The smallest positive integer $$(abc)_{10}$$, in base 10, $$a, b, c$$ are digits, with $$500 > (abc)_{10} > 150$$ such that the number formed by all possible arrangements (for example $$(bac)_{10}$$, $$(cba)_{10}$$) of all the digits of $$(abc)_{10}$$, is prime, is

Every arrangement must be prime, so no digit may be even or equal to 5, leaving digits from $$\{1, 3, 7, 9\}$$. Testing the candidates between 150 and 500 in increasing order, numbers such as 173 and 179 fail because $$371 = 7 \times 53$$ and $$791 = 7 \times 113$$ are composite. The first number that survives is 199, since $$199$$, $$919$$ and $$991$$ are all prime.