Something curious, remarkable, abnormal,...

kar_bon · 2008-12-17, 10:55

Something curious, remarkable, abnormal,...

Here is the place for things that looks curious about Riesel-Primes or something remarkable not shown anywhere.

Perhaps you found a rule to calculate primes or a special k-value generating most primes, Sophie-Germain or twins (will post this the next days).

Some information spotted from the RieselPrime Database but not seen anywhere before so here is the place to post.

kar_bon · 2008-12-17, 12:16

Different k-values with similiar n-values

The first note here is about the appearance of primes with high n (same n or near by) and different k-values.

I've compared all k<2000 with all (so far shown) primes in the Database (from 2008-12-13).

I post here the k-n-pairs with a difference in the n-value from 0 to 9 for n>300k and some differences >10 for n>500k.

Code:


Diff=0
 193-439107  945-349086  235-315773
1389-439107 1425-349086  309-315773

Diff=1
 843-578747    9-503893  395-464002  169-458099   735-390212   875-301010
 675-578746  923-503892  865-464001  975-458098  1321-390211  1395-301009

Diff=2
 591-530223  381-469371  639-343305  831-314002  1299-313352  395-307796   885-305689
 565-530221  379-469369  489-343303  167-314000   585-313350  941-307794  1785-305687

Diff=3
1313-398514  677-357334  155-310622
 381-398511  973-357331  129-310619

Diff=4
1559-491984   159-442189   355-433303   475-402945  1515-391554  1285-386579   755-382168   999-374459   501-371430   241-336197
 563-491980  1587-442185  1791-433299  1587-402941  1253-391550  1471-386575  1337-382164   531-374455   161-371426   769-336193

Diff=5
277-490805   605-393234   363-383088   391-362657   489-307544
179-490800  1101-393229   153-383083  1299-362652  1143-307539

Diff=6
269-628904   377-463086  1051-425263  1099-408371   123-389052   89-369628
281-628898  1245-463080   259-425257   459-408365  1197-389046  513-369622

Diff=7
 75-814857  1343-402952   395-379448   987-358020   685-340547    63-340463   741-317479
105-814850   475-402945* 1371-379441   715-358013   579-340540   909-340456  1377-317472

Diff=8
 795-322014  187-645401  641-496430  1511-459846  1339-404859  1253-391550  181-334579  195-311215
1025-322006  861-645393   23-496422   987-459838  1461-404851  1427-391542  711-334571  661-311207
1091-321998

Diff=9
1741-392029  121-334257   639-320616   681-316819  423-316412
 989-392020  525-334248  1393-320607  1115-316810  259-316403


Higher differences:
diff pairs
 19  465-668701   177-668682
 17  229-586795  1415-586778  /  207-507833  179-507816
 15   25-587585   161-587570
 14   95-968636   143-968622
 13   19-645555    45-645542
 10  199-531357   291-531347


Smallest difference for any pair n>1.0M:
diff=492 1-1257787 49-1257295

So the last found prime 675*2⁵⁷⁸⁷⁴⁶-1 is quite remarkable: such high n-value and a difference of 1 (843*2⁵⁷⁸⁷⁴⁷-1).

Others mentionable:
- the 3er-pair chain (diff 8-8): 1091-321998 -> 1025-322006 -> 795-322014
- the 3er-pair chain (diff 4-7): 1587-402941 -> 475-402945 -> 1343-402952

If more data available, I will update this post then.

kar_bon · 2008-12-17, 12:39

k-values with most primes listed

The k-value for k*2ⁿ-1 with the most primes listed in the Database so far is:

k=37850187375 (Nash-weight=7261)

158 primes found upto n=515k; the 100th prime is at n=13719; tested by T.Ritschel.

kar_bon · 2008-12-17, 12:41

k-values with most twins listed

The k-value for k*2ⁿ-1 with the most twins listed in the Database so far is:

k=7985650262654529465 (Nash-weight=7081)

13 twins found upto n=10k tested by R.Smith.

kar_bon · 2008-12-17, 12:44

k-values with most Sophie-Germain pairs listed

The k-value for k*2ⁿ-1 with the most Sophie-Germain pairs listed in the Database so far is:

k=2037910875 (Nash-weight=6013)

