Atopic Dermatitis (Eczema): A common condition characterized by itchy, inflamed skin, often appearing as a papular rash. The absence of fever and URI symptoms, along with the distribution on the trunk, arms, and legs, makes this a plausible diagnosis.

Contact Dermatitis: An allergic or irritant reaction to substances coming into contact with the skin, leading to a papular rash. This could be due to new soaps, detergents, or exposure to certain plants.
Pityriasis Rosea: A skin condition that starts with a single, large spot (herald patch) followed by an eruption of smaller spots, typically on the trunk, arms, and legs. It's often preceded by a viral infection, but the absence of fever and URI symptoms doesn't rule it out entirely.
Drug Eruption: A reaction to a medication, which can manifest as a papular rash on various parts of the body. Given the widespread nature of the rash, this is a consideration, especially if the patient has recently started new medications.

Syphilis (Secondary): Although less common, syphilis can present with a rash on the trunk, arms, and legs, among other symptoms. It's crucial to consider this due to its serious implications if left untreated.
Lyme Disease: Characterized by a distinctive "bull's-eye" rash (erythema migrans) in early stages, but can also present with a more generalized papular rash in later stages. Given the potential for serious neurological and cardiac complications, it's essential not to miss this diagnosis.
Meningococcemia: A severe infection that can cause a rash, although it's typically accompanied by fever and other systemic symptoms. The absence of these doesn't entirely rule it out, especially in early stages.

Kawasaki Disease: An illness mainly affecting children, characterized by fever, rash, swelling of hands and feet, and other symptoms. Although fever is a key component, its absence in early stages or in atypical cases means it shouldn't be entirely dismissed.
Papular Urticaria: A skin reaction to insect bites or other allergens, leading to itchy papules. While not typically life-threatening, it's a consideration in the differential diagnosis due to its potential to cause significant discomfort.
Erythema Multiforme: A skin condition characterized by target lesions, often triggered by infections or medications. It can present with a variety of skin manifestations, including papular rashes, and is important to recognize due to its potential association with serious underlying conditions.