Emirp

Short description: Class of prime numbers

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), the largest known emirp is [math]\displaystyle{ 10^{10006}+941,992,101\times 10^{4999}+1, }[/math] 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 [math]\displaystyle{ x_m }[/math], such that $x$ is the $y$ th prime, and [math]\displaystyle{ x_m }[/math] is the [math]\displaystyle{ y_m }[/math]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 [math]\displaystyle{ 10^{499} + 174,295,123,052 \pm 1. }[/math] ^[7]

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

References

↑ ^1.0 ^1.1 Weisstein, Eric W.. "Emirp". http://mathworld.wolfram.com/Emirp.html.
↑ Rivera, Carlos. "Problems & Puzzles: Puzzle 20.- Reversible Primes". Retrieved on December 17, 2007.
↑ The Prime Pages - Prime Curios. "The Largest Found Emirp".
↑ "emirpimeses". https://www.numbersaplenty.com/set/emirpimes/.
↑ Prime Curios! Curio for 71
↑ The OEIS sequence for twin emirps
↑ Prime Puzzles. Puzzle 973
↑ OEIS sequence for twin emirps equal partial sum of twin emirps

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Original source: https://en.wikipedia.org/wiki/Emirp. Read more

[mathworld-1] 1.0 ^1.1 Weisstein, Eric W.. "Emirp". http://mathworld.wolfram.com/Emirp.html.

[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". https://www.numbersaplenty.com/set/emirpimes/.

[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

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v t e Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Carol $(2 n - 1) 2 - 2$ Kynea $(2 n + 1) 2 - 2$ Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient
Base-dependent	Happy Dihedral Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n - 1, n + 1, 2 n - 1, 2 n + 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k−Tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Safe ( $p, (p - 1)/2$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Titanic (1,000+ digits) Gigantic (10,000+ digits) Mega (1,000,000+ digits) Largest known
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Somer–Lucas Strong Carmichael number Almost prime Semiprime Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers