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Old 2020-10-04, 06:09   #1035
sweety439
 
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Quote:
Originally Posted by sweety439 View Post
Sierpinski problem base b (b>=2): Finding and proving the smallest k such that (k*b^n+1)/GCD(k+1, b-1) is not prime for all integers n >= 1

Riesel problem base b (b>=2): Finding and proving the smallest k such that (k*b^n-1)/GCD(k-1, b-1) is not prime for all integers n >= 1
Notes:

All n must be >= 1.

k-values that make a full covering set with all or partial algebraic factors are excluded from the conjectures.

k-values that are a multiple of base (b) and where (k+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is not prime are included in the conjectures but excluded from testing. Such k-values will have the same prime as k / b.
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Old 2020-10-04, 06:14   #1036
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This project is proving the Sierpinski/Riesel problems base b for all 2<=b<=128 and b = 256, 512, 1024 (will be extended to all 2<=b<=1024 and b = 2048, 4096, 8192, 16384, 32768, 65536 in the future), proving them all is not possible but we aim to prove many of them.

Last fiddled with by sweety439 on 2020-10-04 at 06:14
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Old 2020-10-04, 18:02   #1037
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For integer triple (k,b,c) such that k>=1, b>=2, c != 0, gcd(k,c) = 1, gcd(b,c) = 1, if there exist (positive or negative or 0) integer n such that

* either numerator((k*b^n+c)/gcd(k+c,b-1)) = +-1, or all prime factors of numerator((k*b^n+c)/gcd(k+c,b-1)) are factors of b

* for all integer r>1, k*b^n and -c are not both r-th powers of rational numbers

* k*b^n*c is not of the form 4*m^4 with rational number m

then there are (conjectured) infinitely many primes of the form (k*b^n+c)/gcd(k+c,b-1) with n>=1, since the sequence of the smallest prime factor of (k*b^n+c)/gcd(k+c,b-1) when n runs the integers such that (k*b^n+c)/gcd(k+c,b-1) has no algebra factors, is unbounded above (note that this does not include the case such that (k*b^n+c)/gcd(k+c,b-1) has algebra factors for all n, e.g. 4*9^n-1 and 9*4^n-1, since the upper bound of the empty set is -infinity, but a set is unbounded above if and only if the upper bound of this set is +infinitely, and -infinity != +infinity)
Examples:

2*581^n-1 (the value of n=0 is 1, and all prime factors of 1 are also prime factors of 581)

3*718^n+1 (the value of n=0 is 4, and all prime factors of 4 are also prime factors of 718)

294*213^n-1 (the value of n=-1 is 27/71, and all prime factors of 27 are also prime factors of 213)

122*123^n+1 (the value of n=0 is 123, and all prime factors of 123 are also prime factors of 123)

267*268^n-1 (the value of n=-1 is -1/268, and all prime factors of 1 are also prime factors of 268)

106*214^n+1 (the value of n=0 is 107, and all prime factors of 107 are also prime factors of 214)

859*430^n+1 (the value of n=0 is 860, and all prime factors of 860 are also prime factors of 430)

354*352^n-1 (the value of n=-1 is 1/176, and all prime factors of 1 are also prime factors of 352)
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Old 2020-10-04, 18:38   #1038
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Examples:

S801 k=2: the value of n=0 is 3, and all prime factors of 3 are also prime factors of 801

S829 k=2: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 829

R581 k=2: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 581

S718 k=3: the value of n=0 is 4, and all prime factors of 4 are also prime factors of 718

R588 k=3: the value of n=0 is 2, and all prime factors of 2 are also prime factors of 588

R107 k=3: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 107

R338 k=5: the value of n=0 is 4, and all prime factors of 4 are also prime factors of 338

S398 k=7: the value of n=0 is 8, and all prime factors of 8 are also prime factors of 398

S341 k=10: the value of n=0 is 11, and all prime factors of 11 are also prime factors of 341

R88 k=17: the value of n=0 is 16, and all prime factors of 16 are also prime factors of 88

R110 k=17: the value of n=0 is 16, and all prime factors of 16 are also prime factors of 110

S248 k=31: the value of n=0 is 32, and all prime factors of 32 are also prime factors of 248

R926 k=65: the value of n=0 is 64, and all prime factors of 64 are also prime factors of 926

S108 k=127: the value of n=0 is 128, and all prime factors of 128 are also prime factors of 108

