Bruise or Contusion: This is the most likely diagnosis given the history of a bump to the arm 3 weeks ago. A bruise or contusion typically results from trauma to the soft tissues, leading to blood leakage into the tissues, which causes the characteristic discoloration and swelling.

Hematoma: A collection of blood outside of blood vessels, which could occur after a significant trauma to the arm. It might present as a lump or swelling.
Fracture: Although less likely if the patient has been mobile and not reporting severe pain, a fracture could still be a possibility, especially if the trauma was significant.
Soft Tissue Injury (Sprain/Strain): Injury to the muscles or ligaments around the arm could also present with pain and swelling after a bump.

Compartment Syndrome: A serious condition that occurs when pressure within the muscles builds to dangerous levels, decreasing blood flow and potentially leading to nerve and muscle damage. It's crucial to identify and treat promptly.
Infection (Abscess): If the skin was broken during the trauma, there's a risk of infection. Signs might include increased redness, warmth, swelling, and pus.
Vascular Injury: Damage to the arteries or veins, which could lead to significant complications, including limb ischemia or thrombosis.

Osteomyelitis: An infection of the bone, which could occur if the trauma introduced bacteria into the bone tissue. It's rare but serious and requires prompt antibiotic treatment.
Bone Cyst or Tumor: Pre-existing conditions that could be exacerbated by trauma, leading to increased symptoms. However, these would be less directly related to the acute trauma.
Nerve Damage: Direct injury to the nerves in the arm, which could result in numbness, tingling, or weakness in the affected limb.