# Further theorising

A project log for Pisano-carry Checksum algorithm

Add X to Y and Y to X, says the song. And carry on.

Yann Guidon / YGDES 05/14/2021 at 21:230 Comments

PS: this log has bogus data so forget about it... but 6 looks interesting anyway because it has 6 orbits of equal lengths.

-o-O-0-O-o-

More than two weeks of computations for w=26 and w=27 have given some time to think on the larger scale and a better look at the available data. But it's getting reeeeeaaaaallllllllyyyyyyy long now and I'm mostly wishing that the w=26 and/or w=27 fail soon. At only 5%, 27 can still fail but so far 26 looks promising.

For now we have this list of potentially valid sizes :

`1 2 3 4 6 10 16 (26?)   -> 6 is false...`

It already struck me that 6 + 10 = 16, and now 10 + 16 = 26. Going backwards, 10 = 6 + 4 and 6 = 4 + 2 (hello, kids' maths). Now it's obviously the Fibonaccy sequence, but doubled ! But what happens with 1 and 3 ? Why do they work but not 5 ?

From there, we can postulate some predictions/hypothesis:

• 32 is not a valid size. Too bad ! 16 is valid because it is the double of 8 which is a Fibonaccy number. No other number in the sequence is a power of two, sadly. Finding 16 on the first try was a very lucky draw!
• The next valid sizes would be 42 (of course !), 68, 110 ...

These speculations need to be verified but at least the theory progresses, and it is dearly needed. At this rate, completing w=26 will take two more months (and I don't imagine for w=27). I must find a trick to make it only one month. Furthermore, the longer it takes, the higher the chances of flukes/miscalculations accumulate. They are rare but not impossible at this scale

So even halving the run time is very important and I'm back to the concepts of the last logs, where 2 cores compute roughly one half of the orbit and meet "somewhere" near the middle. And I think I have solved one major issue I had with the previous ideas: there is no need to store all the paths, we only need to compare the last N values. Indeed, the orbit is closed at one point, not potentially random values, and the sequence is run backwards and forward. Now the question is : how many points and which should be kept in the history ?

We can synchronise the 2 threads so they know what the other has "consumed", and discard the elements that have been compared. My favourite sync method is not semaphores or read/write flags, but a pair of mailboxes where one thread write and the other can only read, which simplifies the cache coherency and updates.