R30 k=25: the value of n=-1 is -1/6, and all prime factors of 1 are also prime factors of 30 (note that although n=0 also satisfies this condition (the value of n=0 is 24, and all prime factors of 24 are also prime factors of 30), but n=0 cannot be used since this n has algebra factorization, all even n for all square k for all Riesel bases have algebra factorization)

R60 k=36: the value of n=-1 is -2/5, and all prime factors of 2 are also prime factors of 60

R213 k=294: the value of n=-1 is 27/71, and all prime factors of 27 are also prime factors of 213

S123 k=122: the value of n=0 is 123, and all prime factors of 123 are also prime factors of 123

S202 k=201: the value of n=0 is 202, and all prime factors of 202 are also prime factors of 202

R128 k=127: the value of n=-1 is -1/128, and all prime factors of 1 are also prime factors of 128

R268 k=267: the value of n=-1 is -1/268, and all prime factors of 1 are also prime factors of 268

R208 k=209: the value of n=0 is 208, and all prime factors of 208 are also prime factors of 208; also the value of n=-1 is 1/208, and all prime factors of 1 are also prime factors of 208

R575 k=576: the value of n=-1 is 1/575, and all prime factors of 1 are also prime factors of 575 (note that although n=0 also satisfies this condition (the value of n=0 is 575, and all prime factors of 575 are also prime factors of 575), but n=0 cannot be used since this n has algebra factorization, all even n for all square k for all Riesel bases have algebra factorization)

S70 k=68: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 70

R152 k=1: the value of n=1 is 1, and all prime factors of 1 are also prime factors of 152

R243 k=81: the value of n=-1 is -1/3, and all prime factors of 1 are also prime factors of 243

S214 k=106: the value of n=0 is 107, and all prime factors of 107 are also prime factors of 214

R216 k=55: the value of n=0 is 54, and all prime factors of 54 are also prime factors of 216

R217 k=50: the value of n=0 is 49, and all prime factors of 49 are also prime factors of 217

S430 k=859: the value of n=0 is 860, and all prime factors of 860 are also prime factors of 430

S447 k=148: the value of n=0 is 149, and all prime factors of 149 are also prime factors of 447

R44 k=46: the value of n=-1 is 1/22, and all prime factors of 1 are also prime factors of 44

R352 k=354: the value of n=-1 is 1/176, and all prime factors of 1 are also prime factors of 352

R430 k=432: the value of n=-1 is 1/215, and all prime factors of 1 are also prime factors of 430

S312 k=12: the value of n=0 is 13, and all prime factors of 13 are also prime factors of 312; also the value of n=-1 is 27/26, and all prime factors of 27 are also prime factors of 312

S340 k=199: the value of n=0 is 200, and all prime factors of 200 are also prime factors of 340

S468 k=191: the value of n=0 is 192, and all prime factors of 192 are also prime factors of 468

S239 k=6: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 239

S576 k=9: the value of n=0 is 2, and all prime factors of 2 are also prime factors of 576

S729 k=6: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 729

R10 k=109: the value of n=-2 is 1/100, and all prime factors of 1 are also prime factors of 10

R10 k=2251: the value of n=0 is 250, and all prime factors of 250 are also prime factors of 10

R12 k=298: the value of n=0 is 27, and all prime factors of 27 are also prime factors of 12

S16 k=23: the value of n=0 is 8, and all prime factors of 8 are also prime factors of 16

S22 k=461: the value of n=0 is 22, and all prime factors of 22 are also prime factors of 22

S28 k=146: the value of n=0 is 49, and all prime factors of 49 are also prime factors of 28

R40 k=157: the value of n=0 is 4, and all prime factors of 4 are also prime factors of 40

R40 k=520: the value of n=-1 is 4, and all prime factors of 4 are also prime factors of 40

S45 k=24: the value of n=0 is 25, and all prime factors of 25 are also prime factors of 45

S46 k=17: the value of n=0 is 2, and all prime factors of 2 are also prime factors of 46

R46 k=86: the value of n=-1 is -4/43, and all prime factors of 4 are also prime factors of 46

R46 k=93: the value of n=0 is 92, and all prime factors of 92 are also prime factors of 46

S64 k=11: the value of n=0 is 4, and all prime factors of 4 are also prime factors of 64

S66 k=269: the value of n=0 is 54, and all prime factors of 54 are also prime factors of 66

R70 k=376: the value of n=0 is 125, and all prime factors of 125 are also prime factors of 70

S72 k=647: the value of n=0 is 648, and all prime factors of 648 are also prime factors of 72

R73 k=79: the value of n=-1 is 1/73, and all prime factors of 1 are also prime factors of 73

