laws.

2. Secondly, the known proofs of Bertrand's theorem are longer and the requirement of closed orbits leads to a rationality condition that appears to have no counterpart in the converse Newton's shell theorem.

It seems to be a coincidence.

1. Firstly, the solutions are different:

• On one hand, Bertrand's theorem only works for a $1/r^2$ force law and Hooke's law but not for non-trivial linear combinations thereof.

• On the other hand, the converse Newton's shell theorem also works for linear combinations thereof.

2. Secondly, the known proofs of Bertrand's theorem are longer and the requirement of closed orbits leads to a rationality condition that appears to have no counterpart in the converse Newton's shell theorem.