5777 &
 5993
are the only known odd numbers that are not prime and cannot be written as sum of a prime and twice a square (p + 2*n^2, n integer).
Prime numbers with latter property are 'Stern primes' 2, 3, 17, 137, 227, 977, 1187, 1493 (Moritz Abraham Stern, 1807–1894), no more found, checked until ∼ 10^9 (M.F. Hasler 2007). It was one of Goldbach's conjectures.
 
 
more interesting 4-digit numbers:

1061
1105
2520
4225
5525
6174
8125
8128
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(background-colors: #005777, #005993, #057770, #059930, #577700, #599300)