Wash and scrub the potatoes.

Put potatoes in a pot and cover completely with cold water and 1 tsp salt.

Bring potatoes to a boil, then turn down to a simmer. Cook potatoes 15-20 minutes, or until just tender when pierced with a fork. (You can boil the eggs while the potatoes simmer.)

Drain potatoes in a colander and let cool just slightly.

Meanwhile, in a small bowl stir together vinegar and 1 tsp salt.

When the potatoes are no longer burning to the touch, cut into cubes and put in a large bowl. Using a rubber spatula, toss potatoes gently with the vinegar/salt mixture.

Let potatoes cool to room temperature.

When potatoes are cool, gently stir in chopped eggs, green onions, celery, mayonnaise, pepper, and additional salt to taste.

Serve immediately at room temperature, or cover and chill in the refrigerator.