9 SG's found upto n=50k tested by L.Soule.

kar_bon · 2011-11-17, 08:43

To bring this thread on top (the last one minutes ago :-) and according to the lastest TPS Debut Rally: November 15-16 and the related Operation Megabit Twin, I've done some work on these:

Collecting some series of first primes of the form k*2^n+/-1 with special exponents.
For now I got these:

Exponents are of the form '1234567...' or ''10987654321...:

Code:

n	Riesel		Proth		n	Riesel		Proth
1	3		1		1	3		1
12	5		3		21	7		11
123	123		101		321	91		51
1234	1451		363		4321	2617		1913
12345	7305		3239		54321	42105 $		38621 $
123456	122577 +	13303 +		654321	125115 ?	?
1234567	?		?		7654321	?		?

Exponents are of the form Pi: '314159265358979...' or e: '271828182...':

Code:

n	Riesel		Proth		n	Riesel		Proth
3	1		5		2	1		1
31	1		35		27	21		15
314	203		63		271	21		65
3141	1771		2201		2718	2271		1257
31415	5905		14451 +		27182	19787 +		3927
314159	234531 +	160707 $	271828	?		147103 $
3141592	?		?		2718281	?		?

Exponent is of the form 10^n:

Code:

n		Riesel		Proth
1		3		1
10		5		13
100		77		165
1000		915		13
10000		42449		6657
100000		62967 #		182271 +
1000000		?		?

So for example, the first k-value where k*2^123456-1 is prime is k=122577.

The recently found primes for n=1000000 (lowest k-value is 20667679305) can only be an upper range of such search, because those n-values were only sieved for twins, but there could exist (I think so) a lower k-value.

Perhaps someone will extend those tables (send me results and searched ranges, please) or suggest other special n-values.

Exponents are of the form '1111111...' or '2222222...':

Code:

n	Riesel		Proth		n	Riesel		Proth
1	3		1		2	1		1
11	3		9		22	33		25
111	75		95		222	93		75
1111	1275		849		2222	6063		2817
11111	7539		1065		22222	61817 +		6325
111111	35839 $		7521		222222	134853 ?	374157 $
1111111	5834751 ?	487419 ?	2222222	?		?

Exponents are of the form '3333333...' or '4444444...':

Code:

n	Riesel		Proth		n	Riesel		Proth
3	1		5		4	3		1
33	31		9		44	35		15
333	231		45		444	1029		261
3333	5911		1685		4444	567		2833
33333	48061 +		15561 #		44444	9495 +		21813 $
333333	49371 ?		123173 ?	444444	1947629 ?	160935 ?
3333333	?		?		4444444	?		?

Exponents are of the form '5555555...' or '6666666...':

Code:

n	Riesel		Proth		n	Riesel		Proth
5	1		3		6	3		3
55	3		5		66	63		3
555	625		141		666	395		375
5555	6163		7185		6666	7173		6399
55555	4189		9921		66666	59057 $		19497 #
555555	135345 ?	180701 ?	666666	192201 ?	?
5555555	?		?		6666666	?		?

Exponents are of the form '7777777...' or '8888888...':

Code:

n	Riesel		Proth		n	Riesel	Proth
7	1		5		8	5		1
77	55		29		88	129		61
777	1089		299		888	23		147
7777	21		2421		8888	3675		2193
77777	154311 $	8289		88888	44163 $		21411 $
777777	1155681 ?	?		888888	7195377 ?	?
7777777	?		?		8888888	?		?

Exponent is of the form '9999999...' :

Code:

n	Riesel	Proth
9	7	15
99	9	219
999	1581	347
9999	3421	14999 +
99999	4641 $	59271 $
999999	?	?
9999999	?	?

