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Re: A new prime series and Goldbach's conjecture

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  • eharsh82
    1663=c disproves the third conjecture for the + side, though I found lots( ~500) primes in the process. The third conjecture for the combined series still
    Message 1 of 5 , Sep 26, 2003
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      1663=c disproves the third conjecture for the + side, though I found
      lots( ~500) primes in the process.
      The third conjecture for the combined series still holds

      Harsh


      --- In primenumbers@yahoogroups.com, "eharsh82" <harsh@u...> wrote:
      > I have checked the +,- series upto N=13 for c=2^N so start at n=14.
      >
      > Each n has atleast 1 prime for the two series.
      >
      > I expect the 2^n conjecture to be true and the prime conjecture to
      be
      > false. There are probably infinitely many primes in this series. If
      > the 2^n conjecture is true then one can find a prime of n digits by
      > searching n candidates. Which is the goal of finding primes.
      >
      > Harsh Aggarwal
      >
      > --- In primenumbers@yahoogroups.com, "eharsh82" <harsh@u...> wrote:
      > >
      > > In general numbers(c) that are highly composite are candiadates
      > that
      > > give poor primes.
      > >
      > > Consider this,
      > > Suppose I choose 6 then
      > > the pairs can be
      > > 1 5
      > > 2 4
      > > 3 3
      > > 4 2
      > > 5 1
      > > for the the - case the even ones will become a^2-1 and hence are
      > > composite
      > > same is the case if each of the two terms have a common factor.
      > >
      > > For the + side on the other hand if both the terms have a common
      > > factor then you get a^2+1 which is generalized fermat.
      > > It is already known that a^3+1 and others non gen fermats can
      never
      > > be prime.
      > >
      > > Also if the two terms have a common factor then (2^x)^n %p=1 and
      > (3^y)
      > > ^n %p =1 has high probabily to happen hence 2^x)^n * (3^y)^n -1
      > will
      > > also be composite.
      > > For the + side one term can be -1 and the other one +1, which is
      > also
      > > common.
      > >
      > > Hence composites C are generally do not produce primes.
      > >
      > > The exception to this is when c=2^n which leads to my two new
      > > conjectures
      > > There is always altleast a prime for the + side if c=2^n and
      > atleast
      > > one for the - side.
      > >
      > > It can be easily shown that all primes on the + sides will be
      > > generalized fermats and thus have specialized factors.
      > > For the - side only the odd tems can produce primes as explained
      > > above.
      > >
      > >
      > > Harsh Aggarwal
      > >
      > >
      > > --- In primenumbers@yahoogroups.com, "Mark Underwood"
      > > <mark.underwood@s...> wrote:
      > > >
      > > > Hi Harsh,
      > > >
      > > > Amazingly I was looking at almost exactly the same thing about
      a
      > > > month ago (!) but I restricted it to the postive case. I think
      > (and
      > > > hope!) you are correct in the case where c is a prime.
      > > >
      > > >
      > > > Below were my (very cursory) observations on my data, followed
      by
      > > > some sample data. I never had time to pursue it...
      > > >
      > > > Mark
      > > >
      > > >
      > > > Primes of the form 2^n*3^m + 1, k = n+m
      > > >
      > > >
      > > > Observations:
      > > > If k is a multiple of 6, and especially 12 it is very poor in
      > > primes!
      > > > If k = (2n+1)^2 - 1, it is VERY poor in primes for n>1 !
      > > > If k is prime, it is usually rich in primes!
      > > > There tends to be proportionally many of (a,b) where a-b or b-a
      =
      > 1
      > > > Multiples of 11 seem to turn up alot in a,b. Also |a-b| is
      often
      > a
      > > > multiple of 11.
      > > > |b-a| never seems to be 10 ?
      > > > |b-a| is 9 only once?
      > > > If k is of the form 4n + 4, a and b are even. If k is of the
      form
      > > 4n
      > > > + 2, a and b are odd.
