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196 AND OTHER LYCHREL NUMBERS

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Welcome to p196.org!

Great notes from Pierre Laurent.
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FIRST EMAIL:
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Hi

Lychrel research is not useless as stated in your site

http://home.cfl.rr.com/p196/faq.html

"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.

Pierre

(From france, working in Ireland)

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SECOND EMAIL:
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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

e.g.
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.

Pierre