The dual Sierpinski/Riesel problem

sweety439 · 2017-01-21, 11:01

There is a "Five or bust!" project, to search the primes of the form 2^n+k for all odd numbers k<78557, this is because 2^n+k is the dual of k*2^n+1. For k*2^n-1, the dual of it is |2^n-k|.

The project found the probable primes 2^1518191+75353, 2^2249255+28433, 2^4583176+2131, 2^5146295+41693, and 2^9092392+40291. (see http://mersenneforum.org/showthread.php?t=10761)

There is a "mixed Sierpinski theorem", which is that for all odd numbers k<78557, there is a prime of the form either k*2^n+1 or 2^n+k. At past, there was 3 k's < 78557 such that there is neither known prime of the form k*2^n+1 nor known prime of the form 2^n+k, namely 19249, 28433 and 67607 (for 67607, there was only probable prime known at that time, this probable prime is 2^16389+67607, but this number is now proven prime). Several years ago, a probable prime 2^551542+19249 was found, since it is only a probable prime and not a proven prime, we can not actually say that 19249 can be removed from the list. However, people were looking for prime for the only remaining number, 28433, and found the prime 28433*2^7830457+1, since it is a proven prime, we can remove 28433 from the list. In March 2007, the prime 19249*2^13018586+1 was found, and "the mixed Sierpinski problem" become a theorem as soon as this prime was found. (The smallest odd k such that there is no known prime of the form k*2^n+1 is 21181, and the smallest odd k such that there is no known proven prime of the form 2^n+k is 2131, it has only probable prime known: 2^4583176+2131, all k's < 78557 has a known (probable) prime of the form 2^n+k)

For the mixed Riesel problem, there are still 5 odd k's < 509203 (these k's are 2293, 342847, 344759, 386801, 444637) with neither known prime of the form k*2^n-1 nor known (probable) prime of the form |2^n-k|, the first such k is 2293. (note that 2293 is also the smallest odd k such that there is no known prime of the form k*2^n-1, 2293 is also the smallest odd k such that there is no known (probable) prime of the form |2^n-k|, and all k's < 2293 has a known proven prime of the form |2^n-k|) For k = 363343 and 384539, only probable prime of the form |2^n-k| is known, the probable primes are |2^13957-363343| and |2^32672-384539|.

See http://mersenneforum.org/showthread.php?t=10754 for the dual Sierpinski problem and see http://mersenneforum.org/showthread.php?t=6545 for the dual Riesel problem.

For generalized Sierpinski/Riesel conjecture to base b, the dual of k*b^n+1 is (b^n+k)/gcd(b^n, k), and the dual of k*b^n-1 is (|b^n-k|)/gcd(b^n, k).

These are the dual (probable) primes that I found of the Sierpinski/Riesel conjectures that has only one k remaining for bases b<=144: (See http://mersenneforum.org/showpost.ph...42&postcount=1)

Code:

form             dual form             least prime for the "dual form"    dual n
2036*9^n+1       9^n+2036              9^4+2036                           4
7666*10^n+1     (10^n+7666)/2         (10^67+7666)/2                      67
244*17^n+1       17^n+244              17^838+244                         838
5128*22^n+1     (22^n+5128)/8         (22^11+5128)/8                      11
398*27^n+1       27^n+398              27^7+398                           7
166*43^n+1       43^n+166              ?                                  (>1000)
17*68^n+1       (68^n+17)/17          (68^1+17)/17                        1
1312*75^n+1      75^n+1312             ?                                  (>1000)
8*86^n+1        (86^n+8)/8            (86^205+8)/8                        205
32*87^n+1        87^n+32               ?                                  (>1000)
1696*112^n+1    (112^n+1696)/32       (112^44+1696)/32                    44
48*118^n+1      (118^n+48)/16         (118^57+48)/16                      57
34*122^n+1      (122^n+34)/2          (122^2+34)/2                        2
40*128^n+1      (128^n+40)/8          (128^2+40)/8                        2

Code:

form             dual form             least prime for the "dual form"    dual n
1597*6^n-1      |6^n-1597|            |6^3-1597|                          3
4421*10^n-1     |10^n-4421|           |10^212-4421|                       212
3656*22^n-1    (|22^n-3656|)/8         ?                                  (>1000)
404*23^n-1      |23^n-404|            |23^568-404|                        568
706*27^n-1      |27^n-706|            |27^2-706|                          2
424*93^n-1      |93^n-424|            |93^1-424|                          1
29*94^n-1       |94^n-29|             |94^2-29|                           2
924*103^n-1     |103^n-924|           |103^1-924|                         1
84*109^n-1      |109^n-84|            |109^6-84|                          6
24*123^n-1     (|123^n-24|)/3        (|123^5-24|)/3                       5
926*133^n-1     |133^n-926|           |133^2-926|                         2

sweety439 · 2017-04-21, 16:07

The n's such that the dual forms are (probable) primes:

9^n+2036: 4, 208, ...
(10^n+7666)/2: 67, 103, ...
17^n+244: 838, ...
(22^n+5128)/8: 11, 63, 519, ...
27^n+398: 7, 375, ...
43^n+166: (no known such n)
(68^n+17)/17: 1, 7, ...
75^n+1312: (no known such n)
(86^n+8)/8: 205, ...
87^n+32: (no known such n)
(112^n+1696)/32: 44, ...
(118^n+48)/16: 57, ...
(122^n+34)/2: 2, 98, ...
(128^n+40)/8: 2, 8, ...

|6^n-1597|: 3, 1731, .,.
|10^n-4421|: 212, 284, ...
(|22^n-3656|)/8: (no known such n)
|23^n-404|: 568, ...
|27^n-706|: 2, 10, 786, ...
|93^n-424|: 1, 133, 151, 397, ...
|94^n-29|: 2, ...
|103^n-924|: 1, 97, ...
|109^n-84|: 6, 18, 20, 362, ...
(|123^n-24|)/3: 5, 84, ...
|133^n-926|: 2, 111, 155, ...

sweety439 · 2017-04-21, 16:28

Thus, the "mix Sierpinski/Riesel problem" of these bases are proven except S43, S75, S87 and R22.

sweety439 · 2017-05-20, 15:41

For generalized Sierpinski/Riesel conjecture to base b, the dual of k*b^n+1 is (b^n+k)/d, where d is the largest number that divides b^n+k for all enough large n, and the dual of k*b^n-1 is (|b^n-k|)/d, where d is the largest number that divides |b^n-k| for all enough large n. If this number is not a integer, then we choose its numerator (this number is always a rational number :-) )

sweety439 · 2017-06-01, 14:05

Quote:

Originally Posted by sweety439

The n's such that the dual forms are (probable) primes:

9^n+2036: 4, 208, ...
(10^n+7666)/2: 67, 103, ...
17^n+244: 838, ...
(22^n+5128)/8: 11, 63, 519, ...
27^n+398: 7, 375, ...
43^n+166: (no known such n)
(68^n+17)/17: 1, 7, ...
75^n+1312: (no known such n)
(86^n+8)/8: 205, ...
87^n+32: (no known such n)
(112^n+1696)/32: 44, ...
(118^n+48)/16: 57, ...
(122^n+34)/2: 2, 98, ...
(128^n+40)/8: 2, 8, ...

|6^n-1597|: 3, 1731, .,.
|10^n-4421|: 212, 284, ...
(|22^n-3656|)/8: (no known such n)
|23^n-404|: 568, ...
|27^n-706|: 2, 10, 786, ...
|93^n-424|: 1, 133, 151, 397, ...
|94^n-29|: 2, ...
|103^n-924|: 1, 97, ...
|109^n-84|: 6, 18, 20, 362, ...
(|123^n-24|)/3: 5, 84, ...
|133^n-926|: 2, 111, 155, ...

Written the dual forms with standard forms (i.e. (a*b^n+c)/gcd(a+c,b-1), with integers a, b, c, a>=1, b>=2, gcd(a,c) = 1, gcd(b,c) = 1):

9^n+2036 is already standard
(10^n+7666)/2 ---> 5*10^n+3833 (this n turns to be n-1)
17^n+244 is already standard
(22^n+5128)/8 ---> 1331*22^n+641 (this n turns to be n-3)
27^n+398 is already standard
43^n+166 is already standard
(68^n+17)/17 ---> 4*68^n+1 (this n turns to be n-1)
75^n+1312 is already standard
(86^n+8)/8 ---> 79507*86^n+1 (this n turns to be n-3)
87^n+32 is already standard
(112^n+1696)/32 ---> 392*112^n+53 (this n turns to be n-2)
(118^n+48)/16 ---> 12117361*118^n+3 (this n turns to be n-4)
(122^n+34)/2 ---> 61*122^n+17 (this n turns to be n-1)
(128^n+40)/8 ---> 16*128^n+5 (this n turns to be n-1)

6^n-1597 is already standard
10^n-4421 is already standard
(22^n-3656)/8 ---> 1331*22^n-457 (this n turns to be n-3)
23^n-404 is already standard
27^n-706 is already standard
93^n-424 is already standard
94^n-29 is already standard
103^n-924 is already standard
109^n-84 is already standard
(123^n-24)/3 ---> 41*123^n-8 (this n turns to be n-1)
133^n-926 is already standard

(now, we allow negative primes, such as -5 and -47, thus we can omit the absolute value sign)

sweety439 · 2017-10-11, 17:35

This file is for the dual Riesel problem for k=2 with bases 2<=b<=400.