      > > >
      > > >
      > > > Primes of the form 2^n*3^m + 1, k = n+m ; (n,m)
      > > >
      > > > k = 1: (1,0)
      > > > k = 2: (2,0) (1,1)
      > > > k = 3: (2,1) (1,2)
      > > > k = 4: (4,0) (2,2)
      > > > k = 5: (3,2) (2,3) (1,4)
      > > > k = 6: (5,1) (1,5)
      > > > k = 7: (6,1) (4,3) (1,6)
      > > > k = 8: (8,0) (6,2) (4,4) (2,6)
      > > > k = 9: (8,1) (7,2) (5,4) (4,5)
      > > > k = 10: (7,3) (3,7) (1,9)
      > > > k = 11: (7,4) (3,8)
      > > > k = 12: No primes!
      > > > k = 13: (12,1) (11,2) (6,7) (5,8) (3,10)
      > > > k = 14: (5,9)
      > > > k = 15: (7,8)
      > > > k = 16: (16,0) (14,2) (12,4) (10,6) (4,12) (2,14)
      > > > k = 17: (12,5) (11,6) (8,9) (6,11) (2,15) (1,16)
      > > > k = 18: (13,5) (1,17)
      > > > k = 19: (18,1) (17,2) (16,3) (15,4) (6,13)
      > > > k = 20: (16,4)
      > > > k = 21: (11,10)
      > > > k = 22: (19,3) (15,7)
      > > > k = 23: (20,3) (18,5) (13,10) (12,11) (11,12)
      > > > k = 24: No primes!
      > > > k = 25: (22,3) (21,4) (3,22)
      > > > k = 26: (11,15)
      > > > k = 27: (19,8) (16,11)
      > > > k = 28: (20,8) (4,24)
      > > > k = 29: (26,3) (25,4) (24,5) (17,12)
      > > > k = 30: (23,7) (11,19)
      > > > k = 31: (30,1) (27,4) (21,10) (15,16) (1,30)
      > > > k = 32: (18,14) (14,18) (12,20)
      > > > k = 33: No primes!
      > > > k = 34: (13,21)
      > > > k = 35: (33,2) (27,8) (19,16) (18,17) (11,24) (8,27)
      > > > k = 36: (32,4) (28,8)
      > > > k = 37: (36,1) (30,7) (25,12) (21,16) (17,20) (10,27) (6,31)
      > (5,32)
      > > > k = 38: (21,17) (15,23)
      > > > k = 39: (35,4) (17,22)
      > > > k = 40: (36,4) (28,12) (4,36)
      > > > k = 41: (35,6) (33,8) (32,9) (31,10) (23,18) (21,20) (12,29)
      > (5,36)
      > > > (2,39)
      > > > k = 42: (41,1)
      > > > k = 43: (40,3) (39,4) (36,7) (31,12) (30,13) (28,15) (24,19)
      > > > k = 44: (42,2) (24,20) (18,26) (8,36)
      > > > k = 45: (43,2) (28,17) (19,26) (8,37)
      > > > k = 46: No primes!
      > > > k = 47: (44,3) (36,11) (32,15) (31,16) (7,40) (6,41)
      > > > k = 48: No primes!
      > > > k = 49: (46,3) (37,12) (12,37)
      > > > k = 50: (47,3) (39,11) (11,39)
      > > > k = 51: (43,8) (13,38)
      > > > k = 52: (48,4) (46,6) (44,8) (38,14) (30,22) (22,30) (8,44)
      > > > k = 53: (50,3) (30,23) (26,27) (21,32) (20,33) (9,44) (8,45)
      > (5,48)
      > > > k = 54: (49,5)
      > > > k = 55: (19,36) (17,38) (3,52) (1,54)
      > > > k = 56: (40,16) (34,22) (12,44) (8,48)
      > > > k = 57: (29,28)
      > > > k = 58: (9,49) (1,57)
      > > > k = 59: (56,3) (53,6) (50,9) (43,16) (39,20) (29,30)
      > > > k = 60: No primes!
