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 2020-08-09, 15:06 #936 sweety439     Nov 2016 226710 Posts S13: 2 (2) 8 (4) 11 (564) 29 (10574) 281? (>5000) S14: 2 (1) 6 (6) 22 (16) 29 (23) 61 (126) 73 (1182) 208 (>5000) S15: 2 (1) 5 (2) 13 (10) 29 (30) 49 (112) 189 (190) 197 (464) 219 (1129) 341 (>5000) S16: 2 (1) 3 (2) 5 (3) 18 (4) 23 (1074) 89 (>20000) R13: 1 (5) 20 (10) 25 (15) 43 (77) 127 (95) 154 (469) 288 (109217) 337? (>5000) R14: 1 (3) 2 (4) 5 (19698) 617? (>5000) R15: 1 (3) 14 (14) 39 (16) 47 (>5000) R16: 2 (1) 11 (2) 18 (3) 31 (12) 48 (15) 74 (638) 322 (4624) 443 (>1500000) Last fiddled with by sweety439 on 2020-08-14 at 14:15
 2020-08-09, 16:40 #937 sweety439     Nov 2016 2,267 Posts Sierpinski k=2: 3 (1) 12 (3) 17 (47) 38 (2729) 101 (192275) 218 (333925) 365? (>200000) Sierpinski k=3: 2 (1) 5 (2) 18 (3) 28 (7) 43 (171) 79 (875) 83 (>8000) Sierpinski k=4: 3 (1) 5 (2) 17 (6) 23 (342) 53 (>1610000) Sierpinski k=5: 2 (1) 3 (2) 16 (3) 19 (78) 31 (1026) 137 (>2000) Sierpinski k=6: 2 (1) 4 (2) 14 (6) 19 (14) 20 (15) 48 (27) 53 (143) 67 (4532) 108 (16317) 129 (16796) 212 (>400000) Riesel k=1: 2 (2) 3 (3) 7 (5) 11 (17) 19 (19) 35 (313) 39 (349) 51 (4229) 91 (4421) 152 (270217) 185? (>66337) Riesel k=2: 2 (1) 5 (4) 20 (10) 29 (136) 67 (768) 107 (21910) 170 (166428) 581 (>200000) Riesel k=3: 2 (1) 3 (2) 23 (6) 31 (18) 42 (2523) 107 (4900) 295 (5270) 347 (>25000) Riesel k=4: 2 (1) 7 (3) 23 (5) 43 (279) 47 (1555) 72 (1119849) 178? (>5000) Riesel k=5: 2 (2) 8 (4) 14 (19698) 31? (>6000) Riesel k=6: 2 (1) 13 (2) 21 (3) 48 (294) 119 (665) 154 (1989) 234 (>400000) Last fiddled with by sweety439 on 2020-08-10 at 03:40
2020-08-10, 06:48   #938
sweety439

Nov 2016

2,267 Posts

Update PARI program files.
Attached Files
 program of Sierpinski.txt (381 Bytes, 9 views) program of Riesel.txt (515 Bytes, 7 views)

