Today is 18 day of the year.
18 = 2 x 3^2
18 = 7 + 11 i.e. the sum of two consecutive primes.
18 is an abundant number.
18 is the fourth triangular matchstick number. All triangular matchstick numbers are 3 times the equivalent triangular number.
The difference between an emirp pair is divisible by 18.
2 is not an emirp therefore all emirps are odd.
Assume that x and y are an emir pair and x > y.
Since they are odd we know that x = 2a + 1, y = 2b + 1 with a, b as whole numbers and a > b.
Calculate the difference between x and y:
x - y = (2a + 1) - (2b + 1) = 2a + 1 -2b - 1 = 2a -2b = 2(a - b)
therefore this difference is divisible by 2.
Assume that x, y are a 4 digit emirp pair.
Let x = 1000a + 100b + 10c + d with a, b, c, d, all single digits.
Thus y = 1000d + 100c + 10b + a.
The difference between them is:
x - y = (1000a + 100b + 10c + d) - (1000d + 100c + 10b + a)
= 1000a + 100b + 10c + d - 1000d - 100c - 10b - a
= 1000a - a + 100b - 10b + 10c - 100c + d - 1000d
= 999a +90b -90c - 999d
= 9(111a + 10b -10c - 111d)
Therefore the difference is divisible by 9.
Since 9 and 2 are coprime we have shown that the difference between any four digit emirp pair must be divisible by 18.
It is left as an exercise for the reader to convince themselves that this proof can be extended to an emirp pair of n digits and thus all emirp pairs.