Remarks:
"+": found by me
"#": verified by me (all others were already in my Database)
"$": done by others

further:
R 54321: next is k=55519
P 54321: next are k=53061, 74355, 75105

2011-11-21: added 14999*2^9999+1 (kar_bon)
2011-11-21: added 61817*2^22222-1 (kar_bon)
2011-11-22: added 9495*2^44444-1 (kar_bon)
2011-11-22: verfied 15561*2^33333+1 (kar_bon)
2011-11-22: added 4641*2^99999-1 (axn)
2011-11-22: added 59057*2^66666-1 (axn)
2011-11-22: added 42105*2^54321-1 (axn)
2011-11-22: added 44163*2^88888-1 (axn)
2011-11-23: added 38621*2^54321+1 (axn)
2011-11-23: added 21813*2^44444+1 (axn)
2011-11-23: verified 35839*2^111111-1 (axn)
2011-11-23: verified 19497*2^66666+1 (kar_bon)
2011-11-23: added 154311*2^77777-1 (axn)
2011-11-24: verified 59271*2^99999+1 (axn)
2011-11-24: verified 21411*2^88888+1 (axn)
2011-11-24: added 160707*2^314159+1 (axn)
2011-12-10: added 147103*2^271828+1 (axn)
2011-12-24: added 374157*2^222222+1 (axn)
2012-08-27: added 1111111-, 1111111+

Reservations:
none

SaneMur · 2011-12-22, 15:36

Quote:

Originally Posted by kar_bon

k-values with most primes listed

The k-value for k*2ⁿ-1 with the most primes listed in the Database so far is:

k=37850187375 (Nash-weight=7261)

158 primes found upto n=515k; the 100th prime is at n=13719; tested by T.Ritschel.

I have 163 so far, still calculating.

axn · 2011-12-24, 02:51

374157*2^222222+1 is prime!

2008-12-17, 10:55	#1
kar_bon Mar 2006 Germany 2·11·131 Posts	Something curious, remarkable, abnormal,... Something curious, remarkable, abnormal,... Here is the place for things that looks curious about Riesel-Primes or something remarkable not shown anywhere. Perhaps you found a rule to calculate primes or a special k-value generating most primes, Sophie-Germain or twins (will post this the next days). Some information spotted from the RieselPrime Database but not seen anywhere before so here is the place to post. Last fiddled with by kar_bon on 2008-12-17 at 12:28

2008-12-17, 12:44	#5
kar_bon Mar 2006 Germany 2·11·131 Posts	k-values with most Sophie-Germain pairs listed The k-value for k2ⁿ-1 with the most Sophie-Germain pairs listed in the Database so far is: k=2037910875 (Nash-weight=6013) 9 SG's found upto n=50k tested by L.Soule. Last fiddled with by kar_bon on 2009-08-31 at 16:12*

