Talk:Lychrel number

Evidence

I will try and find evidence that the term 'Lychrel' is widely accepted.

Quest

Added the "196 palindrome quest". Should this be a separate article? Also did some general capitalisation fixes. --Bruce1ee (Talk) 12:12, 16 May 2005 (UTC)

I think it should be. Note that Wade's page (http://www.p196.org) has a lot of information collaborated together, but neglects to mention the work done before 1,000,000 digits was reached. In terms of exponential increase in CPU power, the work done before, say to 77,000 digit, is just as important. (Likewise, work done from 240,000,000 to 241,000,000 digits is NOT as important as the first million.) CPU and memory limitations have to be taken into account to properly understand this. Regardless, we needn't get into such details... it should just list everyone involved. Jason's site (http://www.jasondoucette.com) collaborates a lot of information found on the 'net regarding all work done prior to his 2 to 13 million digit work, including many sources prior to John Walker's first million digits. Please read over both of these sites during creation of such a page, if it is made. 137.186.22.225 01:06, 18 November 2005 (UTC)

Frames

Links like: http://home.cfl.rr.com/p196/lychrel%20records.html do not work because Wade's page uses frames, and anything not loaded in a frame forwards to the main page. This makes links to the inside impossible, unfortunately. He should really update his site without frames. 137.186.22.225 01:06, 18 November 2005 (UTC)

Proof not found

The text states "Vaughn Suite devised a program that only saves the first and last few digits of each iteration, enabling testing of the digit patterns in millions of iterations to be performed without having to save each entire iteration to a file [5]." Um... I think this is wrong. Saving the first and last digits only was a hunch that perhaps a pattern could be found from these digits, and it would be nice to have JUST these digits, and a lot of them (iterations, that is), rather than have all of the tons of digits inbetween that served no purpose for this particulat pattern searching idea. Vaughne made a program to do this. It had no effect on the algorithm for the 196 quest, or Most Delayed Palindromic Number quest, as far as I know. Please refer to that reference again, and if that reference is unclear, the author of the reference may have it wrong. 137.186.22.182 01:22, 30 November 2005 (UTC)

Huh?

This article seems to show persistent confusion as to what exactly a Lychrel number is (probably because the principal reference at p196.com seems to have the same confusion). If a Lychrel number is defined as in the first sentence of this article, then no example of a Lychrel number is known. The "Proof not found" section is occasionally careful to note that 196 is only a "candidate Lychrel", but then also calls it a "seed Lychrel", which by definition is a Lychrel. And then "Benjamin Despres' program has identified all Lychrel seeds numbers of each digit length starting with 1-digit numbers up to and including all 16-digit numbers have been tested so far" - apart from making little grammatical sense, this sentence implies (actually it states directly) that Lychrel seeds are known. Please correct me if I'm wrong, but the definition and exposition in the article say that no Lychrel number is known, and therefore no seed, no kin, and no thread Lychrel number is known. Like I said, this confusion exists throughout p196.com as far as I can tell.

I additionally don't like the tone of "it is not currently possible to prove that a number will never form a palindrome". Mathematicians would just say that no proof is known -- not that no proof is "currently possible", whatever that means. Staecker 17:58, 28 January 2007 (UTC)

Other Bases

It's been proven that Lychrel numbers exist in other bases; 10110 in binary (22) has been proven to be Lychrel. Where would be a good place to add that?

Oops, forgot to timestamp. Ralphmerridew 13:22, 7 February 2007 (UTC)

Persistence

Would it be informative to add a "See also" link to Persistence of a number? Not a mathematician so not completely sure this is a fruitful connection. Theoh (talk) 22:19, 1 May 2017 (UTC)

Most delayed palindrome

1,999,291,987,030,606,810 is not the largest known most-delayed palindrome.