S81 k=431: the value of n=0 is 27, and all prime factors of 27 are also prime factors of 81

R85 k=61: the value of n=0 is 5, and all prime factors of 5 are also prime factors of 85

R85 k=169: the value of n=-1 is 1/85, and all prime factors of 1 are also prime factors of 85

R100 k=133: the value of n=0 is 4, and all prime factors of 4 are also prime factors of 100; also the value of n=-1 is 1/100, and all prime factors of 1 are also prime factors of 100

S106 k=69: the value of n=0 is 2, and all prime factors of 2 are also prime factors of 106

R110 k=23: the value of n=0 is 22, and all prime factors of 22 are also prime factors of 110

R110 k=55: the value of n=-1 is -1/2, and all prime factors of 1 are also prime factors of 110

S120 k=89: the value of n=0 is 90, and all prime factors of 90 are also prime factors of 120

R120 k=4320: the value of n=0 is 5, and all prime factors of 5 are also prime factors of 120

S256 k=11: the value of n=0 is 4, and all prime factors of 4 are also prime factors of 256

Last fiddled with by sweety439 on 2020-10-07 at 13:59
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Old 2020-10-04, 19:01   #1039
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This including:

* k=1 for all Sierpinski base not of the form m^r with odd r>1 nor of the form 4*m^4
* k=1 for all Riesel base not of the form m^r with r>1
* k=2 for all Sierpinski base not == 2 mod 3
* k=2 for all Sierpinski base of the form 2^r-2 or 3*2^r-2
* k=2 for all Riesel base
* k=3 for all Sierpinski base not == 3 mod 4
* k=3 for all Sierpinski base of the form 3^r-3 or 2*3^r-3 or 4*3^r-3
* k=3 for all Riesel base
* k=5 for all Sierpinski base not == 2 mod 3
* k=5 for all Sierpinski base of the form 5^r-5 or 2*5^r-5 or 3*5^r-5 or 6*5^r-5
* k=5 for all Riesel base not == 3 mod 4
* k=5 for all Riesel base of the form 5^r+5 or 2*5^r+5 or 4*5^r+5
* k=6 for all Sierpinski base == 0, 1 mod 7
* k=6 for all Sierpinski base of the form 2^r*3^s-6 or 7*2^r*3^s-6
* k=6 for all Riesel base == 0, 1 mod 5
* k=6 for all Riesel base of the form 2^r*3^s+6 or 5*2^r*3^s+6
* k=7 for all Sierpinski base not == 3, 5, 7 mod 8
* k=7 for all Sierpinski base of the form 7^r-7 or 2*7^r-7 or 3*7^r-7 or 6*7^r-7
* k=7 for all Riesel base not == 2 mod 3
* k=7 for all Riesel base of the form 7^r+7 or 2*7^r+7 or 4*7^r+7 or 8*7^r+7
* k=b-2 for all Sierpinski base b where k is not of the form m^r with odd r>1 nor of the form 4*m^4
* k=b-2 for all Riesel base b
* k=b-1 for all Sierpinski base b where k is not of the form m^r with odd r>1 nor of the form 4*m^4
* k=b-1 for all Riesel base b
* k=b+1 for all Riesel base b
* k=b+2 for all Riesel base b
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Old 2020-10-05, 06:30   #1040
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S726 has CK = 10923176
R726 has CK = 12751579
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Old 2020-10-06, 17:12   #1041
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PARI program for the smallest Sierpinski number base n: (enter "c(n)" to get)

Code:
is(n)=forprime(p=2,50000,if(n%p==0,return(0)));1
a(n,k,b)=(n*b^k+1)/gcd(n+1,b-1)
b(n,r)=for(k=1,3000,if(is(a(n,k,r)),return(0)));1
c(n)=for(k=1,10^12,if(b(k,n),return(k)))
PARI program for the smallest Riesel number base n: (enter "c(n)" to get)
Code:
is(n)=forprime(p=2,50000,if(n%p==0,return(0)));1
a(n,k,b)=(n*b^k-1)/gcd(n-1,b-1)
b(n,r)=for(k=1,3000,if(is(a(n,k,r)),return(0)));1
c(n)=for(k=1,10^12,if(b(k,n),return(k)))
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Old 2020-10-06, 17:30   #1042
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Quote:
Originally Posted by sweety439 View Post
Examples:

S801 k=2: the value of n=0 is 3, and all prime factors of 3 are also prime factors of 801

S829 k=2: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 829

R581 k=2: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 581

S718 k=3: the value of n=0 is 4, and all prime factors of 4 are also prime factors of 718