 2020-08-11, 16:42 #939 sweety439     Nov 2016 2,267 Posts The "ground base" of (k*b^n+-1)/gcd(k+-1,b-1) is the smallest value of b' such that * b' is a root of b (or b itself) * b'-1 is divisible by gcd(k+-1,b-1) * The relative complement of (k*b^n+-1)/gcd(k+-1,b-1) in (k'*b'^n+-1)/gcd(k'+-1,b'-1) contain no primes e.g. the "ground base" of (k*4^n+1)/gcd(k+1,4-1) is 2 for k == 1 mod 3 (since for these k, the relative complement of (k*4^n+1)/gcd(k+1,4-1) in (k*2^n+1)/gcd(k+1,2-1) is (2*k*4^n+1)/gcd(k+1,4-1), and all numbers of the form (2*k*4^n+1)/gcd(k+1,4-1) is divisible by 3, thus not prime), but it is 4 for k == 0 or 2 mod 3, similarly, the "ground base" of (k*4^n-1)/gcd(k-1,4-1) is 2 for k == 2 mod 3, but 4 for k == 0 or 1 mod 3 More examples: the "ground base" for (k*36^n-1)/gcd(k-1,36-1) is 6 for k == 6 mod 7, but 36 for other k the "ground base" for (k*25^n-1)/gcd(k-1,25-1) is 5 for k == 2 mod 3 or k == 5 mod 8, but 25 for other k the "ground base" for 222*625^n+1 is 25 (since all primes of the form 222*25^n+1 are of the form 222*625^n+1, numbers of the form 5550*625^n+1 are divisible by 13, thus cannot be prime, and 25 is the smallest possible base, numbers of the form 1110*25^n+1 can be prime) the "ground base" for 366*625^n-1 and 9150^625^n-1 are both 625 (since both of them can contain primes) the "ground base" for 29*1024^n-1 and 74*1024^n-1 are both 32 the "ground base" for 40*25^n+1 and (61*25^n+1)/2 are both 5 the "ground base" for 2036*9^n+1, 302*9^n-1, 386*9^n-1, and 744*9^n-1 are all 9 the "ground base" for 443*16^n-1, 2297*16^n-1, 13380*16^n-1, 13703*16^n-1, 2908*16^n+1, 6663*16^n+1, and 10183*16^n+1 are all 16 the "ground base" for 9519*16^n-1 and 19464*16^n-1 are both 4 the "ground base" for 18344*16^n-1, 23669*16^n-1, 31859*16^n-1, and 21181*16^n+1 are all 2 the "ground base" for 244*529^n+1, 376*529^n+1, and 394*529^n+1 are all 23 the "ground base" for 426*529^n+1 is 529 the "ground base" for 62*576^n+1, 227*576^n+1, and 1077*576^n+1 are all 576 the "ground base" for 656*576^n+1, 1851*576^n+1, and 2351*576^n+1 are all 24 the "ground base" for 94*8^n+1 is 2 (this is a case for a non-square power, since 47*8^n+1 has covering set {3, 5, 13} and 188*8^n+1 has trivial factor of 7, only 94*8^n+1 can be prime) the "ground base" for 37*8^n-1 is 8 (since both 37*8^n-1 and 74*8^n-1 can be prime) the "ground base" for 450*100^n-1 is 10 the "ground base" for 653*100^n-1, 74*100^n-1, and 470*100^n-1 are all 100 the "ground base" for 5*196^n-1 is 14 the "ground base" for 198*196^n-1 is 196 Last fiddled with by sweety439 on 2020-08-16 at 21:04
2020-08-11, 16:57   #940
sweety439

