Strobogrammatic number

In the entertainment mathematics is a strobogrammatische number (from the English strobogrammatic number ) is a number which remains the same when it rotates through 180 °. The number is thus a special type of ambigram . A strobogrammatic prime number is a strobogrammatic number that is also a prime number .

Entertaining math enthusiasts will be interested in this concept, while professional mathematicians in general will not bother with it. As with the repunits and the palindrome numbers, strobogrammatic numbers depend on their base . Usually the base is considered, i.e. the decimal system . ${\displaystyle b}$ ${\displaystyle b=10}$

Examples

With a 7-segment display, the digits 0, 1, 2, 5 and 8 are strobogrammatic

In general it has to be mentioned that it depends on which font you use to be able to determine whether a number is strobogrammatic. For example, the number 1 is not strobogrammatic if you write it with primer. Without painting it looks like an I and is then very well strobogrammatic. Usually, however, this number is viewed as strobogrammatic.

• The following three digits do not change when you turn them 180 °, so they are single-digit strobogrammatic numbers:

0, 1, 8

• If you turn the number 6 by 180 °, you get the number 9 and vice versa. These two digits are therefore not strobogrammatic numbers themselves, but are suitable as building blocks for multi-digit strobogrammatic numbers.
• The smallest strobogrammatic numbers are the following:

0, 1, 8, 11, 69, 88, 96, 101, 111, 181, 609, 619, 689, 808, 818, 888, 906, 916, 986, 1001, 1111, 1691, 1881, 1961, 6009, 6119, 6699, 6889, 6969, 8008, 8118, 8698, 8888, 8968, 9006, 9116, 9696, 9886, 9966, 10001, 10101, 10801, 11011, 11111, 11811, 16091, ... (sequence A000787 in OEIS )

• The smallest strobogrammatic prime numbers are the following:

11, 101, 181, 619, 16091, 18181, 19861, 61819, 116911, 119611, 160091, 169691, 191161, 196961, 686989, 688889, 1008001, 1068901, 1160911, 1180811, 1190611, 1191611, 1681891, 1690691, 1880881, 1881881, 1898681, 1908061, 1960961, 1990661, 6081809, 6100019, 6108019, ... (follow A007597 in OEIS )

• The last three strobogrammatic years were 1691, 1881 and 1961. The next strobogrammatic year is 6009. In the English-speaking world, such a year is called an upside down year .
• In the Indian fonts Devanagari or Gurmukhi (and many other fonts) there are no strobogrammatic numbers.
• In the case of a 7-segment display, as is common with older pocket calculators, for example, the three strobogrammatic digits 0, 1 and 8 also the two digits 2 and 5 are strobogrammatic.

Strobogrammatic numbers in other number systems

• In the dual system there are only the digits 0 and 1. Since both digits are strobogrammatic, all palindromic numbers are also strobogrammatic at the same time. The smallest of them are:

0, 1, 11, 101, 111, 1001, 1111, 10001, 10101, 11011, 11111, 100001, ...

In the decimal system these are the following numbers:

0, 1, 3, 5, 7, 9, 15, 17, 21, 27, 31, 33, ... (sequence A006995 in OEIS )

Example:

The 9th position in the list above is the number 10101. This number, when converted to the decimal system, is the number 21:

10101 = ${\displaystyle {\underline {1}}\cdot 2^{4}+{\underline {0}}\cdot 2^{3}+{\underline {1}}\cdot 2^{2}+{\underline {0}}\cdot 2^{1}+{\underline {1}}\cdot 2^{0}=16+4+1=21}$

• All Fermat numbers in the dual system have the form with a one at the beginning and at the end and zeros in the middle, are obviously palindromic numbers and thus strobogrammatic numbers in the dual system (for example is a strobogrammatic number). ${\displaystyle F_{n}=2^{2^{n}}+1}$ ${\displaystyle 100\ldots 001}$ ${\displaystyle 2^{n}-1}$ ${\displaystyle F_{3}=2^{2^{3}}+1=257_{10}=100000001_{2}}$
• All Mersenne numbers have the form with ones in the dual system , are also palindromic numbers and thus strobogrammatic numbers in the dual system (for example is a strobogrammatic number). ${\displaystyle M_{n}=2^{n}-1}$ ${\displaystyle 111\ldots 111}$ ${\displaystyle n}$ ${\displaystyle M_{6}=2^{6}-1=63_{10}=111111_{2}}$
• In the duodecimal system , i.e. in the number system with the base , there are the digits 0123456789↊↋, so the decimal number 10 is written as ↊, 11 as ↋ and only the decimal number 12 is the number 10 in the duodecimal system and thus two-digit. So the digits 0, 1 and 8 are still strobogrammatic, but the digits 2 and 3 (next to 6 and 9), rotated by 180 °, also result in suitable digits. The following numbers are strobogrammatic in the duodecimal system: ${\displaystyle b=12}$

0, 1, 8, 11, 2↊, 3↋, 69, 88, 96, ↊2, ↋3, 101, 111, 181, 20↊, 21↊, 28↊, 30↋, 31↋, 38↋, 609, 619, 689, 808, 818, 888, 906, 916, 986, ↊02, ↊12, ↊82, ↋03, ↋13, ↋83, ...

• The first strobogrammatic primes in the duodecimal system are the following:

11, 3↋, 111, 181, 30↋, 12↊1, 13↋1, 311↋, 396↋, 3↊2↋, 11111, 11811, 130↋1, 16191, 18881, 1↋831, 3000↋, 3181↋, 328↊↋, 331↋↋, 338↋↋, 3689↋, 3818↋, 3888↋, ...

Example:

In the 10th position in the list above is the number 3↊2↋. In fact, when converted to the decimal system, this number is prime:

3↊2↋ = ${\displaystyle {\underline {3}}\cdot 12^{3}+{\underline {10}}\cdot 12^{2}+{\underline {2}}\cdot 12^{1}+{\underline {11}}\cdot 12^{0}=3\cdot 1728+10\cdot 144+2\cdot 12+11\cdot 1=5184+1440+24+11=6659\in \mathbb {P} }$

• The two digits 0 and 1 are the only digits that are strobogrammatic digits in any number system, provided that a suitable font is used.

Trivia

The American satirical magazine Mad parodied the upside-down year in March of the strobogrammatic year 1961.

See also

• Dihedral prime number
• Tetradic number

Individual evidence

1. ↑ Chris K. Caldwell: strobogrammatic. The Prime Glossary, accessed February 4, 2020 . 
2. ^ Heinrich Hemme : Mathematics for breakfast: 89 mathematical puzzles with detailed solutions. Vandenhoeck & Ruprecht , p. 71 , accessed on February 4, 2020 . 
3. ↑ ^{a b} Mad Magazine March 1961 # 61 Upside-down year spy vs spy