For a 5-gallon bucket, you should plant 1-2 seed potatoes maximum.
Here's the calculation based on standard potato spacing requirements: A 5-gallon bucket typically measures about 12 inches (30 cm) in diameter, giving you roughly 0.07 square meters of growing space. According to CIP and USDA guidelines, standard potato spacing requires 20-30 cm between plants in commercial production, with ware crops planted at 40,000-55,000 plants per hectare.
Converting this to container growing: each potato plant needs approximately 0.04-0.09 square meters of space (based on the 60 cm × 20 cm spacing used in Indian standards cited by ICAR-CPRI). Your 5-gallon bucket provides just enough space for one healthy plant, though you could potentially fit two smaller seed potatoes if you're targeting smaller tubers.
The key consideration is the tuber-size tradeoff noted in the FAO and CIP data: denser planting produces more but smaller tubers, while wider spacing yields fewer but larger tubers. In a confined container like a bucket, overcrowding will result in very small potatoes and increased competition for nutrients.
For best results, plant one quality seed potato cut into 2-3 pieces (each with at least one eye), following the standard planting depth of 8-12 cm as recommended by University Extension Services. This will give you the best chance of producing decent-sized potatoes in your limited growing space.