#Lychrel numbers

[the-adhd-sorcerer]

196 is the smallest number conjectured to be a Lychrel number in base 10; the process has been carried out for over a billion iterations without finding a palindrome, but no one has ever proven that it will never produce one.

So why did they choose the number 196?

[mythoughtsverbatim]

About 80% of all numbers under 10,000 resolve into a palindrome in four or fewer steps; about 90% of those resolve in seven steps or fewer. Here are a few examples of non-Lychrel numbers:

[l-since-l]

A006960 - Reverse and Add! sequence starting with 196

196, 887, 1675, 7436, 13783, 52514, 94039, 187088, 1067869, 10755470, 18211171, 35322452, 60744805, 111589511, 227574622, 454050344, 897100798, 1794102596, 8746117567, 16403234045, 70446464506, 130992928913, 450822227944, 900544455998, 1800098901007, 8801197801088, 17602285712176 

196 is conjectured to be the smallest initial term which does not lead to a palindrome. John Walker, Tim Irvin and others have extended the trajectory of 196 to millions of digits without finding a palindrome.

From A.H.M. Smeets, Jan 31 2019: (Start)

Palindromes for a(9)/2, a(14)/2 and a(20)/2.

Observed:  It seems that most, but not all, Lychrel numbers (seeds given in A063048) have a trajectory term that, divided by 2, becomes palindromic. Note that 196 is the first Lychrel number (A063048(1)). (End)

[kagaya25]

Project Euler Problem#55 Lychrel numbers

[mathematicianadda]

Aperiodical Round Up 11: more than you could ever need, want or be able to know https://ift.tt/2Es2Zjf

Hello, my name’s Christian Lawson-Perfect and my main mathematical interest is “everything”.

Before The Aperiodical existed as its own thing, the only outlet I had for my mathematical eclecticism was a series of posts on the Acme Science blog called Aperiodical Round-Up. Eventually I stopped writing them, as work and family took up more of my time. This post has been sitting in The Aperiodical’s drafts folder for six years. Time to finish it!

Let’s begin with the first 10,000 digits of π dialled on a rotary phone.

10,000 digits is more than I could ever remember. I’m glad I’ve got a reference to come back to if I ever need anything after the first six I’ve memorised. References are handy to have, and it’s hard to find an area of maths that doesn’t have at least one.

There’s an Encyclopedia of Finite Graphs. If you need slightly more but not a lot more whimsy, there’s the House of Graphs and its database of interesting graphs. If graphs aren’t your bag, there’s also a database of L-functions and modular forms.

Counterexamples are harder to categorise, but also worth collecting. Math Counterexamples features such handy objects as a module without a basis.

Giovanni Resta’s Numbers aplenty is an Aladdin’s cave of number facts. It’s like maximalist Number Gossip. There’s a convention to use wordplay when naming families of numbers to help you remember what they mean, like Achilles (powerful but not a perfect power) or emirp (prime forwards and backwards). You really see the limits of this approach when they’re gathered together on one page: any idea what an iban number is when it’s not an International Bank Account Number? Anyway, like Number Gossip, you can type in a number and receive a list of facts about it that are interesting to varying degrees. Weirdly, it thinks it’s notable that $3435 = 3 + 3^5 + 4^3+ 5^5$, but not that $3435 = 3^3 + 4^4 + 3^3 + 5^5$.

If you want an online database of mathematical facts that’s neither authoritative nor useful, but could be mistaken from a distance for an occult conspiracy libel, “196 and other Lychrel numbers” is the site for you. Or rather, it was – it seems to have disappeared since I noted it down a few years ago. Ask yourself this: who stands to gain from its disappearance?

Once you start looking, you see menacing mathematics everywhere: “Kepler, Dürer and the mystery of the forbidden tilings” is a subheading to set the heart racing. Implied conspiracy? No, Imperfect congruence: an essay in four parts and a few appendices by Kevin Jardine, about tilings and the fact that no edge-to-edge regular polygon tiling of the plane can include a pentagon.

Once you start looking, you find that mathematics really is everywhere. Some walls in Leiden have formulas on them; you can thank Jan van der Molen and Ivo van Vulpen for that. Other walls in Leiden have poems on them; it’s important to have a varied diet.

Which Springer Verlag graduate text in mathematics are you? I’m Saunders Mac Lane’s Categories for the Working Mathematician.

Antonio Marquez-Raygoza has made a lovely page about constructing the Sierpiński triangle, with interactive diagrams. If you’ve read and absorbed all of that – I don’t think you have, by the way – you can visit Tadao Ito’s Hyperbolic Non-Euclidean World and Figure-8 Knot. Pack a bag – I reckon I’d need to spend at least a week in Hyperbolic Non-Euclidean World to see all the sights.

For some mathematical enrichment closer to home, Yutaka Nishiyama has thought up 55 examples of mathematics in daily life.

Now, a sequence of links to pictures.

The Picture this maths blog, run by Rachael Boyd and Anna Seigal, features essays on an eclectic range of topics. The schtick is – maybe you can guess – there are lots of pictures.

DivulgaMAT has a collection of Instantáneas Matemáticas – snapshots of maths in old works of art, which I’ve always got time for.

Dataisnature collects examples of the “hand-drawn yet precise diagrams with sparse lines and the occasional dot” aesthetic. A post linking John Cage’s music and nomography has reignited my fascination with nomograms, leading me to Ron Doerfler’s book on The Lost Art of Nomography, and his fabulous 2010 graphical computing calendar. Excitingly, there’s a program called pyNomo which sufficiently powerful wizards can use to produce their own nomograms!