The formula is (b^n-2)/gcd(b,2)

There are 7 bases remain: 278, 296, 305, 338, 353, 386, 397.

The original primes for these bases are:

Code:

2*278^43908-1
2*296^36-1
2*305^2-1
2*338^12-1
2*353^2-1
2*386^2-1
2*397^18-1

In fact, the "mixed Riesel conjecture for k=2" is now a theorem for bases 2<=b<=1024, since all bases 2<=b<=1024 have either a prime of the form 2*b^n-1 or a (probable) prime of the form (b^n-2)/gcd(b,2)

The dual primes for the remain bases 2<=b<=1024 are:

Code:

581^18-2
(992^90-2)/2
1019^4-2

Also see the OEIS sequence A255707 and the website http://www.primepuzzles.net/puzzles/puzz_887.htm.

sweety439 · 2017-10-13, 13:18

Also update the dual Sierpinski k=2 file, also for bases 2<=b<=400. (in fact, this is "dual extended Sierpinski")

The formula is (b^n+2)/gcd(b,2)/gcd(b-1,3)

Note: b=128 has no possible prime.

Also see the OEIS sequence A138066

sweety439 · 2017-10-13, 13:19

See OEIS sequence A067760 and A252168 for the dual Sierpinski/Riesel base 2 problem.

sweety439 · 2017-10-15, 14:06

These are files for the dual Sierpinski/Riesel problem base 3. (for k<=1024, k even, k not divisible by 3)

i.e. least n>=1 such that 3^n+k or |3^n-k| is prime.

sweety439 · 2017-11-01, 18:48

305^n-2 is composite for all n<=10000.

File attached.

sweety439 · 2017-11-25, 20:35

Quote:

Originally Posted by sweety439

305^n-2 is composite for all n<=10000.

File attached.

Reserve 305^n-2 to n=20000.

2017-01-21, 11:01	#1
sweety439 Nov 2016 2²×3×5×47 Posts	The dual Sierpinski/Riesel problem There is a "Five or bust!" project, to search the primes of the form 2^n+k for all odd numbers k<78557, this is because 2^n+k is the dual of k2^n+1. For k2^n-1, the dual of it is \|2^n-k\|. The project found the probable primes 2^1518191+75353, 2^2249255+28433, 2^4583176+2131, 2^5146295+41693, and 2^9092392+40291. (see http://mersenneforum.org/showthread.php?t=10761) There is a "mixed Sierpinski theorem", which is that for all odd numbers k<78557, there is a prime of the form either k2^n+1 or 2^n+k. At past, there was 3 k's < 78557 such that there is neither known prime of the form k2^n+1 nor known prime of the form 2^n+k, namely 19249, 28433 and 67607 (for 67607, there was only probable prime known at that time, this probable prime is 2^16389+67607, but this number is now proven prime). Several years ago, a probable prime 2^551542+19249 was found, since it is only a probable prime and not a proven prime, we can not actually say that 19249 can be removed from the list. However, people were looking for prime for the only remaining number, 28433, and found the prime 284332^7830457+1, since it is a proven prime, we can remove 28433 from the list. In March 2007, the prime 192492^13018586+1 was found, and "the mixed Sierpinski problem" become a theorem as soon as this prime was found. (The smallest odd k such that there is no known prime of the form k2^n+1 is 21181, and the smallest odd k such that there is no known proven prime* of the form 2^n+k is 2131, it has only probable prime known: 2^4583176+2131, all k's < 78557 has a known (probable) prime of the form 2^n+k) For the mixed Riesel problem, there are still 5 odd k's < 509203 (these k's are 2293, 342847, 344759, 386801, 444637) with neither known prime of the form k2^n-1 nor known (probable) prime of the form \|2^n-k\|, the first such k is 2293. (note that 2293 is also the smallest odd k such that there is no known prime of the form k2^n-1, 2293 is also the smallest odd k such that there is no known (probable) prime of the form \|2^n-k\|, and all k's < 2293 has a known proven prime of the form \|2^n-k\|) For k = 363343 and 384539, only probable prime of the form \|2^n-k\| is known, the probable primes are \|2^13957-363343\| and \|2^32672-384539\|. See http://mersenneforum.org/showthread.php?t=10754 for the dual Sierpinski problem and see http://mersenneforum.org/showthread.php?t=6545 for the dual Riesel problem. For generalized Sierpinski/Riesel conjecture to base b, the dual of kb^n+1 is (b^n+k)/gcd(b^n, k), and the dual of kb^n-1 is (\|b^n-k\|)/gcd(b^n, k). These are the dual (probable) primes that I found of the Sierpinski/Riesel conjectures that has only one k remaining for bases b<=144: (See http://mersenneforum.org/showpost.ph...42&postcount=1) Code: form dual form least prime for the "dual form" dual n 20369^n+1 9^n+2036 9^4+2036 4 766610^n+1 (10^n+7666)/2 (10^67+7666)/2 67 24417^n+1 17^n+244 17^838+244 838 512822^n+1 (22^n+5128)/8 (22^11+5128)/8 11 39827^n+1 27^n+398 27^7+398 7 16643^n+1 43^n+166 ? (>1000) 1768^n+1 (68^n+17)/17 (68^1+17)/17 1 131275^n+1 75^n+1312 ? (>1000) 886^n+1 (86^n+8)/8 (86^205+8)/8 205 3287^n+1 87^n+32 ? (>1000) 1696112^n+1 (112^n+1696)/32 (112^44+1696)/32 44 48118^n+1 (118^n+48)/16 (118^57+48)/16 57 34122^n+1 (122^n+34)/2 (122^2+34)/2 2 40128^n+1 (128^n+40)/8 (128^2+40)/8 2 Code: form dual form least prime for the "dual form" dual n 15976^n-1 \|6^n-1597\| \|6^3-1597\| 3 442110^n-1 \|10^n-4421\| \|10^212-4421\| 212 365622^n-1 (\|22^n-3656\|)/8 ? (>1000) 40423^n-1 \|23^n-404\| \|23^568-404\| 568 70627^n-1 \|27^n-706\| \|27^2-706\| 2 42493^n-1 \|93^n-424\| \|93^1-424\| 1 2994^n-1 \|94^n-29\| \|94^2-29\| 2 924103^n-1 \|103^n-924\| \|103^1-924\| 1 84109^n-1 \|109^n-84\| \|109^6-84\| 6 24123^n-1 (\|123^n-24\|)/3 (\|123^5-24\|)/3 5 926133^n-1 \|133^n-926\| \|133^2-926\| 2 Last fiddled with by sweety439 on 2017-01-21 at 12:18*