Since 89 reaches the palindrome 8813200023188, 890000000000089 reaches 88132000231888813200023188, doing the same thing for 1999291987030606810: 1999291987030606810 0000000...0000000 1999291987030606810 with 100+n zeros reaches 4456...6544000...0004456...6544 with n zeros in 261 steps. There is no limit to how large n can be and therefore no largest most delayed palindrome. — Preceding unsigned comment added by 71.179.19.89 (talk) 12:47, 31 August 2017 (UTC)

I think I agree with this. Anyone else? Staecker (talk) 11:25, 1 September 2017 (UTC)

I can confirm that 199929198703060681000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001999291987030606810 also forms a palindrome after 261 steps. Checked it on this here computer. So 1999291987030606810 is certainly not the largest known. Staecker (talk) 23:02, 4 September 2017 (UTC)

Interesting observation, but there is a big difference between 89 and 89000..00..0089, as the former is a kind of "prime" delayed palindrome whereas the latter is just a "composite" or "derived" one. There is a straight analogy with prime (originating) and composite (derived) numbers. They both compose infinite sets, but 49th Mersenne prime is the largest presently known prime. Similarly, 1999291987030606810 may well be the largest known non-composite ("originating" or "prime" or "base" or whatever you like most) delayed palindrome.87.236.191.246 (talk) 11:07, 13 September 2017 (UTC)

I think it may be difficult to make this "originating" concept into a real mathematical definition. This "insert 0's" strategy isn't the only way to enlarge a number like this. See http://jasondoucette.com/worldrecords.html, look up "consequence". But anyway it's certainly false that any number is the "largest most delayed". Staecker (talk) 11:36, 13 September 2017 (UTC)

It is not going to be difficult if you define a delayed palindrome as a number that comes to 2D (or 2-fold) symmetry via Reverse-And-Add process. Then 89 is going to be a delayed palindome, whereas 8900..00..0089 will obvioisly have 4D (or 4-fold) symmetry - the clear sign that it is derived from a lower-level order. By adding more zeroes with more delayed palindromes between them one could potentially get symmetries of much higher magnitude than 4D, but all of them will require the initial representations by base palindromes - the numbers that can not be constructed by duplication and zeros insertion.87.236.191.246 (talk) 18:55, 13 September 2017 (UTC)

How do you define the order of the symmetry? (like 2-fold vs 4-fold) Staecker (talk) 11:09, 14 September 2017 (UTC)

89 gives you a 8813200023188 which has 2 symmetrical parts, whereas 890000000000089 gives you 88132000231888813200023188 that gives you 8813200023188 concatenated with the same 8813200023188 or 4 parts if you divide it in the middle between 8888 to get 2 palindromes with 2 symmetrical parts. Now imagine that you constructed smth. like 8900...00...0089..00...0089...00..0089 with appropriate number of 0 between four 89. Then, believe it or not, you get 8813200023188881320002318888132000231888813200023188 which will have 8 symmetrical parts (imagine that you have this number written on a piece of paper that you then fold in 8 symmetrical parts). This does not eliminate the need for 89 - the base delayed palindrome. BTW, this looks a little bit like the modulus math - you reduce everything to the base delayed palindromes and get all others through concatination and zeros insertion. — Preceding unsigned comment added by 87.236.191.246 (talk) 15:09, 14 September 2017 (UTC)

I think we've strayed away from discussion pertaining to improvements to this wikipedia article. I think we both agree that there is no "largest most delayed", which is what this was all about. All mentions of "largest most delayed" have been removed from the article, so everything's good as far as I can see.

In case you really want to continue investigating this: I'm still not convinced this "symmetrical parts" business can be made into a real useful definition. I think it's possible that a "originating" number (not formed by digit insertion from some other number) could still resolve into a palindrome with extra symmetry. But I don't really need to continue this here...

Staecker (talk) 16:37, 14 September 2017 (UTC)

It depends on your definitions and whether you accept that 8900..00...0089 is derived from 89 - the base delayed palindrome. Jason Doucette in his "consequences" discussion counts 96 delayed palindromes with the same properties as 140,669,390. Obviously, in this set 140,669,390 will be the first and the lowest number whereas 193,966,040 will be the largest one. I guess that if you look at 261 step delayed palindromes you will find that this is a limited set with 108864 numbers that starts with 1186060307891929990 and ends with 1999291987030606810. All your "combinational" 261 step delayed palindromes come from one of those 108864 ones and nobody can produce at present anything outside of this set. There are no other easy way to obtain them and no other sets that form 261 (or higher) step palindromes are known. Hopefully, as computers become more powerful people will find the number beyond 1999291987030606810 with the same or higher delay. — Preceding unsigned comment added by 87.236.191.246 (talk) 19:00, 14 September 2017 (UTC)

list of different bases

Has anyone noticed the pattern in 'smallest possible lychrel numbers in different bases' ? — Preceding unsigned comment added by 101.98.178.115 (talk) 21:01, 22 December 2018 (UTC)

Most Delayed Palindrome in Other Bases

When I saw the most delayed palindrome was a 23-digit number taking 288 steps, I was interested in doing the same thing with binary numbers.
The smallest binary Lychrel number is 10110 and the first binary number that is not known to be Lychrel is 1000010100110. After checking all binary numbers up to 16 bits, the member of the longest sequence is 100110111111010, taking 89 iterations. After checking all 24-bit numbers, the longest delay is 273 (the number 110000100111001100010). After checking all 32-bit numbers, the longest delay is 297 (the number 100001000000101000110101110, 27 bits). When I got to 33 bits, the longest delayed palindrome has blasted up to 515 iterations (it's the number 100001001000000001011010101100110, which is the smallest binary number that solves in over 500 iterations). Currently I'm at 35 bits and the longest sequence found was 10000111000010000111111111111100010 (573 iterations).

It is known that there exists at least one binary number (in fact, infinitely many of them) that solves in x iterations for all positive integers x. The binary number 11011000...000101110001 with 9n+2 zeros in the 000...000 resolves into a binary palindrome in 6n+4 iterations. But the job is to find the smallest such binary number solving in x iterations.

71.179.2.145 (talk) 23:15, 10 November 2019 (UTC)

Proven vs conjectured Lychrel numbers in various bases

"Lychrel numbers have been proven to exist in the following bases: 11, 17, 20, 26 and all powers of 2."

However, except for the binary number 10110, no comment is made on what is the smallest proven Lychrel number in any base. I see that the Mathpages reference gives a construction method for Lychrel numbers in bases that are powers of 2. But this doesn't mean nobody has proven the existence of smaller Lychrel numbers than those constructed in this method. Something else that isn't clear is whether, in some of the mentioned bases, it is merely proven that Lychrel numbers exist and no explicit examples are known.

Essentially, for each base there are five possibilities:

1. the smallest Lychrel number is known and proven
2. an explicit example of a Lychrel number is known, but it is unknown what the smallest Lychrel number is
3. Lychrel numbers are proven to exist, but no explicit example is known
4. it is unknown whether Lychrel numbers exist
5. is it proven that Lychrel numbers don't exist

It seems there are examples of cases 1, 2 and 4, there may or may not be examples of case 3, but there are no examples of case 5 as it stands at the moment. It would be useful if we could find information on which is the case for each base, and for case 2 find out the smallest proven Lychrel number, and present this info in the table. — Smjg (talk) 16:13, 6 December 2020 (UTC)

Can a palindrome be a Lychrel number?

We seem to have a contradiction here. The definition of 'seed' is essentially saying a palindrome can be a Lychrel number, and this cited source takes this view (though I realise it doesn't actually use the term). However, the table gives 10202 as the smallest candidate Lychrel number in base 4, apparently taking the view that 3333 isn't a Lychrel number because it's already a palindrome.

But P196 - which seems to be Wade's own website (though he is referred to in the third person in at least one place) - likewise contains contradictory information. On the FAQ page, for a few bases it states that the representation of 196 is already a palindrome, and treats it as being not a Lychrel number on this basis. However, according to the definitions page, a palindrome can be a Lychrel number. If we take this view, the table is wrong in saying that 10202 is the smallest possible Lychrel number in base 4.

What do people think we should do? I think the article needs to address this ambiguity, at the very least. — Smjg (talk) 16:37, 6 December 2020 (UTC)