Conjecture related to Lychrel numbers

Question

My conjecture is as follows:

Take any number. It will become a palindrome eventually through the same reversal process used for Lychrel numbers except if the term (first term is excluded) starts with 1, the 1 is disregarded.

For example: 196+691=887

887+788=1675 => 675

675+576=1251 => 251

251+152=403

403+304=707 is a palindrome.

Can anyone prove this or find a counterexample? Thanks for your time and help!

score 3 · Answer 1 · answered Nov 27 '21 at 09:28

3

The smallest counter-example is $10039547$, which gets into a cycle:

$$ \begin{array}{} \color{red}{84632548} \\ 69156196 \\ 38321392 \\ 67633775 \\ 25367451 \\ 40843803 \\ 71678607 \\ 42366224 \\ \color{red}{84632548} \\ \end{array} $$ Found by exhaustive brute-force search.

answered Nov 27 '21 at 09:28

Vepir

12,516

does this mean all numbers less than $10039547$ end in a palindrome? – miracle173 Nov 27 '21 at 11:56
@miracle173 Yes. Only after $10^7$ we get (a lot of) examples that start cycling. – Vepir Nov 27 '21 at 13:31
@miracle173 I found cycles in other number bases, but I can't find them for all number bases. (See linked question if interested.) – Vepir Dec 01 '21 at 17:05
maybe something to calculate for you https://math.stackexchange.com/questions/4333686/how-to-transform-a-monomial – miracle173 Dec 15 '21 at 22:50

Conjecture related to Lychrel numbers

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