2017-05-20, 15:41	#4
sweety439 Nov 2016 2²×3×5×47 Posts	For generalized Sierpinski/Riesel conjecture to base b, the dual of kb^n+1 is (b^n+k)/d, where d is the largest number that divides b^n+k for all enough large* n, and the dual of kb^n-1 is (\|b^n-k\|)/d, where d is the largest number that divides \|b^n-k\| for all enough large* n. If this number is not a integer, then we choose its numerator (this number is always a rational number :-) ) Last fiddled with by sweety439 on 2017-05-20 at 15:42

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2017-04-21, 16:07	#2
sweety439 Nov 2016 2²·3·5·47 Posts	The n's such that the dual forms are (probable) primes: 9^n+2036: 4, 208, ... (10^n+7666)/2: 67, 103, ... 17^n+244: 838, ... (22^n+5128)/8: 11, 63, 519, ... 27^n+398: 7, 375, ... 43^n+166: (no known such n) (68^n+17)/17: 1, 7, ... 75^n+1312: (no known such n) (86^n+8)/8: 205, ... 87^n+32: (no known such n) (112^n+1696)/32: 44, ... (118^n+48)/16: 57, ... (122^n+34)/2: 2, 98, ... (128^n+40)/8: 2, 8, ... \|6^n-1597\|: 3, 1731, .,. \|10^n-4421\|: 212, 284, ... (\|22^n-3656\|)/8: (no known such n) \|23^n-404\|: 568, ... \|27^n-706\|: 2, 10, 786, ... \|93^n-424\|: 1, 133, 151, 397, ... \|94^n-29\|: 2, ... \|103^n-924\|: 1, 97, ... \|109^n-84\|: 6, 18, 20, 362, ... (\|123^n-24\|)/3: 5, 84, ... \|133^n-926\|: 2, 111, 155, ...

2017-04-21, 16:28	#3
sweety439 Nov 2016 2²×3×5×47 Posts	Thus, the "mix Sierpinski/Riesel problem" of these bases are proven except S43, S75, S87 and R22.

2017-10-13, 13:19	#8
sweety439 Nov 2016 2²×3×5×47 Posts	See OEIS sequence A067760 and A252168 for the dual Sierpinski/Riesel base 2 problem.