Robert Munafo has drawn a charmingly old-hat ASCII fractal drawing of the Feigenbaum point. However, the connoisseur knows that a true fractal must be printed out on a dot matrix or at least produced with a typewriter: anything else is just sparkling self-similarity.

That’s it for now!

from The Aperiodical https://ift.tt/2G7G8Kb Christian Lawson-Perfect from Blogger https://ift.tt/3lstdmk

[dani-malpica]

WSQ08 - Yo soy 196

This assignment is called "Yo soy 196" which can be translated to "I am 196", for this assignment we are working with Palindromes and Lychrel Numbers.

For some context Palindromes can be number, words or even complete sentences that can be read form left to right and vice versa and it will be the same. ex: 11, 666, 181, 373, sexes, Never odd or even

With numbers it exist something we call "Natural palindromes" you can guess what this means by the name (ex: 11, 44, 88, 333). But there is another kind of palindrome when we refer about numbers, in order to find the palindrome of these numbers we need to add their inverse (ex: 12 + 21 = 33). There are numbers that may be so large that after to many repetitions or iterations we cannot prove they are palindromes. These numbers are called Lychrel numbers and the first example is the number 196 because it's the first number that cannot be proved to reach a palindrome after too many iterations.

I found this assignment to be particularly difficult if you don’t understand the previous concepts. In this program the user has to introduce the lower bound and the higher bound of a range of numbers. The program has to display how many natural palindromes are, how many non-Lychrel numbers are, and finally how many Lychrel candidates are. This program is designed to do the iteration process 30 times, so if after this process the program cannot find a palindrome the number will be considered a Lychrel candidate.

[casmong]

Math: Lychrel number

A Lychrel number is a natural number that cannot form a palindrome through the iterative process of repeatedly reversing its digits and adding the resulting numbers. This process is sometimes called the 196-algorithm, after the most famous number associated with the process. In base ten, no Lychrel numbers have been yet proved to exist, but many, including 196, are suspected on heuristic[1] and statistical grounds. The name "Lychrel" was coined by Wade Van Landingham as a rough anagram of Cheryl, his girlfriend's first name.

[mathandnumberystuff]

Lychrel numbers, part 2

This is a follow-up to my earlier post about Lychrel numbers.

It seems very likely that the reverse-and-add process will never reach a palindrome when starting with 196. This is because when a number is on the order of 10^(2*n), the probably of it resolving into a palindrome in the next iteration is very roughly 1 in 2^n. Since the number multiplies by about sqrt(10) after each iteration, the probability of it reaching a palindrome gets decreases exponentially for each iteration, by a factor of about 2^(1/4). For those with a background in cellular automata, the process can be liked to running a pattern in a very barely exploding rule, like B368/S1278, where the probability of eventual stabilization decreases exponentially with the number of unstabilized cells.

However, like the normality of pi or e, or whether any patterns in the aforementioned rule explode, this has not been proven, because the process is an adequate pseudo-random number generator.

On the other hand, in binary it can be proven that some numbers will never reach a palindrome this way, the smallest being 10110.

This comes from the fact that when a binary number is of the form 10[1]n-101[0]n (where [b]n is the digit b repeated n times), the next four steps of the process will give 11[0]n-110[1]n-201, 10[1]n01[0]n, and 10[1]n01[0]n+1. None of these forms are palindromes, and the fourth is just the first with n replaced by n+1. So whenever a number can be written as 10[1]n-101[0]n for some value of n, by induction it will never produce a palindrome. 10110 becomes 1010100 after two steps and thus, it is proven that it is a binary Lychrel number.

[mathandnumberystuff]

Lychrel numbers

One year ago, on July 16, 2015, I got the laptop on which I am writing this now.

First of all, the day July 16 should be familiar to anyone who read my Cheryl’s Birthday post. And when I got my laptop, I wanted to name it after this math problem. At first I thought of naming my laptop Cheryl. But then, I remembered the Lychrel numbers: a set of numbers that was named as a “rough anagram” of the name Cheryl. To make my computer’s name more unique, I decided to name it Lychrel as well, a name that related to two math concepts.

But what are the Lychrel numbers?

Pick a number, like 73. Now add it to its reverse. Keep doing this until the result is a palindrome. In the case of 73, we add 73 + 37 = 110, 110 + 011 = 121, which is a palindrome.

Now, some numbers take a few steps to become a palindrome, like 56. Some take several steps (89 takes 24). And there is strong evidence that some of them never become palindromes.

Numbers that never produce palindromes from this process are called Lychrel numbers. The first Lychrel number is (conjectured to be) 196.

The interesting thing is, though, that no number has been proven to be a Lychrel number in base 10.

Here is a site where a number can be tested for 300 iterations (just replace the 196 in the URL: http://jasondoucette.com/pal/196)

[pep-the-artemis]

because its 1 less than 196 which is the smallest number conjectured to be a Lychrel number in base 10; the process has been carried out for over a billion iterations without finding a palindrome, but no one has ever proven that it will never produce one.

So why did they choose the number 196?

[mythoughtsverbatim]

The first few candidate Lychrel numbers (sequence A023108 in the OEIS) are:

196, 295, 394, 493, 592, 691, 790,

689, 788, 879,

887, 978,

986,

1495, 1497,

1585, 1587,

1675, 1677,

1765, 1767,

1855, 1857,

1945, 1947, 1997.

About 80% of all numbers under 10,000 resolve into a palindrome in four or fewer steps; about 90% of those resolve in seven steps or fewer. Here are a few examples of non-Lychrel numbers.

About 80% of all numbers under 10,000 resolve into a palindrome in four or fewer steps; about 90% of those resolve in seven steps or fewer. Here are a few examples of non-Lychrel numbers: