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Great notes from Pierre Laurent.


Lychrel research is not useless as stated in your site


"Does this have any use? Absolutely none. If you have a use, let us know. Otherwise, it doesn't really have any application. "

Consider that Lychrel "reverse and add" can be written as:

upper half of a number += reverse (lower half of a number)
lower half of a number += reverse (upper half of a number)
rewrite it as A, B and a transform T()

A -> A + T(B)
B -> B + T(A)
In Lychrel case, T() is the base-10 polynomial indices reverse, and the quest stops when A equals T(B).

Lychrel operations are part of a more general problem.

Consider a point (A,B) and 2 transforms S() and T()

Point (non cummutative) addition is
(A,B) + (C,D) -> (A + S(D), B + T(C))

Point doubling is
(A,B) -> (A + S(B), B + T(A))

Point inverse is
(A, B) -> (-S(B), -T(A))

(note that there are a few other ways to rewrite these formula, and apply come constraints on 2 points to make the addition being commutative)

You can find the same concepts in elliptic curve stuff, with a great outcome in number theory and cryptography.


(From france, working in Ireland)

There are many other problems similar to the lychrel reverse'n'add. Simalar problems on the networking area.

Imagine two network pipes joining to form a third pipe. When there is some excess of traffic, an overflow

if the pipe throughput is 10 pkts/s

if, during 1 second, you get 3 packsts from pipe 1 and 4 packets from pipe 2, then the output will be 7 packet/s at this time.

if, during 1 second, you get 5 packsts from pipe 1 and 6 packets from pipe 2, then the output will be 10 packet/s plus a delayed packet on the next timeslot, which can propagate some delays for the packets in the next-next time slot again... and so on..... like a carry propagation.

At that point, the Legendre transform may apply (typical queuing problem), I didn't find exactly how (yet) it could help for Lychrel quest.

Networking problems are very hot topics these times. No big progress made in the last 5 years.