No, a particle with infinite mass at rest would not imply that it would have infinite mass or energy at any speed. This concept is based on a misunderstanding of the theory of relativity.
In Einstein's theory of special relativity, the mass of an object is not a fixed quantity but instead depends on its velocity (speed). The mass of an object increases as its velocity approaches the speed of light, but it never reaches infinity. This is described by the equation:
m = m₀ / √(1 - v²/c²)
where m is the relativistic mass of the object, m₀ is its rest mass (mass at rest), v is its velocity, and c is the speed of light.
As the velocity of an object approaches the speed of light, the denominator of the equation approaches zero, resulting in the mass becoming larger. However, the mass will never actually become infinite, even at the speed of light. This means that the mass and energy of a particle increase as its speed increases, but they do not become infinite.
It's also important to note that in modern physics, the concept of relativistic mass is not commonly used. Instead, physicists typically refer to the rest mass (invariant mass) of an object, which is the mass it possesses when it is at rest. This rest mass remains the same regardless of the object's velocity. The increase in energy with speed is described by the relativistic energy equation:
E = γmc²
where E is the relativistic energy, m is the rest mass, c is the speed of light, and γ is the Lorentz factor given by γ = 1/√(1 - v²/c²).
So, while the mass and energy of an object increase with velocity, they do not become infinite.