Tetradic number

A tetradic number, also known as a four-way number, is a number that remains the same when flipped back to front, flipped front to back, mirrored up-down, or flipped up-down. The only numbers that remain the same which turned up-side-down or mirrored are 0, 1, and 8, so a tetradic number is a palindromic number containing only 0, 1, and 8 as digits. (This is dependent on the use of a handwriting style or font in which these digits are symmetrical, as well on the use of Arabic numerals in the first place.) The first few tetradic numbers are 1, 8, 11, 88, 101, 111, 181, 808, 818, ... (OEIS A006072).^[1][2][3][4]

Tetradic numbers are also known as four-way numbers due to the fact that they have four-way symmetry and can flipped back to front, flipped front to back, mirrored up-down, or flipped up-down and always stay the same. The four-way symmetry explains the name, due to tetra- being the Greek prefix for four. Tetradic numbers are both strobogrammatic and palindromic.^[3][4]

A larger tetradic number can always be generated by adding another tetradic number to each end, retaining the symmetry.

Tetradic primes

Tetradic primes are a specific type of tetradic number defined as tetradic numbers that are also prime numbers. The first few tetradic primes are 11, 101, 181, 18181, 1008001, 1180811, 1880881, 1881881, ... (OEIS A068188).^{[5][6][7][8][9][10]}

The largest known tetradic prime as of April 2010^[update] is

${\displaystyle 10^{180054}+8R_{58567}\cdot 10^{60744}+1,}$ 

where ${\displaystyle R_{n}}$  is a repunit, that is, a number which contains only the digit 1 repeated ${\displaystyle n}$  times. The prime has 180,055 decimal digits.^[3]

References

1. Sloane, N. J. A. Sequences A006072/M4481 in "The On-Line Encyclopedia of Integer Sequences."
2. Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics (2nd ed.). CRC Press. ISBN 978-1420035223.
3. "Tetradic Number". Wolfram MathWorld. Retrieved October 28, 2018.
4. "tetradic number". Everthing2. January 5, 2002. Retrieved October 28, 2018.
5. Sloane, N. J. A. Sequences A068188 in "The On-Line Encyclopedia of Integer Sequences."
6. Caldwell, Chris K. "tetradic prime". The Prime Glossary. The University of Tennessee Martin. Retrieved October 28, 2018.
7. H. Dubner and R. Ondrejka, "A PRIMEr on palindromes," J. Recreational Math., 26:4 (1994) 256–267.
8. R. Ondrejka, "On tetradic or 4-way primes," J. Recreational Math., 21:1 (1989) 21–25.
9. Ondrejka, R. "The Top Ten Prime Numbers" (PDF). The Prime Pages. Retrieved October 28, 2018.
10. Carmody, Phil. "Totally Tetradic!". Fat Phil. Retrieved October 28, 2018.