Answer: I'm afraid that the warm days and cool nights are responsible for the development of powdery mildew on your begonias. It's the end of the growing season for them so I'd pinch off the affected leaves and begin reducing water so the plants can go into dormancy. When the tops die down, cut them off and then dust the tuber with a fungicide before storing for the winter.
Next year you can prevent powdery mildew from attacking your Tuberous Begonias by spraying with a mild fungicide that you can prepare yourself by mixing one tablespoon of baking soda and 2 or 3 drops of insecticidal soap in a gallon of tepid water. Spray this solution every 10 days beginning in August.
General storage guidelines include allowing the plants to grow until frost (usually mid-November) to store energy for the next season. Dig plants grown in the ground with a ball of soil and let them dry out in a shed or on the garage floor. When the stems break free from the tubers, shake off excess soil and allow the tubers to cure in the sun for about 4 days. Then store them in dry peat moss or sand in open flats in a cool (45?-50?F), dry place.
Q&A Library Searching Tips