R588 k=3: the value of n=0 is 2, and all prime factors of 2 are also prime factors of 588

R107 k=3: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 107

R338 k=5: the value of n=0 is 4, and all prime factors of 4 are also prime factors of 338

S398 k=7: the value of n=0 is 8, and all prime factors of 8 are also prime factors of 398

S341 k=10: the value of n=0 is 11, and all prime factors of 11 are also prime factors of 341

S248 k=31: the value of n=0 is 32, and all prime factors of 32 are also prime factors of 248

R926 k=65: the value of n=0 is 64, and all prime factors of 64 are also prime factors of 926

S108 k=127: the value of n=0 is 128, and all prime factors of 128 are also prime factors of 108

R30 k=25: the value of n=-1 is -1/6, and all prime factors of 1 are also prime factors of 30 (note that although n=0 also satisfies this condition (the value of n=0 is 24, and all prime factors of 24 are also prime factors of 30), but n=0 cannot be used since this n has algebra factorization, all even n for all square k for all Riesel bases have algebra factorization)

R60 k=36: the value of n=-1 is -2/5, and all prime factors of 2 are also prime factors of 60

R213 k=294: the value of n=-1 is 27/71, and all prime factors of 27 are also prime factors of 213

S123 k=122: the value of n=0 is 123, and all prime factors of 123 are also prime factors of 123

S202 k=201: the value of n=0 is 202, and all prime factors of 202 are also prime factors of 202

R128 k=127: the value of n=-1 is -1/128, and all prime factors of 1 are also prime factors of 128

R268 k=267: the value of n=-1 is -1/268, and all prime factors of 1 are also prime factors of 268

R208 k=209: the value of n=0 is 208, and all prime factors of 208 are also prime factors of 208; also the value of n=-1 is 1/208, and all prime factors of 1 are also prime factors of 208

R575 k=576: the value of n=-1 is 1/575, and all prime factors of 1 are also prime factors of 575 (note that although n=0 also satisfies this condition (the value of n=0 is 575, and all prime factors of 575 are also prime factors of 575), but n=0 cannot be used since this n has algebra factorization, all even n for all square k for all Riesel bases have algebra factorization)

S70 k=68: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 70

R152 k=1: the value of n=1 is 1, and all prime factors of 1 are also prime factors of 152

R243 k=81: the value of n=-1 is -1/3, and all prime factors of 1 are also prime factors of 243

S214 k=106: the value of n=0 is 107, and all prime factors of 107 are also prime factors of 214

R216 k=55: the value of n=0 is 54, and all prime factors of 54 are also prime factors of 216

R217 k=50: the value of n=0 is 49, and all prime factors of 49 are also prime factors of 217

S430 k=859: the value of n=0 is 860, and all prime factors of 860 are also prime factors of 430

S447 k=148: the value of n=0 is 149, and all prime factors of 149 are also prime factors of 447

R44 k=46: the value of n=-1 is 1/22, and all prime factors of 1 are also prime factors of 44

R352 k=354: the value of n=-1 is 1/176, and all prime factors of 1 are also prime factors of 352

R430 k=432: the value of n=-1 is 1/215, and all prime factors of 1 are also prime factors of 430

S312 k=12: the value of n=0 is 13, and all prime factors of 13 are also prime factors of 312; also the value of n=-1 is 27/26, and all prime factors of 27 are also prime factors of 312

S340 k=199: the value of n=0 is 200, and all prime factors of 200 are also prime factors of 340

S468 k=191: the value of n=0 is 192, and all prime factors of 192 are also prime factors of 468

S239 k=6: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 239

S576 k=9: the value of n=0 is 2, and all prime factors of 2 are also prime factors of 576

S729 k=6: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 729

R10 k=109: the value of n=-2 is 1/100, and all prime factors of 1 are also prime factors of 10

R10 k=2251: the value of n=0 is 250, and all prime factors of 250 are also prime factors of 10

R12 k=298: the value of n=0 is 27, and all prime factors of 27 are also prime factors of 12

S16 k=23: the value of n=0 is 8, and all prime factors of 8 are also prime factors of 16

S22 k=461: the value of n=0 is 22, and all prime factors of 22 are also prime factors of 22

S28 k=146: the value of n=0 is 49, and all prime factors of 49 are also prime factors of 28

R40 k=157: the value of n=0 is 4, and all prime factors of 4 are also prime factors of 40

R40 k=520: the value of n=-1 is 4, and all prime factors of 4 are also prime factors of 40

