Tetradic number

In entertainment mathematics , a tetradic number (from the English tetradic number , also four-way number ) is a number with the property that the four numbers ${\displaystyle n}$

• ${\displaystyle n}$
• ${\displaystyle n}$ rotated by 180 °
• ${\displaystyle n}$ mirrored horizontally
• ${\displaystyle n}$ flipped vertically

always gives the same number . ${\displaystyle n}$

The number 1 must be written as I and with the number 8 both loops must be the same size. The word tetra is the Greek prefix for the number 4 and gives it its name because tetradic numbers have four symmetries in the manner mentioned above .

A tetradic number that is prime is called a tetradic prime number .

As with the strobogrammatic numbers , tetradic numbers depend on their base . Usually the base is considered, i.e. the decimal system . ${\displaystyle b}$ ${\displaystyle b=10}$

Examples

• The only digits that can appear in tetradic numbers are the digits 0, 1 and 8.
• The smallest tetradic numbers are the following:

0, 1, 8, 11, 88, 101, 111, 181, 808, 818, 888, 1001, 1111, 1881, 8008, 8118, 8888, 10001, 10101, 10801, 11011, 11111, 11811, 18081, 18181, 18881, 80008, 80108, 80808, 81018, 81118, 81818, 88088, 88188, 88888, 100001, 101101, 108801, 110011, ... (sequence A006072 in OEIS )

• The smallest tetradic prime numbers are the following:

11, 101, 181, 18181, 1008001, 1180811, 1880881, 1881881, 100111001, 100888001, 108101801, 110111011, 111010111, 111181111, 118818811, 180101081, 181111181, 181888181, 188010881, 188888881, 10008180001, 10081818001, ... (sequence A068188 in OEIS )

• The next list shows how many -digit tetradic numbers (in ascending order ) there are: ${\displaystyle n}$ ${\displaystyle n=1,2,3,\ldots }$

3, 2, 6, 6, 18, 18, 54, 54, 162, 162, 486, 486, 1458, 1458, 4374, 4374, 13122, 13122, 39366, 39366, 118098, 118098, 354294, 354294, 1062882, 1062882, 3188646, 3188646, 9565938, 9565938, 28697814, 28697814, 86093442, 86093442, 258280326, 258280326, 774840978, ... ( continuation A225367 in OEIS )

Example:

The fourth number in the top list is 6. This means that there are 6 four-digit tetradic numbers (specifically 1001, 1111, 1881, 8008, 8118 and 8888).

• There is no greatest tetradic number. One can always find a larger tetradic number by adding any other tetradic number on either side of a given tetradic number so that symmetry is preserved.

Example:

The number 8008 is a tetradic number. For example , if you add the tetradic number 1001 to both sides , you get 1001 8008 1001 , which in turn is a tetradic number.

• The largest known tetradic prime number is the following (as of February 5, 2020):

${\displaystyle p=10^{180054}+8\cdot {\frac {10^{58567}-1}{9}}\cdot 10^{60744}+1}$

It was discovered by Darren Bedwell in 2009 and has 180,055 jobs.

useful information

• In contrast to the dihedral prime numbers, the digits 2 and 5 must not appear in the tetradic numbers.
• Every tetradic prime number is also a dihedral prime number at the same time.
• Prime number palindromes that only contain the digits 0, 1 and 8 are tetradic numbers.
• Every tetradic number is a prime number palindrome in which only the digits 0, 1 and 8 appear (the reverse of the sentence above).
• Tetradic numbers are both strobogrammatic numbers and number palindromes .
• Each repunit is a tetradic number.
• The prime number is the only tetradic prime number that has an even number of digits. All other tetradic primes have an odd number of digits. ${\displaystyle p=11}$

(All other tetradic prime numbers with an even number of digits are divisible by 11.)

Tetradic numbers in other number systems

• In the dual system , i.e. in the number system with a base , all prime number palindromes are tetradic numbers. The smallest are the following, written in the dual system: ${\displaystyle b=2}$

0, 1, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 11111, 100001, 101101, 110011, ... (sequence A057148 in OEIS )

(This follows from the above-mentioned theorem that prime number palindromes in which only the digits 0, 1 and 8 occur are tetradic numbers. Since only zeros and ones occur in the binary system, this condition is fulfilled.)

• If you replace the number 8 in the tetradic numbers with the number 2, you get the number palindromes for the base . The smallest number palindromes for the base are the following (written in this number system, the ternary system ): ${\displaystyle b=3}$ ${\displaystyle b=3}$

0, 1, 2, 11, 22, 101, 111, 121, 202, 212, 222, 1001, 1111, 1221, 2002, 2112, 2222, 10001, 10101, 10201, 11011, 11111, 11211, 12021, 12121, 12221, 20002, 20102, 20202, 21012, 21112, 21212, 22022, 22122, 22222, 100001, 101101, 102201, 110011, ... (follow A118594 in OEIS )

However, these numbers are not tetradic numbers for the base , because the number 2 must not appear in tetradic numbers. The smallest tetradic numbers for the base are the following written in the ternary system: ${\displaystyle b=3}$ ${\displaystyle b=3}$

0, 1, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 11111, 100001, 101101, 110011, ... (sequence A057148 in OEIS )

See also

• Dihedral prime number
• Strobogrammatic number

Web links

• Eric W. Weisstein : Tetradic Number . In: MathWorld (English).
• tetradic number. everything2, accessed February 8, 2020 . 
• Chris K. Caldwell: tetradic prime. Prime Pages, accessed February 8, 2020 . 
• Róbert Ondrejka: The top ten prime numbers. The Prime Pages, accessed February 8, 2020 . 
• Phil Carmody: Totally Tetradic! Retrieved February 8, 2020 . 

Individual evidence

1. ↑ Patrick De Geest: Palindromic Primes, September 2 , 2007. World! Of Numbers, accessed February 8, 2020 . 
2. ↑ Comments on OEIS A006072