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Three Unique Primes

What is the largest integer less than 100 which cannot be written as the sum
of three unique prime numbers? Less than 1000? Less than 1 million?
For example,

10 = 5+3+2, all unique.

11 = 7+2+2 = 5+3+3, but neither sum has
all three primes unique.

*Extension*: The problem above actually has been investigated
thoroughly. Try also finding the largest integer in each range which
can be written as exactly one such sum.

Source: Original.

Solutions were received from
Jeremy Galvagni,
Joseph DeVincentis,
Philippe Fondanaiche,
Denis Borris,
Saw L.B.,
Paul Botham,
Carlos Rivera,
and Sagar Chivate.
It was quickly pointed out to me that this question is a rewording of
Goldbach's Conjecture. The answer in all cases is 17. For the extension,
the answer in all cases is 40.

Other sites to investigate:

http://mathworld.wolfram.com/WaringsPrimeNumberConjecture.html

www.utm.edu/research/primes/notes/conjectures

Mail to Ken