Dihedral prime
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A dihedral prime or dihedral calculator prime is a prime number that still reads like itself or another prime number when read in a seven-segment display, regardless of orientation (normally or upside down), and surface (actual display or reflection on a mirror). The first few base 10 dihedral primes are
- 2, 11, 101, 181, 1181, 1811, 18181, 108881, 110881, 118081, 120121, 121021, 121151, 150151, 151051, 151121, 180181, 180811, 181081 (sequence A038136 in OEIS)
The smallest dihedral prime that reads differently with each orientation and surface combination is 120121.
The digits 0, 1 and 8 remain the same regardless of orientation or surface. 2 and 5 turn into each other when viewed upside down or reflected on a mirror. In the display of a calculator that can handle hexadecimal, 3 would become E reflected, but E being an even digit, the 3 can't be used as the first digit because the reflected number will be even. Though 6 and 9 become each other upside down, they are not valid digits when reflected, at least not in any of the numeral systems pocket calculators usually operate in. In binary, all primes are dihedral.
Strobogrammatic primes that don't use 6 or 9 are dihedral primes.
Every repunit prime is a dihedral prime. Also, every prime of the form 10n + 1 is a dihedral prime, but the only known primes of that form appear to be 2, 11, and 101. It appears to be unknown whether there exist infinitely many dihedral primes, but this would follow from the conjecture that there are infinitely many repunit primes.
[edit] References
- Mike Keith. Puzzle 39.- The Mirrorable Numbers. The prime puzzles & problems connection.
- Eric W. Weisstein. Dihedral Prime. MathWorld – A Wolfram Web Resource.
