Penultimate Lucas-Lehmer step - Page 4

kriesel · 2018-08-06, 04:02

For s0=4, is M50 + or - in the p-3 iteration? (Tiebreaker...)

ATH · 2018-08-06, 13:21

It is a +. I only posted it back in the M50 discussion thread and forgot to update this one:

Code:

		     S[0]=4  S[0]=10 S[0]=2/3
M2:  p=3		+	-	-
M3:  p=5		+	-	-
M4:  p=7		-	-	-
M5:  p=13		+	+	+
M6:  p=17		-	+	+
M7:  p=19		-	+	-
M8:  p=31		+	+	-
M9:  p=61		+	+	+
M10: p=89		-	+	+
M11: p=107		-	+	-
M12: p=127		+	+	-
M13: p=521		-	+	+
M14: p=607		-	-	-
M15: p=1279		-	-	-
M16: p=2203		+	-	-
M17: p=2281		-	+	+
M18: p=3217		-	+	+
M19: p=4253		+	+	+
M20: p=4423		-	-	-
M21: p=9689		-	+	+
M22: p=9941		+	+	+
M23: p=11213		-	-	+
M24: p=19937		+	-	+
M25: p=21701		-	-	+
M26: p=23209		+	-	+
M27: p=44497		-	+	+
M28: p=86243		+	-	-
M29: p=110503		+	+	-
M30: p=132049		+	-	+
M31: p=216091		-	+	-
M32: p=756839		+	+	-
M33: p=859433		-	+	+
M34: p=1257787		-	+	-
M35: p=1398269		+	+	+
M36: p=2976221		+	+	+
M37: p=3021377		+	-	+	
M38: p=6972593		+	-	+
M39: p=13466917		+	+	+	
M40: p=20996011		+	-	-	
M41: p=24036583		-	-	-
M42: p=25964951		+	+	-
M43: p=30402457		-	+	+
M44: p=32582657		-	+	+
M45: p=37156667		+	-	-
M46: p=42643801		-	+	+
M47: p=43112609		+	-	+
M48: p=57885161		-	+	+
M49: p=74207281		-	+	+
M50: p=77232917		+	+	+

		       25+ 	30+	29+
		       24-	19-	20-
		-----------------------------
		       49	49	49

penultimateresultsS0=4.txt

kriesel · 2018-08-06, 14:21

Quote:

Originally Posted by ATH

It is a +. I only posted it back in the M50 discussion thread and forgot to update this one:

...
M50: p=77232917 + + +

25+ 30+ 29+
24- 19- 20-
-----------------------------
49 49 49[/CODE]penultimateresultsS0=4.txt

Thanks for the update and quick response.

Note http://www.hoegge.dk/mersenne/penult...esultsS0=4.txt does not include M50 yet.

Attached is a chart of the ratio of +/- versus maximum Mp.

ATH · 2018-08-06, 16:48

Quote:

Originally Posted by kriesel;493279Note [URL

http://www.hoegge.dk/mersenne/penultimateresultsS0=4.txt[/URL] does not include M50 yet.

It does contain M50, try refreshing the page. I uploaded a new version just before I posted 3.5 hours ago.

kriesel · 2018-08-06, 21:19

Quote:

Originally Posted by ATH

It does contain M50, try refreshing the page. I uploaded a new version just before I posted 3.5 hours ago.

Ack! Confirmed. (Never trust a browser to do the right thing. SHIFT-refresh page)

ATH · 2018-12-09, 20:55

The penultimate iteration is a minus this time for S0=4 and S0=10 and a plus for S0=2/3, which leads to a very symmetrical result: 25/25, 30/20, 30/20

Code:

		     S[0]=4  S[0]=10 S[0]=2/3
M2:  p=3		+	-	-
M3:  p=5		+	-	-
M4:  p=7		-	-	-
M5:  p=13		+	+	+
M6:  p=17		-	+	+
M7:  p=19		-	+	-
M8:  p=31		+	+	-
M9:  p=61		+	+	+
M10: p=89		-	+	+
M11: p=107		-	+	-
M12: p=127		+	+	-
M13: p=521		-	+	+
M14: p=607		-	-	-
M15: p=1279		-	-	-
M16: p=2203		+	-	-
M17: p=2281		-	+	+
M18: p=3217		-	+	+
M19: p=4253		+	+	+
M20: p=4423		-	-	-
M21: p=9689		-	+	+
M22: p=9941		+	+	+
M23: p=11213		-	-	+
M24: p=19937		+	-	+
M25: p=21701		-	-	+
M26: p=23209		+	-	+
M27: p=44497		-	+	+
M28: p=86243		+	-	-
M29: p=110503		+	+	-
M30: p=132049		+	-	+
M31: p=216091		-	+	-
M32: p=756839		+	+	-
M33: p=859433		-	+	+
M34: p=1257787		-	+	-
M35: p=1398269		+	+	+
M36: p=2976221		+	+	+
M37: p=3021377		+	-	+	
M38: p=6972593		+	-	+
M39: p=13466917		+	+	+	
M40: p=20996011		+	-	-	
M41: p=24036583		-	-	-
M42: p=25964951		+	+	-
M43: p=30402457		-	+	+
M44: p=32582657		-	+	+
M45: p=37156667		+	-	-
M46: p=42643801		-	+	+
M47: p=43112609		+	-	+
M48: p=57885161		-	+	+
M49: p=74207281		-	+	+
M50: p=77232917		+	+	+
M51: p=82589933		-	-	+

		       25+ 	30+	30+
		       25-	20-	20-
		-----------------------------
		       50	50	50

Batalov · 2018-12-09, 21:04

Well, we know e(2/3) = 1 because p₅₁ = 1 (mod 4), as announced by George.
And now we know that the p₅₁ = 5 (mod 8) - because see post #31

M29 · 2018-12-15, 05:07

I think the table deserves to be included in the Mersenne Prime Wikipedia article.

GP2 · 2018-12-15, 11:18

Quote:

Originally Posted by M29

I think the table deserves to be included in the Mersenne Prime Wikipedia article.

It might be a little too specialized for that article.

However, there is a separate Lucas-Lehmer primality test article, where it would fit well.

That article would also need a discussion of alternate starting values s₀. I read once that there is actually a range of possibilities other than 4, 10, and 2/3.

science_man_88 · 2018-12-15, 13:08

Quote:

Originally Posted by GP2

I read once that there is actually a range of possibilities other than 4, 10, and 2/3.

A018844 normal test
A002812 alternate form of test

GP2 · 2018-12-19, 18:23

Quote:

Originally Posted by M29

I think the table deserves to be included in the Mersenne Prime Wikipedia article.

Quote:

Originally Posted by GP2

It might be a little too specialized for that article.

However, there is a separate Lucas-Lehmer primality test article, where it would fit well.

Actually, the table isn't strictly speaking necessary, since the values for s0=4 are given by OEIS sequence A123271, and then values for s0=10 can also be derived from them, and the values for s0=2/3 as well are also easily determined, as per Batalov's post.

2018-08-06, 04:02	#34
kriesel "TF79LL86GIMPS96gpu17" Mar 2017 US midwest 31·173 Posts	M50? For s0=4, is M50 + or - in the p-3 iteration? (Tiebreaker...)

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2018-12-09, 21:04	#40
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3⁶·13 Posts	Well, we know e(2/3) = 1 because p₅₁ = 1 (mod 4), as announced by George. And now we know that the p₅₁ = 5 (mod 8) - because see post #31

2018-12-15, 05:07	#41
M29 Dec 2003 164₁₀ Posts	I think the table deserves to be included in the Mersenne Prime Wikipedia article.