Penultimate Lucas-Lehmer step

[ewmayer]

Originally Posted by maxal

Thanks, ATH.
Please submit more terms to http://www.research.att.com/~njas/sequences/A123271

Max, since you apparently have edit privileges to the above page (I assume it was you who added the link to this thread), why don't you just add ATH's newly-reported terms yourself?

[m_f_h]

Originally Posted by ATH

I'm not sure how to submit terms, can you do it? The missing terms are:
-1, -1, 1, 1, 1, 1, 1, 1, -1, 1, -1, -1

You have to open the link "Contribute new seq. or comment" (1st link on 2nd line at bottom), at best in a new window, and then fill in the form (in fact most of the work consists in emptying the form....)
(there are no "editing privileges" - submitting the form sends an email to N.Sloane who will incorporate the specially formatted data within some days, and mention the contributor & date for any new data, comment etc.).

Btw it doesn't say that the sequence starts for n=2 or is that what "OFFSET 2,1" means?

Indeed.

[m_f_h]

Can't we exploit the fact that D3=X(X^2-3)
successively gives all elements of the tree at given height,
and commutes with LL=X^2-2 ?

[ATH]

M37156667 interim Wd1 residue 833185DE0326B429 at iteration 37156663
M37156667 interim Wd1 residue 21F10ABD65A37DF4 at iteration 37156664
M37156667 interim Wd1 residue 36E916675C4F0334 at iteration 37156665
M37156667 interim Wd1 residue 0000000000000000 at iteration 37156666
M37156667 interim Wd1 residue 0000000000000000 at iteration 37156667
M37156667 is prime! Wd1: EA38283B,00000000

M43112609 interim Wd4 residue F042423AF370FBF0 at iteration 43112605
M43112609 interim Wd4 residue C395044781A6672D at iteration 43112606
M43112609 interim Wd4 residue 138F82F361A9E5B9 at iteration 43112607
M43112609 interim Wd4 residue 0000000000000000 at iteration 43112608
M43112609 interim Wd4 residue 0000000000000000 at iteration 43112609
M43112609 is prime! Wd4: 5B377150,00000000

So:
M45: p=37156667 s[p-3]== 2^((p+1)/2) (mod Mp).
M46: p=43112609 s[p-3]== 2^((p+1)/2) (mod Mp).

Now we have 24+ and 21-.

penultimate.txt

[jinydu]

Isn't the next-to-last residue of a Mersenne prime never supposed to be 0? Because then the final residue would be 0^2 -2 = -2. Or am I forgetting something?

[jrk]

Originally Posted by jinydu

Isn't the next-to-last residue of a Mersenne prime never supposed to be 0? Because then the final residue would be 0^2 -2 = -2. Or am I forgetting something?

The next-to-last residue is not zero. It only looks that way in some of the results.

If the next-to-last residue is positive, it is of the form:

2^((p+1)/2)

In binary, a power of two has only a single '1' bit and the rest are zeroes.

When p is bigger than or equal to 127, then the '1' bit in the p-1 residue will be outside the 64-bit residue, making the 64-bit residue have all-zeroes, but the full residue is not zero.

Look at the p-1 residue of M61: 0000000080000000 or 2^31.

[ATH]

Originally Posted by jinydu

Isn't the next-to-last residue of a Mersenne prime never supposed to be 0? Because then the final residue would be 0^2 -2 = -2. Or am I forgetting something?

Yeah its either -2^(p+1)/2 or +2^(p+1)/2. If its + in binary it will be 10000...millions of zeros..00000 and the 64bit residue will be 0x0000000000000000. If its - the residue will be 2^^p-1 - 2^(p+1)/2, which in binary is 1111...millions of 1's...111101111...millions of 1's...11111, so the 64 bit residue will in hex be 0xFFFFFFFFFFFFFFFF.

[ATH]

M42643801 interim Wd4 residue 56D4857CEECAE826 at iteration 42643797
M42643801 interim Wd4 residue 1FE903F0A3186A6F at iteration 42643798
M42643801 interim Wd4 residue DAACA15663F030C8 at iteration 42643799
M42643801 interim Wd4 residue FFFFFFFFFFFFFFFF at iteration 42643800
M42643801 interim Wd4 residue 0000000000000000 at iteration 42643801
M42643801 is prime! Wd4: B7E878A9,00000000

M46: p=42643801 s[p-3]== - 2^((p+1)/2) (mod Mp).

Now there is 24+ and 22- :

penultimate.txt
resultspenultimate.txt

[ewmayer]

Thanks for the table addition, ATH.

Now, anyone want to trot out a curious-though-likely-spurious statistical correlation, perhaps with the phase of the moon or one of the major stock market indices at time of discovery?

[henryzz]

Originally Posted by ATH

Now there is 24+ and 22- :

hopefully that means that we have missed two with - below the highest prime

[plandon]

Originally Posted by henryzz

hopefully that means that we have missed two with - below the highest prime

The sequence above is for the discovered primes.

Maybe it is best just to submit the entries up to the completely checked range and add the rest as a comment to Sloanes. Otherwise A123271 will have to be corrected if another prime is discovered out of sequence.