S45 k=24: the value of n=0 is 25, and all prime factors of 25 are also prime factors of 45

S46 k=17: the value of n=0 is 2, and all prime factors of 2 are also prime factors of 46

R46 k=86: the value of n=-1 is -4/43, and all prime factors of 4 are also prime factors of 46

R46 k=93: the value of n=0 is 92, and all prime factors of 92 are also prime factors of 46

S64 k=11: the value of n=0 is 4, and all prime factors of 4 are also prime factors of 64

S66 k=269: the value of n=0 is 54, and all prime factors of 54 are also prime factors of 66

S72 k=647: the value of n=0 is 648, and all prime factors of 648 are also prime factors of 72

R73 k=79: the value of n=-1 is 1/73, and all prime factors of 1 are also prime factors of 73

S81 k=431: the value of n=0 is 27, and all prime factors of 27 are also prime factors of 81

R100 k=133: the value of n=0 is 4, and all prime factors of 4 are also prime factors of 100; also the value of n=-1 is 1/100, and all prime factors of 1 are also prime factors of 100

S106 k=69: the value of n=0 is 2, and all prime factors of 2 are also prime factors of 106

R110 k=23: the value of n=0 is 22, and all prime factors of 22 are also prime factors of 110

S120 k=89: the value of n=0 is 90, and all prime factors of 90 are also prime factors of 120

R120 k=4320: the value of n=0 is 5, and all prime factors of 5 are also prime factors of 120

S256 k=11: the value of n=0 is 4, and all prime factors of 4 are also prime factors of 256
Non-examples: (because of algebra factors)

S426 k=8: the value of n=0 is 9, and all prime factors of 9 are also prime factors of 426 -- however, n=0 has algebra factorization, all n divisible by 3 for all cube k for all Sierpinski&Riesel bases have algebra factorization, thus n=0 cannot be used

S163 k=8: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 163 -- however, n=0 has algebra factorization, all n divisible by 3 for all cube k for all Sierpinski&Riesel bases have algebra factorization, thus n=0 cannot be used

R721 k=8: the value of n=0 is 7, and all prime factors of 7 are also prime factors of 721 -- however, n=0 has algebra factorization, all n divisible by 3 for all cube k for all Sierpinski&Riesel bases have algebra factorization, thus n=0 cannot be used

R372 k=8: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 372 -- however, n=0 has algebra factorization, all n divisible by 3 for all cube k for all Sierpinski&Riesel bases have algebra factorization, thus n=0 cannot be used

S155 k=4: the value of n=0 is 5, and all prime factors of 5 are also prime factors of 230 -- however, n=0 has algebra factorization, all n divisible by 4 for all k of the form 4*m^4 for all Sierpinski bases have algebra factorization, thus n=0 cannot be used

S230 k=4: the value of n=0 is 5, and all prime factors of 5 are also prime factors of 230 -- however, n=0 has algebra factorization, all n divisible by 4 for all k of the form 4*m^4 for all Sierpinski bases have algebra factorization, thus n=0 cannot be used

S335 k=4: the value of n=0 is 5, and all prime factors of 5 are also prime factors of 335 -- however, n=0 has algebra factorization, all n divisible by 4 for all k of the form 4*m^4 for all Sierpinski bases have algebra factorization, thus n=0 cannot be used

S266 k=4: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 266 -- however, n=0 has algebra factorization, all n divisible by 4 for all k of the form 4*m^4 for all Sierpinski bases have algebra factorization, thus n=0 cannot be used

R72 k=4: the value of n=0 is 3, and all prime factors of 3 are also prime factors of 72 -- however, n=0 has algebra factorization, all even n for all square k for all Riesel bases have algebra factorization, thus n=0 cannot be used

R178 k=4: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 178 -- however, n=0 has algebra factorization, all even n for all square k for all Riesel bases have algebra factorization, thus n=0 cannot be used

R438 k=4: the value of n=0 is 3, and all prime factors of 3 are also prime factors of 438 -- however, n=0 has algebra factorization, all even n for all square k for all Riesel bases have algebra factorization, thus n=0 cannot be used

R378 k=9: the value of n=0 is 8, and all prime factors of 8 are also prime factors of 378 -- however, n=0 has algebra factorization, all even n for all square k for all Riesel bases have algebra factorization, thus n=0 cannot be used

R438 k=9: the value of n=0 is 8, and all prime factors of 8 are also prime factors of 438 -- however, n=0 has algebra factorization, all even n for all square k for all Riesel bases have algebra factorization, thus n=0 cannot be used

