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 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, 12:16 #2 kar_bon     Mar 2006 Germany 288210 Posts 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. Last fiddled with by kar_bon on 2009-04-26 at 21:40
 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.
 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 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. 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 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 Last fiddled with by kar_bon on 2012-08-27 at 16:02 Reason: added some more
2011-12-22, 15:36   #7
SaneMur

Jul 2011

1658 Posts

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.

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

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