3 Interesting 3-digit Numbers

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Some numbers have really curious properties! Below are 3 such 3-digit numbers.

196-Algorithm

A palindromic number is a number (in some base b) that is the same when written forwards or backwards. Like 16461, for example, it is “symmetrical”. The term palindromic is derived from palindrome, which refers to a word (such as rotor or racecar) whose spelling is unchanged when its letters are reversed.
Take any positive integer of two digits or more, reverse the DIGITS , and add to the original number. Now repeat the procedure with the sum so obtained. This procedure quickly produces palindromic numbers for most integers. For example, starting with the number 5280 produces (5280, 6105, 11121,23232). The value for 89 is especially large, being 8813200023188. This process is sometimes called the 196-algorithm, after the most famous number associated with the process.
The first few numbers not known to produce Palindromes are 196, 887, 1675, 7436, 13783, which are simply the numbers obtained by iteratively applying the algorithm to the number 196. This number therefore lends itself to the name of the Algorithm. A Lychrel number is a natural number that cannot form a palindrome through this process. The name “Lychrel” was coined by Wade Van Landingham as a rough anagram of Cheryl, his girlfriend’s first name.
In 1990, a programmer named John Walker computed 2,415,836 iterations of the algorithm for the number 196, yielding a non-palindromic number with a million digits in length. This result has continually been improved over the years. In 1995, Tim Irvin used a supercomputer and reached the two million digit mark in only three months without finding a palindrome. Jason Doucette then followed suit and reached 12.5 million digits in May 2000. Wade VanLandingham used Jason Doucette’s program to reach 13 million digits. Since June 2000, Wade VanLandingham has been carrying the flag using programs written by various enthusiasts. By 1 May 2006, VanLandingham had reached the 300 million digit mark. Using distributed processing, in 2011 Romain Dolbeau completed a billion iterations to produce a number with 413,930,770 digits, and in February 2015 his calculations reached a number with billion digits. A palindrome has yet to be found.
With this insight, a more general question to be asked is: Do Lychrel numbers exist at all? If they do, their existence seems to be few and far between. In fact, through computational verification, about 90% of all natural numbers less than 10,000 are not Lychrel numbers. Of course, no matter how intuitive these results seem to be, in mathematics, computational calculation is not always the same as a proof.

The Sisyphus String: 123

Suppose we start with any natural number, regarded as a string, such as 9,288,759. Count the number of even digits, the number of odd digits, and the total number of digits. These are 3 (three evens), 4 (four odds), and 7 (seven is the total number of digits), respectively. So, use these digits to form the next string or number, 347. Now repeat with 347, counting evens, odds, total number, to get 1, 2, 3, so write down 123. If we repeat with 123, we get 123 again. The number 123 with respect to this process and the universe of numbers is a mathemagical black hole. All numbers in this universe are drawn to 123 by this process, never to escape.
But will every number really be sent to 123? Try a really big number now, say 122333444455555666666777777788888888999999999(or pick one of your own). The numbers of evens, odds, and total are 20, 25, and 45, respectively. So, our next iterate is 202,545, the number obtained from 20, 25, 45. Iterating for 202,545 we find 4, 2, and 6 for evens, odds, total, so we have 426 now. One more iteration using 426 produces 303, and a final iteration from 303 produces 123. At this point, any further iteration is futile in trying to get away from the black hole of 123, since 123 yields 123 again.

257-gon

Euclidean Constructions are those constructions that can be completed using only a straight edge and a collapsing compass which closes when it is picked up. This collapsability causes problems because it means we cannot simply move a distance with a compass. There are three basic constructions that can be completed in Euclidean constructions. These constructions are:

  1. The straight line by connecting two points,
  2. A circle of a given radius centered at a given point or
  3. Continuing a segment infnitely.

From these given constructions a number of other constructions can be created. A constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regular pentagon is constructible with compass and straightedge while a regular heptagon is not.

The nth Fermat number is defines as:

F_n = 2^{2^n} + 1

The 17th-Century French lawyer and mathematician Pierre de Fermat incorrectly conjectured that all these numbers were primes, although he had no proof. The first 5 Fermat numbers: 3, 5, 17, 257, 65537 corresponding to n = 0,1,2,3,4 are all primes. These are called Fermat primes.
Some regular polygons are easy to construct with compass and straightedge; others are not. The ancient Greek mathematicians knew how to construct a regular polygon with 3, 4, or 5 sides and they knew how to construct a regular polygon with double the number of sides of a given regular polygon. This led to the question being posed: is it possible to construct all regular n-gons with compass and straightedge? If not, which n-gons are constructible and which are not?
Carl Friedrich Gauss proved the constructibility of the regular 17-gon in 1796. Five years later, he developed the theory of Gaussian periods in his Disquisitiones Arithmeticae. This theory allowed him to formulate a sufficient condition for the constructibility of regular polygons:
A regular n-gon can be constructed with compass and straightedge if n is the product of a power of 2 and any number of distinct Fermat primes (including none).
Although Gauss proved that the regular 17-gon is constructible, he did not actually show how to do it. The first construction is due to Erchinger, a few years after Gauss’ work. The first explicit constructions of a regular 257-gon were given by Magnus Georg Paucker (1822) and Friedrich Julius Richelot (1832). A construction for a regular 65537-gon was first given by Johann Gustav Hermes (1894). The construction is very complex; Hermes spent 10 years completing the 200-page manuscript.

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