Despite what previous answers have been given, I can confirm that the true figure is around 18,000 US soldiers who died in the Vietnam War. The reason for my assurance is that I have recently returned from a holiday in the US. Part of my visit was to see the Vietnam War Memorial in Washington. Whilst there, our tour guide said this was the figure that was finalised at the end of the war. This was confirmed by a beautiful huge marble wall which lists all of the dead soldiers beginning in 1959 (not 1964/65) and the wall continues through the years right up until 1975. It was also mentioned by our tour guide that the youngest soldier to lose his life was only 16 years old. He had been on duty for a mere three weeks...
If ever you get the chance to go visit, I totally recommend it. It's a truly humbling experience.