Nov 2016

43338 Posts

Quote:
 Originally Posted by sweety439 The "ground base" of (k*b^n+-1)/gcd(k+-1,b-1) is the smallest value of b' such that * b' is a root of b (or b itself) * b'-1 is divisible by gcd(k+-1,b-1) * The relative complement of (k*b^n+-1)/gcd(k+-1,b-1) in (k'*b'^n+-1)/gcd(k'+-1,b'-1) contain no primes e.g. the "ground base" of (k*4^n+1)/gcd(k+1,4-1) is 2 for k == 1 mod 3 (since for these k, the relative complement of (k*4^n+1)/gcd(k+1,4-1) in (k*2^n+1)/gcd(k+1,2-1) is (2*k*4^n+1)/gcd(k+1,4-1), and all numbers of the form (2*k*4^n+1)/gcd(k+1,4-1) is divisible by 3, thus not prime), but it is 4 for k == 0 or 2 mod 3, similarly, the "ground base" of (k*4^n-1)/gcd(k-1,4-1) is 2 for k == 2 mod 3, but 4 for k == 0 or 1 mod 3 More examples: the "ground base" for (k*36^n-1)/gcd(k-1,36-1) is 6 for k == 6 mod 7, but 36 for other k the "ground base" for (k*25^n-1)/gcd(k-1,25-1) is 5 for k == 2 mod 3 or k == 5 mod 8, but 25 for other k the "ground base" for 222*625^n+1 is 25 (since all primes of the form 222*25^n+1 are of the form 222*625^n+1, numbers of the form 5550*625^n+1 are divisible by 13, thus cannot be prime, and 25 is the smallest possible base, numbers of the form 1110*25^n+1 can be prime) the "ground base" for 366*625^n-1 and 9150^625^n-1 are both 625 (since both of them can contain primes) the "ground base" for 29*1024^n-1 and 74*1024^n-1 are both 32 the "ground base" for 40*25^n+1 and (61*25^n+1)/2 are both 5 the "ground base" for 2036*9^n+1, 302*9^n-1, 386*9^n-1, and 744*9^n-1 are all 9 the "ground base" for 443*16^n-1, 2297*16^n-1, 13380*16^n-1, 13703*16^n-1, 2908*16^n+1, 6663*16^n+1, and 10183*16^n+1 are all 16 the "ground base" for 9519*16^n-1 and 19464*16^n-1 are both 4 the "ground base" for 18344*16^n-1, 23669*16^n-1, 31859*16^n-1, and 21181*16^n+1 are all 2 the "ground base" for 244*529^n+1, 376*529^n+1, and 394*529^n+1 are all 23 the "ground base" for 426*529^n+1 is 529 the "ground base" for 62*576^n+1, 227*576^n+1, and 1077*576^n+1 are all 576 the "ground base" for 656*576^n+1, 1851*576^n+1, and 2351*576^n+1 are all 24
This means that, if finding a (probable) of the form (k'*b'^n+-1)/gcd(k'+-1,b'-1) is completely the same as finding a (probable) of the form (k*b^n+-1)/gcd(k+-1,b-1), then the "ground base" of (k*b^n+-1)/gcd(k+-1,b-1) is the smallest possible value of b'

2020-08-11, 17:07   #941
sweety439

Nov 2016

8DB16 Posts

Quote:
 Originally Posted by sweety439 Like Bunyakovsky conjecture, it is conjectured that for all integer triples (k, b, c) satisfying these conditions: 1. k>=1, b>=2, c != 0 2. gcd(k, c) = 1, gcd(b, c) = 1 3. there is no finite set {p_1, p_2, p_3, ..., p_u} (all p_i (1<=i<=u) are primes) and finite set {r_1, r_2, r_3, ..., r_s} (all r_i (1<=i<=s) are integers > 1) such that for every integer n>=1: either (k*b^n+c)/gcd(k+c, b-1) is divisible by at least one p_i (1<=i<=u) or k*b^n and -c are both r_i-th powers for at least one r_i (1<=i<=s) or one of k*b^n and c is a 4th power, another is of the form 4*t^4 with integer t 4. the triple (k, b, c) is not in this case: c = 1, b = q^m, k = q^r, where q is an integer not of the form t^s with odd s > 1, and m and r are integers having no common odd prime factor, and the exponent of highest power of 2 dividing r >= the exponent of highest power of 2 dividing m, and the equation 2^x = r (mod m) has no solution Then there are infinitely many integers n>=1 such that (k*b^n+c)/gcd(k+c, b-1) is prime.
Conjecture: If there is at least one prime of the form (k*b^n+c)/gcd(k+c, b-1) (k>=1, b>=2, c != 0, gcd(k, c) = 1, gcd(b, c) = 1) with n>=1, then (k*b^n+c)/gcd(k+c, b-1) has no covering set

If this conjecture is true, then k*2^n+1 has no covering set for all k<78557, since all odd k<78557 have at least one known proven prime of the form either k*2^n+1 (n>=1) or of the form 2^n+k (n>=1), and if k*2^n+1 has covering set, then 2^n+k must have the same covering set, since the dual of (k*b^n+c)/gcd(k+c, b-1) is (|c*b^n+k|)/gcd(k+c, b-1)/gcd(k, b^n), and if a form has covering set, then the dual form must have the same covering set

Last fiddled with by sweety439 on 2020-08-11 at 17:14

2020-08-12, 16:34   #942
sweety439

Nov 2016

226710 Posts

fixed the program so it can check whether (k*b^n+-1)/gcd(k+-1,b-1) has:

