Emirp
An emirp (prime spelled backwards) is a prime number that results in a different prime when its decimal digits are reversed. This definition excludes the related palindromic primes. The term reversible prime is used to mean the same as emirp, but may also, ambiguously, include the palindromic primes.
The sequence of emirps begins 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, ... (sequence A006567 in the OEIS).
The difference in all pairs of emirps is always a multiple of 18. Unique pairs of numbers whose reversed version is also prime (sorted by the first number, excluding palindromes): (13,31), 18 (17,71), 54 (37,73), 36 (79,97), 18 (107,701), 594 (113,311), 198 (149,941), 792 (157,751), 594 (167,761), 594 (179,971), 792 (199,991), 792 (337,733), 396 (347,743), 396 (359,953), 594 (389,983), 594 (709,907), 198 (739,937), 198 (769,967), 198 (1009,9001), 7992 (1021,1201), 180 (1031,1301), 270 (1033,3301), 2268 (1061,1601), 540 (1069,9601), 8532 (1091,1901), 810 (1097,7901), 6804 (1103,3011), 1908 (1109,9011), 7902 (1151,1511), 360 (1153,3511), 2358 (1181,1811), 630 (1193,3911), 2718 (1213,3121), 1908 (1217,7121), 5904 (1223,3221), 1998 (1229,9221), 7992 (1231,1321), 90 (1237,7321), 6084 (1249,9421), 8172 (1259,9521), 8262 (1279,9721), 8442 (1283,3821), 2538 (1381,1831), 450 (1399,9931), 8532 (1409,9041), 7632 (1429,9241), 7812 (1439,9341), 7902 (1453,3541), 2088 (1471,1741), 270 (1487,7841), 6354 (1499,9941), 8442 (1523,3251), 1728 (1559,9551), 7992 (1583,3851), 2268 (1597,7951), 6354 (1619,9161), 7542 (1657,7561), 5904 (1669,9661), 7992 (1723,3271), 1548 (1733,3371), 1638 (1753,3571), 1818 (1789,9871), 8082 (1847,7481), 5634 (1867,7681), 5814 (1879,9781), 7902 (1913,3191), 1278 (1933,3391), 1458 (1949,9491), 7542 (1979,9791), 7812 (3019,9103), 6084 (3023,3203), 180 (3049,9403), 6354 (3067,7603), 4536 (3083,3803), 720 (3089,9803), 6714 (3109,9013), 5904 (3163,3613), 450 (3169,9613), 6444 (3257,7523), 4266 (3299,9923), 6624 (3319,9133), 5814 (3343,3433), 90 (3347,7433), 4086 (3359,9533), 6174 (3373,3733), 360 (3389,9833), 6444 (3407,7043), 3636 (3463,3643), 180 (3467,7643), 4176 (3469,9643), 6174 (3527,7253), 3726 (3583,3853), 270 (3697,7963), 4266 (3719,9173), 5454 (3767,7673), 3906 (3889,9883), 5994 (3917,7193), 3276 (3929,9293), 5364 (7027,7207), 180 (7057,7507), 450 (7177,7717), 540 (7187,7817), 630 (7219,9127), 1908 (7229,9227), 1998 (7297,7927), 630 (7349,9437), 2088 (7457,7547), 90 (7459,9547), 2088 (7529,9257), 1728 (7577,7757), 180 (7589,9857), 2268 (7649,9467), 1818 (7687,7867), 180 (7699,9967), 2268 (7879,9787), 1908 (7949,9497), 1548 (9029,9209), 180 (9349,9439), 90 (9479,9749), 270 (9679,9769), 90
All non-palindromic permutable primes are emirps.
As of November 2009, the largest known emirp is found by Jens Kruse Andersen in October 2007.
The term "emirpimes" (singular) is used also in places to treat semiprimes in a similar way. That is, an emirpimes is a semiprime that is also a (distinct) semiprime upon reversing its digits.
It is an open problem whether there are infinitely many emirps. (sequence A178545 in the OEIS)