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 is

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

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

References

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

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[1] Sloane, N. J. A. Sequences A006072/M4481 in "The On-Line Encyclopedia of Integer Sequences."

[:0-2] Weisstein, Eric W. (2002). CRC Concise Encyclopedia of Mathematics (2nd ed.). CRC Press. ISBN 978-1420035223.

[:1-3] 1 2 3 "Tetradic Number". Wolfram MathWorld. Retrieved October 28, 2018.

[:2-4] 1 2 "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.

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Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
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 $(10 n - 1)/9$ Permutable Circular Truncatable Minimal Delicate Primeval Full reptend Unique Happy Self Smarandache–Wellin Strobogrammatic Dihedral Tetradic
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n \pm 1, 2 n \pm 1, 4 n \pm 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k-tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain/Safe ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
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