An emirp (which is "prime" spelled backward) is a prime number that results in a different prime number when its decimal digits are reversed.
For example, the number 13 is an emirp because when reversed, it becomes 31, which is also a prime.
Examples of Emirp
First Few Emirps are:
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, 1009, 1021, 1031, 1033, 1061, 1069, 1091, 1097, 1103, 1109, 1151, 1153, 1181, 1193, 1201, . . .
Here’s a small table showing the first few Emirps along with their reversals:
| Emirp | Reversed | Prime Status |
|---|---|---|
| 13 | 31 | Both are prime |
| 17 | 71 | Both are prime |
| 37 | 73 | Both are prime |
| 71 | 17 | Both are prime |
| 107 | 701 | Both are prime |
| 149 | 941 | Both are prime |
Important Points
- It’s important to note that palindromic primes, such as 101 (which reads the same forwards and backward), are excluded from this classification.
- The difference in all pairs of emirps is always a multiple of 18. (except for 2)
- All non-palindromic permutable primes are emirps.
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Conclusion
Emirps represent a fascinating intersection of prime numbers and numerical manipulation. They highlight the unique properties of numbers and their reversals, which is an intriguing aspect of number theory. Understanding emirps not only enriches our appreciation of prime numbers but also illustrates the playful nature of mathematics.
