Emirp

Class of prime numbers

An emirp (an anadrome of prime) 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. This follows from all primes bigger than 2 being odd (making their differences even, i.e. multiples of 2) and from differences between pairs of natural numbers with reversed digits being multiples of 9 (which itself is a consequence of  10 n − 1 {\displaystyle 10^{n}-1} being a multiple of 9 for every non-negative integer  n {\displaystyle n} ).

All non-palindromic permutable primes are emirps.

It is an open problem whether there are infinitely many emirps.

## References

[edit]

1. ^ a b Weisstein, Eric W. "Emirp". MathWorld.

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Prime number classes

By formula|

* Fermat (22n + 1)
* Mersenne (2p − 1)
* Double Mersenne (22p−1 − 1)
* Wagstaff (2p + 1)/3
* Proth (k·2n + 1)
* Factorial (n! ± 1)
* Primorial (pn# ± 1)
* Euclid (pn# + 1)
* Pythagorean (4n + 1)
* Pierpont (2m·3n + 1)
* Quartan (x4 + y4)
* Solinas (2m ± 2n ± 1)
* Cullen (n·2n + 1)
* Woodall (n·2n − 1)
* Cuban (x3 − y3)/(x − y)
* Leyland (xy + yx)
* Thabit (3·2n − 1)
* Williams ((b−1)·bn − 1)
* Mills (⌊A3n⌋)

By integer sequence|

* Fibonacci
* Lucas
* Pell
* Newman–Shanks–Williams
* Perrin

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
* Unique

Base-dependent|

* Palindromic
* Emirp
* Repunit (10n − 1)/9
* Permutable
* Circular
* Truncatable
* Minimal
* Delicate
* Primeval
* Full reptend
* Unique
* Happy
* Self
* Smarandache–Wellin
* Strobogrammatic
* Dihedral
* Tetradic

Patterns|

| k-tuples|

* Twin (p, p + 2)
* Triplet (p, p + 2 or p + 4, p + 6)
* Quadruplet (p, p + 2, p + 6, p + 8)
* Cousin (p, p + 4)
* Sexy (p, p + 6)
* Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...)
* Balanced (consecutive p − n, p, p + n)

|

* Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …)
* Chen
* Sophie Germain/Safe (p, 2p + 1)
* Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...)

By size|

* Mega (1,000,000+ digits)
* Largest known

* list

Complex numbers|

* Eisenstein prime
* Gaussian prime

Composite numbers|

* Pseudoprime
* Catalan
* Elliptic
* Euler
* Euler–Jacobi
* Fermat
* Frobenius
* Lucas
* Perrin
* Somer–Lucas
* Strong
* Carmichael number
* Almost prime
* Semiprime
* Sphenic number
* 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

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