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Is 196 a Lychrel Number?


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Photo: Karen Carr

Photo: Karen Carr

Mathematical revelation is often manifested in the emergence of imagination inspired by simple truths.   From cups of coffee and the clacking of chalk, generalizations leap from eraser-smudged boards into the fabric of physical reality.  The legendary mathematician Carl Friedrich Gauss once said, “Mathematics is concerned only with the enumeration and comparison of relations.”   This notion echoing from the prolific 19th century genius provided a guiding light into the dark regions of non-intuitive abstraction.

In particular, number theorists live out Gauss’ words by exploring the nature of numbers in hopes of exposing unique patterns and hidden connections. Such a concept is demonstrated in the examination of palindromic numbers.  These unique numbers exhibit symmetry and can be read the same way forwards as well as backwards.  For example, 1001 is a palindromic number since it is the same number from left to right as it is from right to left.

Special numbers such as these are frequently related to unsolved problems in mathematics. Palindromic Numbers, in particular, are firmly related to an open problem with an algorithmic flavor.  If we define a process in which we reverse the digits for any positive integer with two or more digits and add the new number to the original, repeating this process again and again, and terminating when a palindromic number is yielded, then the process is known as the 196-Algorithm.  For instance, take the number 25 and reverse its digits to obtain 52.   Then add the two numbers together to obtain 25 + 52 = 77; yielding a palindromic number that ends the process.

The 196-Algorithm works for many natural numbers.  However, there are a few, with 196 being the least, in which the process has yet to yield a palindromic number.  Numbers that do not terminate to a palindromic number after an iterative process of reversing and adding digits are known as Lychrel numbers.   With this definition in mind, the following unanswered question can be proposed: Is 196 a Lychrel number?

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.  So much so, that in 2012, it was determined that if the iterative process yields a palindromic number for 196, then the resulting palindromic number would have more than 600 million digits.

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.

Many times, mathematicians seek to prove general questions in which special cases follow.   In the case of the 196-Algorithm, proving the existence or non-existence of Lychrel numbers determines whether 196, and candidates like it, terminate to palindromic numbers under the algorithmic process. Through the looking glass of mathematical revelation, generalizations for the existence of Lychrel numbers may come from a clever manipulation of profound ideas.  However, it is just as likely to be distilled from a logical derivation of simple truths.

Avery Carr About the Author: Avery Carr is a graduate student studying mathematics at Emporia State University. He holds a B.S. in mathematics from the University of Memphis and is currently serving as an Editor for the AMS Graduate Student Blog and Editor for the problems section for the 100th Anniversary Issue of the Pi Mu Epsilon Journal. Inspired by the late Scientific American “Mathematical Games” columnist Martin Gardner and the prolific mathematician Paul Erdős, Avery has recently started the blog Math With Coffee to celebrate discrete and recreational mathematics. Follow on Twitter @ATCarrMath.

The views expressed are those of the author and are not necessarily those of Scientific American.






Comments 3 Comments

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  1. 1. brynn217 10:04 am 10/1/2013

    Umm – 90% of all natural numbers less than 10,000 are not Lychrel numbers…. Doesnt this mean that 1000 numbers ARE Lychrel numbers..

    However, doesnt the existance of 1 Lychrel number simply disprove the 196-Algorithm? How can we have an Algorithm that only works 90% of the time?

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  2. 2. kebil 10:41 am 10/1/2013

    The remaining 10% may, or may NOT be Lychrel numbers. Because there is no general proof that a Lychrel number exists, you would have to calculate, through to infinity, the sums in order to prove that no palindrome can be made.

    Since calculation to infinity is not possible, we currently don’t have a way of knowing if any particular number is a Lychrel number.

    So no, 90% of natural numbers is the lower limit of the number numbers that are not Lychrel. These can be shown to not be because they can be made into palindromic numbers. To show that an number can not be made palindromic is to prove the negative, which will require a mathematical insight as pure computation does not seem up to the challenge.

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  3. 3. ATCarrMath 10:54 am 10/1/2013

    brynn217 and kebil,

    Thank you for your comments. kebil is right. A computational verification that 90% of all integers less than 10,000 are not Lychrel numbers does not necessarily imply that the remaining 10% are Lychrel numbers. They are only Lychrel candidates until proven to be or not to be so.

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