Emirp Numbers: Primes That Reverse to Another Prime

The number 13 has an unusual two-way property: 13 is prime, and reversing its decimal digits gives 31, which is also prime. Because the reversed value is both prime and different from the original, 13 is an emirp number.

The name is literal: emirp is the word prime written backward. The condition that the reversed number must be different matters because it excludes palindromic primes such as 101.

The Three Conditions for an Emirp

Digit reversal alone does not make a number an emirp. The original integer and its reversal must pass separate primality tests, and the reversal must produce a different value.

The last example prevents a common mistake. Although reversing 91 produces the prime number 19, the starting number is composite because 91 = 7 × 13. Both sides of the reversal must be prime.

Emirps and Palindromic Primes Are Different

The phrase reversible prime is sometimes used for an emirp, although usage can be less precise because palindromic primes may also be included under that wording. Emirp specifically requires a different prime after reversal.

The First Emirp Pairs

Emirps naturally occur in two-way pairs. If reversing one prime produces another prime, reversing the second number restores the first.

The emirp sequence therefore begins:

• 13
• 17
• 31
• 37
• 71
• 73
• 79
• 97
• 107
• 113
• 149
• 157
• 167
• 179
• 199

There are exactly eight two-digit emirps: 13, 17, 31, 37, 71, 73, 79, and 97. They form four reversal pairs.

Why Emirps Always Come in Pairs

Digit reversal is reversible. If R(13) = 31, then applying the same operation again gives R(31) = 13.

For an emirp, both p and R(p) are prime and unequal. That automatically makes each number the emirp partner of the other.

This is why 149 and 941 should not be treated as unrelated examples. They represent the same reversal relationship viewed from opposite directions.

Digit Rules That Eliminate Many Candidates

A multi-digit emirp has stronger digit restrictions than an arbitrary integer. These restrictions can reject many candidates before a full primality test is needed.

The First and Last Digits Must Be 1, 3, 7, or 9

Every prime greater than 5 ends in 1, 3, 7, or 9. An emirp is prime, so its final digit must belong to that set.

The first digit has the same restriction for a different reason. After reversal, the original first digit becomes the final digit of the reversed number. Since that reversed number must also be prime, a multi-digit emirp cannot begin with 2, 4, 5, 6, or 8.

For example, 23 is prime, but its first digit is 2. Reversal moves that 2 to the units place:

23 → 32

The result is even and greater than 2, so it is composite.

Digit Sum Is Preserved by Reversal

Reversing digits does not change their sum. For 149 and 941:

1 + 4 + 9 = 14
9 + 4 + 1 = 14

Divisibility by 3 depends on the digit sum, so a number and its reversal always have the same divisibility status with respect to 3. A candidate greater than 3 whose digits sum to a multiple of 3 cannot be an emirp.

Every Emirp Pair Differs by a Multiple of 18

The differences in the pair table follow a fixed divisibility rule. Reversing decimal digits does not change a number modulo 9 because every power of 10 is congruent to 1 modulo 9.

Therefore n − R(n) is divisible by 9.

Every multi-digit emirp is greater than 2, so both members of the pair are odd primes. The difference between two odd integers is even. The difference is therefore divisible by both 9 and 2.

The first pairs show the rule directly:

• 31 − 13 = 18
• 71 − 17 = 54 = 3 × 18
• 73 − 37 = 36 = 2 × 18
• 701 − 107 = 594 = 33 × 18
• 941 − 149 = 792 = 44 × 18

Being separated by a multiple of 18 does not prove that two reversed numbers are emirps. It is a necessary consequence of an emirp pair, not a sufficient primality test.

Testing 149 and 941

The pair 149 ↔ 941 gives a more useful primality example than the small pair 13 ↔ 31.

Checking 149

The square root of 149 is about 12.21. If 149 were composite, it would have a prime divisor no greater than √149. The relevant possible divisors are therefore 2, 3, 5, 7, and 11.

None divides 149, so 149 is prime.

Checking the Reversal

Reversing the digits produces 941. Its square root is about 30.68, so trial division only needs prime divisors up to that point:

• 2
• 3
• 5
• 7
• 11
• 13
• 17
• 19
• 23
• 29

None divides 941. The reversed value is therefore prime as well.

Why Prime Numbers Fail the Emirp Test

A prime can fail after reversal in several distinct ways. Separating these cases makes the definition clearer than treating every failed example as the same type.

The 91 → 19 case is especially useful because it shows that the property cannot be checked in only one direction. A prime reversal does not repair a composite starting value.

How Common Are Emirps Among Small Primes?

Emirps form only a subset of prime numbers. Exact counts in small decimal ranges show how the property behaves without implying a formula for larger ranges.

The counts treat each emirp as a number in its own right. Thus 13 and 31 both contribute to the two-digit total. These finite ranges describe the listed intervals only; they do not establish a general density formula for emirps.

Base 10 Is Part of the Definition

An emirp is normally defined by reversing its decimal digits. That detail is mathematically important because digit reversal depends on the numeral system used to write the integer.

The integer itself does not change when another base is used, but its sequence of written digits can change completely. Reversing that new representation can therefore produce a different integer than decimal reversal produces.

Zeros Can Appear Inside an Emirp

A zero inside the decimal representation causes no problem. The pair 107 ↔ 701 is one of the earliest examples.

A multi-digit prime cannot end in 0, so a decimal emirp cannot create an ambiguity from removing a leading zero after reversal. Its final digit must already be one of 1, 3, 7, or 9.

There are also no one-digit emirps under the standard definition. The one-digit primes 2, 3, 5, and 7 reverse to themselves, so each fails the requirement that the reversed prime be different from the original.