* covering set (for primes <= 50000, check to exponent = 5000)
* full algebra factors (k and b are both r-th powers for (r>1 Riesel case) (odd r>1 Sierpinski case)) (4*k and b are both 4-th powers Sierpinski case)
* partial algebra factors for algebra factors of difference of squares for (even/odd) n and one fixed prime factor for (odd/even) n for Riesel case

It current cannot check these cases:

* Partial algebra factors for algebra factors of difference of squares for (even/odd) n and covering set of >=2 primes for (odd/even) n for Riesel case:

** 1369*30^n-1 ({7, 13, 19} for odd n)
** (400*88^n-1)/3 ({3, 7, 13} for odd n)
** 324*95^n-1 ({7, 13, 229} for odd n)
** 93025*498^n-1 ({13, 67, 241} for odd n)
** 61009*540^n-1 ({17, 1009} for odd n)

** Partial algebra factors with period = 2 for square Sierpinski base:

** 114244*225^n+1

* Partial algebra factors for period > 2:

** (343*10^n-1)/9 (period = 3)
** 3511808*63^n+1 (period = 3)
** 27000000*63^n+1 (period = 3)
** (64*847^n-1)/9 (period = 3)
** 64*957^n-1 (period = 3)
** 2500*13^n+1 (period = 4)
** 2500*55^n+1 (period = 4)
** 16*200^n+1 (period = 4)
** (324*1101^n+1)/5 (period = 4)
** 324*2070^n+1 (period = 4)
** 64*936^n-1 (period = 6)

* b = q^m, k = q^r, where q is not of the form t^s with odd s>1, and m and r have no common odd prime factor, and the exponent of highest power of 2 dividing r >= the exponent of highest power of 2 dividing m, and the equation 2^x = r (mod m) has no solution

** 8*128^n+1
** 32*128^n+1
** 64*128^n+1
** (27*2187^n+1)/2
** (243*2187^n+1)/2
** (729*2187^n+1)/2
** 64*16384^n+1
** 1024*16384^n+1
** 4096*16384^n+1
** 128*32768^n+1
** 2048*32768^n+1
** 8192*32768^n+1
** 16384*32768^n+1
** 8*131072^n+1
** 32*131072^n+1
** 64*131072^n+1
** 128*131072^n+1
** 1024*131072^n+1
** 2048*131072^n+1
** 4096*131072^n+1
** 16384*131072^n+1

But since these cases are very rarely happen, these programs are >99% sufficient
Attached Files
 program of Sierpinski.txt (380 Bytes, 7 views) program of Riesel.txt (696 Bytes, 7 views)

Last fiddled with by sweety439 on 2020-08-12 at 16:39

 2020-08-14, 14:09 #943 sweety439     Nov 2016 1000110110112 Posts A PRP number (k*b^n+-1)/gcd(k+-1,b-1) with large n can be proven be prime if and only if: either * gcd(k+-1,b-1) = 1 or * gcd(k+-1,b-1) = k'+-1 (k' is the reduced k, i.e. k' = k/(b^r) which r is largest number such that this number is integer (i.e. r = valuation(k,b) in PARI), if k is not multiple of b (MOB), then k' = k) and the number b^n-1 (= prod_(d|n)Phi_d(b), where Phi is the cyclotomic polynomial) has >= 33.3333% factored Last fiddled with by sweety439 on 2020-08-14 at 14:10
2020-08-14, 14:22   #944
sweety439

Nov 2016

226710 Posts

Re-update the files for all Sierpinski/Riesel conjectures for bases 2<=b<=128 and bases b = 256, 512, 1024

Also see https://github.com/xayahrainie4793/E...el-conjectures for these files online.
Attached Files
 extend SR conjectures.zip (1.48 MB, 4 views)