R705 k=9: the value of n=0 is 1, and all prime factors of 1 are also prime factors of 705 -- however, n=0 has algebra factorization, all even n for all square k for all Riesel bases have algebra factorization, thus n=0 cannot be used

R666 k=74: the value of n=-1 is -8/9, and all prime factors of 8 are also prime factors of 666 -- however, n=-1 has algebra factorization, all odd n for all k such that k*base is square for all Riesel bases have algebra factorization, thus n=-1 cannot be used

Last fiddled with by sweety439 on 2020-10-06 at 17:32
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Old 2020-10-07, 20:35   #1043
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Conjecture 1 (the strong Sierpinski conjecture): For b>=2, k>=1, if there is an n such that:

(1) k*b^n is neither a perfect odd power (i.e. k*b^n is not of the form m^r with odd r>1) nor of the form 4*m^4.
(2) gcd((k*b^n+1)/gcd(k+1,b-1),(b^(9*2^s)-1)/(b-1)) = 1 for all s, i.e. for all s, every prime factor of (k*b^n+1)/gcd(k+1,b-1) does not divide (b^(9*2^s)-1)/(b-1). (i.e. ord_p(b) is not of the form 2^r (r>=0 if p = 2 or p = 3, r>=1 if p>=5), 3*2^r (r>=0) or 9*2^r (r>=0) for every prime factor p of (k*b^n+1)/gcd(k+1,b-1)).
(3) this k is not excluded from this Sierpinski base b by the post #265. (the first 6 Sierpinski bases with k's excluded by the post #265 are 128, 2187, 16384, 32768, 78125 and 131072)

Then there are infinitely many primes of the form (k*b^n+1)/gcd(k+1,b-1).

Conjecture 2 (the strong Riesel conjecture): For b>=2, k>=1, if there is an n such that:

(1) k*b^n is not a perfect power (i.e. k*b^n is not of the form m^r with r>1).
(2) gcd((k*b^n-1)/gcd(k-1,b-1),(b^(9*2^s)-1)/(b-1)) = 1 for all s, i.e. for all s, every prime factor of (k*b^n-1)/gcd(k-1,b-1) does not divide (b^(9*2^s)-1)/(b-1). (i.e. ord_p(b) is not of the form 2^r (r>=0 if p = 2 or p = 3, r>=1 if p>=5), 3*2^r (r>=0) or 9*2^r (r>=0) for every prime factor p of (k*b^n-1)/gcd(k-1,b-1)).

Then there are infinitely many primes of the form (k*b^n-1)/gcd(k-1,b-1).
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Old 2020-10-07, 20:39   #1044
sweety439
 
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All n must be >= 1.

All MOB k-values need an n>=1 prime (unless they have a covering set of primes, or make a full covering set with all or partial algebraic factors)

MOB k-values such that (k+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is not prime are included in the conjectures but excluded from testing. Such k-values will have the same prime as k / b.

There are some MOB k-values such that (k+-1)/gcd(k+-1,b-1) (+ for Sierpinski, - for Riesel) is prime which do not have an easy prime for n>=1:

GFN's and half GFN's: (all with no known primes)

S2 k=65536
S3 k=3433683820292512484657849089281
S4 k=65536
S5 k=625
S6 k=1296
S7 k=2401
S8 k=256
S8 k=65536
S9 k=3433683820292512484657849089281
S10 k=100
S11 k=14641
S12 k=12
S13 k=815730721
S14 k=196
S15 k=225
S16 k=65536

Other k's:

S2 k=55816: first prime at n=14536
S2 k=90646: no prime with n<=6.6M
S2 k=101746: no prime with n<=6.6M
S3 k=621: first prime at n=20820
S4 k=176: first prime at n=228
S5 k=40: first prime at n=1036
S6 k=90546
S7 k=21: first prime at n=124
S9 k=1746: first prime at n=1320
S9 k=2007: first prime at n=3942
S10 k=640: first prime at n=120
S24 k=17496
S155 k=310
S333 k=1998
R2 k=74: first prime at n=2552
R2 k=674: first prime at n=11676
R2 k=1094: first prime at n=652
R4 k=1524: first prime at n=1994
R4 k=19464: no prime with n<=3.3M
R6 k=103536: first prime at n=6474
R6 k=106056: first prime at n=3038
R7 k=679: no prime with n<=3K
R10 k=450: first prime at n=11958
R10 k=16750: no prime with n<=200K
R11 k=308: first prime at n=444
R14 k=2954: no prime with n<=50K
R15 k=2940: first prime at n=13254
R15 k=8610: first prime at n=5178
R18 k=324: first prime at n=25665
R21 k=84: first prime at n=88
R23 k=230: first prime at n=6228
R27 k=594: first prime at n=36624
R40 k=520: no prime with n<=1K
R42 k=1764: first prime at n=1317
R48 k=384: no prime with n<=200K
R66 k=1056: no prime with n<=1K
R78 k=7800: no prime with n<=1K
R88 k=3168: first prime at n=205764
R96 k=9216: first prime at n=3341
R120 k=4320: no prime with n<=1K
R210 k=44100: first prime at n=19817
R306 k=93636: first prime at n=26405
R396 k=156816: no prime with n<=50K
R591 k=1182: first prime at n=1190
R954 k=1908: first prime at n=1476
R976 k=1952: first prime at n=1924
R1102 k=2204: first prime at n=52176
R1297 k=2594: first prime at n=19839
R1360 k=2720: first prime at n=74688
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Old 2020-10-07, 20:55   #1045
sweety439
 
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Quote:
Originally Posted by sweety439 View Post
The test limit of GFN's and half GFN's:

base 2:

According to http://www.prothsearch.com/fermat.html, the first numbers n>=5 such that 2^(2^n)+1 might be primes are 33, 34, 35, 40, 41, 44, 45, 46, 47, 49, 50, 51, 53, 54, 56, ..., and we have:

* For S2 k=65536, the test limit is (2^33-16)-1 = 8589934575
* For S32 k=4, the test limit is (2^33-2)/5-1 = 1717986917
* For S128 k=16, the test limit is (2^35-4)/7-1 = 4908534051 ((2^n-4)/7 is not integer for n = 33, 34)
* For S512 k=2, the test limit is (2^54-1)/9-1 = 2001599834386886 ((2^n-1)/9 is not integer for n = 33, 34, 35, 40, 41, 44, 45, 46, 47, 49, 50, 51, 53)
* For S512 k=4, the test limit is (2^49-2)/9-1 = 62549994824589 ((2^n-2)/9 is not integer for n = 33, 34, 35, 40, 41, 44, 45, 46, 47)
* For S512 k=16, the test limit is (2^44-4)/9-1 = 1954687338267 ((2^n-4)/9 is not integer for n = 33, 34, 35, 40, 41)
* For S1024 k=4, the test limit is (2^33-2)/10-1 = 858993458
* For S1024 k=16, the test limit is (2^34-4)/10-1 = 1717986917 ((2^n-4)/10-1 is not integer for n = 33)

base 3:

None of S3, S9, S27, S81 have half GFN remain

base 5:

None of S5, S25, S125 have half GFN remain

base 6:

According to http://www.prothsearch.com/GFN06.html, the first numbers n>=3 such that 6^(2^n)+1 might be primes are 28, 29, 30, 31, 38, 41, 45, 46, 48, ..., and we have:

* For S6 k=1296, the test limit is (2^28-4)-1 = 268435451
* For S36 k=1296, the test limit is (2^28-4)/2-1 = 134217725

base 7:

Neither S7 nor S49 have half GFN remain

base 10:

According to http://www.prothsearch.com/GFN10.html, the first numbers n>=2 such that 10^(2^n)+1 might be primes are 31, 32, 33, 34, 36, 42, 44, 45, 47, 49, ..., and we have:

* For S10 k=100, the test limit is (2^31-2)-1 = 2147483645

base 11:

Neither S11 nor S121 have half GFN remain

base 12:

According to http://www.prothsearch.com/GFN12.html, the first numbers n>=1 such that 12^(2^n)+1 might be primes are 24, 25, 27, 28, 31, 32, 35, 36, 37, 43, 47, 48, ..., however, according to http://www.primegrid.com/stats_genefer.php, n^(2^22)+1 is composite for all n<169020, and 169020>12^4, thus (12^4)^(2^22)+1 = 12^(2^24)+1 is composite, and the first number n>=1 such that 12^(2^n)+1 might be prime is 25

* For S12 k=12, the test limit is (2^25-1)-1 = 33554430

even bases > 12:

According to http://www.primegrid.com/stats_genefer.php, n^(2^22)+1 is composite for all even n<169020, and 169020^(1/4) = 20.276104663..., thus n^(2^24)+1 is composite for all even n<=20, and 169020^(1/2) = 411.120420315..., thus n^(2^23)+1 is composite for all even n<=411