2011-11-17, 08:43	#6
kar_bon Mar 2006 Germany 2×11×131 Posts	To bring this thread on top (the last one minutes ago :-) and according to the lastest TPS Debut Rally: November 15-16 and the related Operation Megabit Twin, I've done some work on these: Collecting some series of *first primes of the form k2^n+/-1 with special exponents. For now I got these: Exponents are of the form '1234567...' or ''10987654321...: Code: n Riesel Proth n Riesel Proth 1 3 1 1 3 1 12 5 3 21 7 11 123 123 101 321 91 51 1234 1451 363 4321 2617 1913 12345 7305 3239 54321 42105 $ 38621 $ 123456 122577 + 13303 + 654321 125115 ? ? 1234567 ? ? 7654321 ? ? Exponents are of the form Pi: '314159265358979...' or e: '271828182...': Code: n Riesel Proth n Riesel Proth 3 1 5 2 1 1 31 1 35 27 21 15 314 203 63 271 21 65 3141 1771 2201 2718 2271 1257 31415 5905 14451 + 27182 19787 + 3927 314159 234531 + 160707 $ 271828 ? 147103 $ 3141592 ? ? 2718281 ? ? Exponent is of the form 10^n:** Code: n Riesel Proth 1 3 1 10 5 13 100 77 165 1000 915 13 10000 42449 6657 100000 62967 # 182271 + 1000000 ? ? So for example, the first k-value where k2^123456-1 is prime is k=122577. The recently found primes for n=1000000 (lowest k-value is 20667679305) can only be an upper range of such search, because those n-values were only sieved for twins, but there could exist (I think so) a lower k-value. Perhaps someone will extend those tables (send me results and searched ranges, please) or suggest other special n-values. Exponents are of the form '1111111...' or '2222222...':* Code: n Riesel Proth n Riesel Proth 1 3 1 2 1 1 11 3 9 22 33 25 111 75 95 222 93 75 1111 1275 849 2222 6063 2817 11111 7539 1065 22222 61817 + 6325 111111 35839 $ 7521 222222 134853 ? 374157 $ 1111111 5834751 ? 487419 ? 2222222 ? ? Exponents are of the form '3333333...' or '4444444...': Code: n Riesel Proth n Riesel Proth 3 1 5 4 3 1 33 31 9 44 35 15 333 231 45 444 1029 261 3333 5911 1685 4444 567 2833 33333 48061 + 15561 # 44444 9495 + 21813 $ 333333 49371 ? 123173 ? 444444 1947629 ? 160935 ? 3333333 ? ? 4444444 ? ? Exponents are of the form '5555555...' or '6666666...': Code: n Riesel Proth n Riesel Proth 5 1 3 6 3 3 55 3 5 66 63 3 555 625 141 666 395 375 5555 6163 7185 6666 7173 6399 55555 4189 9921 66666 59057 $ 19497 # 555555 135345 ? 180701 ? 666666 192201 ? ? 5555555 ? ? 6666666 ? ? Exponents are of the form '7777777...' or '8888888...': Code: n Riesel Proth n Riesel Proth 7 1 5 8 5 1 77 55 29 88 129 61 777 1089 299 888 23 147 7777 21 2421 8888 3675 2193 77777 154311 $ 8289 88888 44163 $ 21411 $ 777777 1155681 ? ? 888888 7195377 ? ? 7777777 ? ? 8888888 ? ? Exponent is of the form '9999999...' : Code: n Riesel Proth 9 7 15 99 9 219 999 1581 347 9999 3421 14999 + 99999 4641 $ 59271 $ 999999 ? ? 9999999 ? ? Remarks: "+": found by me "#": verified by me (all others were already in my Database) "$": done by others further: R 54321: next is k=55519 P 54321: next are k=53061, 74355, 75105 2011-11-21: added 149992^9999+1 (kar_bon) 2011-11-21: added 618172^22222-1 (kar_bon) 2011-11-22: added 94952^44444-1 (kar_bon) 2011-11-22: verfied 155612^33333+1 (kar_bon) 2011-11-22: added 46412^99999-1 (axn) 2011-11-22: added 590572^66666-1 (axn) 2011-11-22: added 421052^54321-1 (axn) 2011-11-22: added 441632^88888-1 (axn) 2011-11-23: added 386212^54321+1 (axn) 2011-11-23: added 218132^44444+1 (axn) 2011-11-23: verified 358392^111111-1 (axn) 2011-11-23: verified 194972^66666+1 (kar_bon) 2011-11-23: added 1543112^77777-1 (axn) 2011-11-24: verified 592712^99999+1 (axn) 2011-11-24: verified 214112^88888+1 (axn) 2011-11-24: added 1607072^314159+1 (axn) 2011-12-10: added 1471032^271828+1 (axn) 2011-12-24: added 3741572^222222+1 (axn) 2012-08-27: added 1111111-, 1111111+ Reservations: none Last fiddled with by kar_bon on 2012-08-27 at 16:02 Reason: added some more

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2008-12-17, 12:39	#3
kar_bon Mar 2006 Germany 101101000010₂ Posts	k-values with most primes listed The k-value for k*2ⁿ-1 with the most primes listed in the Database so far is: k=37850187375 (Nash-weight=7261) 158 primes found upto n=515k; the 100th prime is at n=13719; tested by T.Ritschel.

2008-12-17, 12:41	#4
kar_bon Mar 2006 Germany 2·11·131 Posts	k-values with most twins listed The k-value for k*2ⁿ-1 with the most twins listed in the Database so far is: k=7985650262654529465 (Nash-weight=7081) 13 twins found upto n=10k tested by R.Smith.

2011-12-24, 02:51	#8
axn Jun 2003 2·2,459 Posts	374157*2^222222+1 is prime!