 2020-08-15, 17:48 #945 sweety439     Nov 2016 8DB16 Posts Some algebra factors for the k's which is not (perfect power for Riesel case) (perfect odd power or of the form 4*m^4 for Sierpinski case): * S48 k=36 (n = 1 mod 3 and n = 2 mod 4 are algebraic) * S200 k=5 (n = 1 mod 3 are algebraic) * S200 k=16 (n = 2 mod 4 are algebraic, and make this k have full covering set with partial algebraic factors) * S200 k=40 (n = 1 mod 3 are algebraic) * S529 k=184 (n = 1 mod 3 are algebraic) * S968 k=11 (n = 1 mod 3 are algebraic) * R12 k=27 (n = 1 mod 2 and n = 0 mod 3 are algebraic, and make this k have full covering set with partial algebraic factors) * R12 k=300 (n = 1 mod 2 are algebraic, and make this k have full covering set with partial algebraic factors) * R18 k=50 (n = 1 mod 2 are algebraic) * R40 k=490 (n = 1 mod 2 are algebraic) * R80 k=10 (n = 2 mod 3 are algebraic) * R88 k=3773 (n = 2 mod 3 are algebraic) * R392 k=7 (n = 1 mod 3 are algebraic) * R392 k=56 (n = 1 mod 3 are algebraic) * R432 k=3 (n = 1 mod 2 are algebraic) * R578 k=2 (n = 1 mod 2 are algebraic) * R588 k=3 (n = 1 mod 2 are algebraic) * R750 k=6 (n = 2 mod 3 are algebraic) * R800 k=5 (n = 2 mod 5 are algebraic) * R800 k=8 (n = 1 mod 2 and n = 0 mod 3 are algebraic) * R800 k=25 (n = 0 mod 2 and n = 4 mod 5 are algebraic) * R972 k=3 (n = 1 mod 2 are algebraic) * R972 k=8 (n = 0 mod 3 and n = 1 mod 5 are algebraic) * R1152 k=2 (n = 1 mod 2 are algebraic) Last fiddled with by sweety439 on 2020-08-15 at 17:55
2020-08-15, 17:55   #946
sweety439

Nov 2016

226710 Posts

Quote:
 Originally Posted by sweety439 Some algebra factors for the k's which is not (perfect power for Riesel case) (perfect odd power or of the form 4*m^4 for Sierpinski case): * S48 k=36 (n = 1 mod 3 and n = 2 mod 4 are algebraic) * S200 k=5 (n = 1 mod 3 are algebraic) * S200 k=16 (n = 2 mod 4 are algebraic, and make this k have full covering set with partial algebraic factors) * S200 k=40 (n = 1 mod 3 are algebraic) * S529 k=184 (n = 1 mod 3 are algebraic) * S968 k=11 (n = 1 mod 3 are algebraic) * R12 k=27 (n = 1 mod 2 are algebraic, and make this k have full covering set with partial algebraic factors) * R12 k=300 (n = 1 mod 2 are algebraic, and make this k have full covering set with partial algebraic factors) * R18 k=50 (n = 1 mod 2 are algebraic) * R40 k=490 (n = 1 mod 2 are algebraic) * R80 k=10 (n = 2 mod 3 are algebraic) * R88 k=3773 (n = 2 mod 3 are algebraic) * R392 k=7 (n = 1 mod 3 are algebraic) * R392 k=56 (n = 1 mod 3 are algebraic) * R432 k=3 (n = 1 mod 2 are algebraic) * R578 k=2 (n = 1 mod 2 are algebraic) * R588 k=3 (n = 1 mod 2 are algebraic) * R750 k=6 (n = 2 mod 3 are algebraic) * R800 k=5 (n = 2 mod 5 are algebraic) * R800 k=8 (n = 1 mod 2 and n = 0 mod 3 are algebraic) * R800 k=25 (n = 0 mod 2 and n = 4 mod 5 are algebraic) * R972 k=3 (n = 1 mod 2 are algebraic) * R972 k=8 (n = 0 mod 3 and n = 1 mod 5 are algebraic) * R1152 k=2 (n = 1 mod 2 are algebraic)
A k-value which does not have covering set is proven composite by full algebraic factors if and only if all n-values are algebraic (e.g. R4 k=1, R4 k=9, R9 k=1, R9 k=4, S8 k=27, S27 k=8)

A k-value which does not have covering set is proven composite by partial algebraic factors if and only if there is covering set for all n-values which is not algebraic (e.g. R12 k=25, R12 k=27, R19 k=4, R28 k=175, R30 k=1369, S55 k=2500)

Both cases of k-values are excluded from the conjectures.

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