* For even bases <= 20, the test limit of GFN's are 2^25-epsilon
* For 21 <= even bases <= 411, the test limit of GFN's are 2^24-epsilon

odd bases > 12:

According to http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt, (n^(2^17)+1)/2 is composite for all odd n<11559, and 11559^(1/2) = 107.512789936..., thus (n^(2^18)+1)/2 is composite for all odd n<=107

* For odd bases <= 107, the test limit of half GFN's are 2^19-epsilon (I also checked the (n^(2^18)+1)/2 for 108 <= odd bases <= 128, they are also all composite, thus (n^(2^18)+1)/2 is composite for all odd n<=128)
test limits for GFN's and half GFN's for 128 < base <= 1024 (except 256, 512, 1024):

base 3:

According to http://www.prothsearch.com/GFN03.html, the first numbers n>=7 such that (3^(2^n)+1)/2 might be primes are 24, 25, 27, 37, 40, 41, 42, 43, 46, 48, 52, 53, ..., and we have:

* For S243 k=27: the test limit is (2^27-3)/5-1 = 26843544 ((2^n-3)/5 is not integer for n = 24, 25)
* S729 has only CK = 31, and no half GFN remain for k<31

base 5:

According to http://www.prothsearch.com/GFN05.html, the first numbers n>=3 such that (5^(2^n)+1)/2 might be primes are 29, 30, 31, 37, 42, 43, 46, 52, 55, 56, 57, 61, ..., and we have:

* S625 has only CK = 185, and no half GFN remain for k<185

base 6:

According to http://www.prothsearch.com/GFN06.html, the first numbers n>=3 such that 6^(2^n)+1 might be primes are 28, 29, 30, 31, 38, 41, 45, 46, 48, ..., and we have:

* For S216 k=36, the test limit is (2^29-2)/3-1 = 178956969 ((2^n-2)/3 is not integer for n = 28)

base 7:

According to http://www.prothsearch.com/GFN07.html, the first numbers n>=3 such that (7^(2^n)+1)/2 might be primes are 23, 24, 26, 27, 28, 29, 35, 37, 38, 39, 43, ..., and we have:

* For S343 k=49, the test limit is (2^23-2)/3-1 = 2796201

base 10:

According to http://www.prothsearch.com/GFN10.html, the first numbers n>=2 such that 10^(2^n)+1 might be primes are 31, 32, 33, 34, 36, 42, 44, 45, 47, 49, ..., and we have:

* For S1000 k=10, the test limit is (2^32-1)/3-1 = 1431655764 ((2^n-1)/3 is not integer for n = 31)

base 11:

No power-of-11 bases between 128 and 1024

base 12:

According to http://www.prothsearch.com/GFN12.html, the first numbers n>=1 such that 12^(2^n)+1 might be primes are 24, 25, 27, 28, 31, 32, 35, 36, 37, 43, 47, 48, ..., however, according to http://www.primegrid.com/stats_genefer.php, n^(2^22)+1 is composite for all n<169020, and 169020>12^4, thus (12^4)^(2^22)+1 = 12^(2^24)+1 is composite, and the first number n>=1 such that 12^(2^n)+1 might be prime is 25

* For S144 k=1, the test limit is (2^24)-1 = 16777215

even bases > 12:

According to http://www.primegrid.com/stats_genefer.php, n^(2^22)+1 is composite for all even n<169020, and 169020^(1/4) = 20.276104663..., thus n^(2^24)+1 is composite for all even n<=20, and 169020^(1/2) = 411.120420315..., thus n^(2^23)+1 is composite for all even n<=411

* For even bases <= 20, the test limit of GFN's are 2^25-epsilon
* For 21 <= even bases <= 411, the test limit of GFN's are 2^24-epsilon
* For 412 <= even bases <= 169019, the test limit of GFN's are 2^23-epsilon

odd bases > 12:

According to http://www.fermatquotient.com/PrimSerien/GenFermOdd.txt, (n^(2^17)+1)/2 is composite for all odd n<11559, and 11559^(1/2) = 107.512789936..., thus (n^(2^18)+1)/2 is composite for all odd n<=107

* For odd bases <= 107, the test limit of half GFN's are 2^19-epsilon (I also checked the (n^(2^18)+1)/2 for 108 <= odd bases <= 128, they are also all composite, thus (n^(2^18)+1)/2 is composite for all odd n<=128)
* For 129 <= odd bases <= 11558, the test limit of half GFN's are 2^18-epsilon

Last fiddled with by sweety439 on 2020-10-07 at 20:57
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