Solution to Problem:

The answer is 5 weights.

There are many possibilities.
(1) You could use a 1-pound weight, a 2-pound weight, a 4-pound weight, an 8-pound weight, and a 16-pound weight to make all the combinations from 1 to 31 pounds (more than the 1 to 17 pounds which was asked for).
(2) You could use two 1-pound weights, a 2-pound weight, a 4-pound weight, and a 9-pound weight to make all the combinations from 1 to 17.
(3) You could use a 1-pound weight, two 2-pound weights, a 3-pound weight, and a 9-pound weight to make all the combinations from 1 to 17.
(4) You could use a 1-pound weight, a 2-pound weight, a 4-pound weight, a 5-pound weight, and a 6-pound weight to make all the combinations from 1 to 17.
And there are others...

Two people sent in solutions with weights being used on both sides, and one of those persons, Keith Mealy, found that it is possible to do the weighings with only four weights!
Here is Keith's solution:

1, 3, 7, 15 works by putting a weight on either side of the scale (+ indicates opposite the object and - on the same side as the object).

1 = +1
2 = 3 - 1
3 = 3
4 = 7 - 3
5 = 7 + 1 -3
6 = 7 -1
7 = 7
8 = 8 + 1
9 = 7 + 3 -1
10= 7 + 3
11= 7 + 3 +1
12=15 - 3
13=15 + 1 - 3
14=15 - 1
15=15
16=15 + 1
17=15 + 3 - 1
(EXTRA CREDIT)
16-26 are just 15 + (numbers formed from 1 to 11), as above.