Emirp

An emirp (prime spelled backwards) is a prime number that results in a different prime when its decimal digits are reversed. ^[1] 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 ). ^[1]

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 ^[update] , the largest known emirp is ${\displaystyle 10^{10006}+941,992,101\times 10^{4999}+1,}$ found by Jens Kruse Andersen in October 2007. ^{[2] [3]}

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. ^[4]

It is an open problem whether there are infinitely many emirps. (sequence A178545 in the OEIS )

Emirps with added mirror properties

There is a subset of emirps x, with mirror ${\displaystyle x_{m}}$, such that x is the yth prime, and ${\displaystyle x_{m}}$ is the ${\displaystyle y_{m}}$th prime. (e.g., 73 is the 21st prime number; its mirror, 37, is the 12th prime number; 12 is the mirror of 21.)

Twin emirp

A twin emirp (or emirp twin) is a pair of emirp such that the smaller one and its reversal is a twin prime. For example, 71 is the smallest twin emirp. 71, 73, 17 and 19 are all different primes, so 71 is a twin emirp. ^[5]

The first fourteen twin emirps are 71, 1031, 1151, 1229, 3299, 3371, 3389, 3467, 3851, 7457, 7949, 9011, 9437, and 10007 (the sequence A175215 in the OEIS). ^[6]

The largest found twin emirp is ${\displaystyle 10^{499}+174,295,123,052\pm 1.}$ ^[7]

The smallest twin emirp that is sum of first twin emirps is ${\displaystyle 71+1031+1151+{\text{...}}+901,814,489=18,036,881,674,937.}$ ^[8]

References

1. Weisstein, Eric W. "Emirp". MathWorld .
2. Rivera, Carlos. "Problems & Puzzles: Puzzle 20.- Reversible Primes". Retrieved on December 17, 2007.
3. The Prime Pages - Prime Curios. "The Largest Found Emirp".
4. "emirpimeses". www.numbersaplenty.com. Retrieved 2022-10-03.
5. Prime Curios! Curio for 71
6. The OEIS sequence for twin emirps
7. Prime Puzzles. Puzzle 973
8. OEIS sequence for twin emirps equal partial sum of twin emirps