It only occurred to me today that my year of birth, 1949, is an emirp or a prime number whose reversal, 9491, is also a prime (but not the same prime). This definition excludes 2, 5 and 7 as well as palindromic primes like 11 and 131. This got me thinking about the distribution of emirps over the past two thousand years or so. There's been a steady stream of them beginning with 13 AD as shown below:
[13, 17, 31, 37, 71, 73, 79, 97, 107, 113, 149, 157, 167, 179, 199, 311, 337, 347, 359, 389, 701, 709, 733, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 991, 1009, 1021, 1031, 1033, 1061, 1069, 1091, 1097, 1103, 1109, 1151, 1153, 1181, 1193, 1201, 1213, 1217, 1223, 1229, 1231, 1237, 1249, 1259, 1279, 1283, 1301, 1321, 1381, 1399, 1409, 1429, 1439, 1453, 1471, 1487, 1499, 1511, 1523, 1559, 1583, 1597, 1601, 1619, 1657, 1669, 1723, 1733, 1741, 1753, 1789, 1811, 1831, 1847, 1867, 1879, 1901, 1913, 1933, 1949, 1979]
However, because all years in the third millenium begin with the digit 2, this one thousand year period contains no emirps and so, after 1979, there are no more until 3011 AD. There are 96 emirps between 13 AD and 1979 AD inclusive and, as I said earlier, my birth year is one of them. I was reading the Wikipedia entry for emirps and discovered the interesting concept of a twin emirp. This is a prime that is the smaller of a twin prime pair such that its emirp is also part of a twin prime pair. The smallest such twin emirp is 71 because it forms a twin prime pair 73 and because its emirp, 17, forms a twin prime pair with 19. In the range between 1 and 1999, the only twin emirps are 71, 1031, 1151 and 1229.
The Online Encyclopaedia of Integer Sequences (OEIS) lists a variety of categories of emirps. One such category is that of reflectable emirps that belong to OEIS A007628. The initial terms are 13, 31, 113, 311, 1031, 1033, 1103, 1181, 1301, 1381, 1811, 1831, 3011, 3083, 3301, 3803, 10333, 11003, 11083, 11833, 18013, 18133, 18803, 30011, 30881, 31033, 31081, 31183, 33013, 33181, 33301, 33811, 38011, 38113. Another category is that of norep emirps: primes with distinct digits which remain prime when reversed. These occuply OEIS A046732 and the initial members are 13, 17, 31, 37, 71, 73, 79, 97, 107, 149, 157, 167, 179, 347, 359, 389, 701, 709, 739, 743, 751, 761, 769, 907, 937, 941, 953, 967, 971, 983, 1069, 1097, 1237, 1249, 1259, 1279, 1283, 1409, 1429, 1439, 1453, 1487, 1523, 1583, 1597, 1657, 1723, 1753. In fact, just typing the word “emirps” into the OEIS search bar brings up 211 different results:
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