      > > > k = 61: (53,8) (40,21) (25,36) (3,58) (1,60)
      > > > k = 62: (59,3) (15,47)
      > > > k = 63: (55,8) (23,40) (19,44) (8,55)
      > > > k = 64: (38,26) (22,42)
      > > > k = 65: (63,2) (14,51)
      > > > k = 66: (1,65)
      > > > k = 67: (66,1) (65,2) (64,3) (52,15) (48,19) (35,32) (31,36)
      > > (15,52)
      > > > (12,55) (11,56) (6,61)
      > > > k = 68: (56,12) (50,18) (14,54) (8,60)
      > > > k = 69: (67,2) (64,5) (19,50)
      > > > k = 70: (19,51)
      > > > k = 71: (63,8) (44,27) (42,29)
      > > > k = 72: No primes!
      > > > k = 73: (66,7) (65,8) (54,19) (52,21) (28,45) (21,52)
      > > > k = 74: No primes!
      > > > k = 75: (68,7) (59,16)
      > > > k = 76: (16,60) (14,62) (8,68)
      > > > k = 77: (69,8) (65,12) (24,53) (20,57)
      > > > k = 78: (19,59)
      > > > k = 79: (72,7) (57,22) (35,44) (28,51) (5,74) (3,76)
      > > > k = 80: No primes!
      > > > k = 81: (56,25) (37,44) (13,68) (4,77)
      > > > k = 82: (75,7) (69,13) (55,27)
      > > > k = 83: (81,2) (78,5) (61,22) (54,29) (43,40) (42,41) (35,48)
      > > (12,71)
      > > > k = 84: (44,40) (8,76)
      > > > k = 85: (54,31) (11,74)
      > > > k = 86: (51,35) (21,65)
      > > > k = 87: (76,11) (52,35)
      > > > k = 88: (50,38)
      > > > k = 89: (37,52) (19,70) (18,71)
      > > > k = 90: (73,17)
      > > > k = 91: (66,25) (15,76)
      > > > k = 92: (54,38) (50,42) (38,54)
      > > > k = 93: (89,4) (83,10) (80,13)
      > > > k = 94: (75,19)
      > > > k = 95: (92,3) (74,21) (14,81)
      > > > k = 96: No primes!
      > > > k = 97: (54,43) (49,48) (45,52) (30,67) (9,88)
      > > > k = 98: (11,87)
      > > > k = 99: (79,20) (65,34)
      > > > k = 100: (34,66)
      > > > k = 100: (34,66)
      > > > k = 101: (73,28) (72,29) (65,36) (61,40) (50,51) (24,77)
      (11,90)
      > > > k = 102: No primes!
      > > > k = 103: (93,10) (89,14) (88,15) (61,42) (60,43) (41,62)
      (22,81)
      > > > k = 104: (36,68) (10,94)
      > > > k = 105: (67,38) (64,41)
      > > > k = 106: (61,45) (27,79)
      > > > k = 107: (9,98)
      > > > k = 108: (104,4)
      > > > k = 109: (99,10) (90,19) (89,20) (47,62) (19,90) (17,92)
      > > > k = 110: (93,17) (87,23) (51,59) (39,71) (17,93)
      > > > k = 111: No primes!
      > > > k = 112: (94,18) (80,32) (58,54) (46,66) (44,68) (32,80)
      > > > k = 113: (60,53) (41,72) (25,88)
      > > > k = 114: (77,37)
      > > > k = 115: (58,57)
      > > > k = 116: (90,26) (88,28) (44,72) (42,74) (38,78)
      > > > k = 117: (101,16) (92,25) (59,58) (37,80) (20,97) (19,98)
      (5,112)
      > > > k = 118: No primes!
      > > > k = 119: (79,40) (60,59) (19,100)
      > > > k = 120: No primes!
      > > > k = 121: (82,39) (67,54) (57,64) (18,103)
      > > > k = 122: (111,11) (39,83) (35,87)
      > > > k = 123: (52,71)
      > > > k = 124: (110,14) (102,22) (58,66) (34,90) (20,104) (10,114)
      > > > k = 125: (121,4)
      > > > k = 126: (85,41)
      > > > k = 127: (108,19) (97,30) (96,31) (90,37) (70,57) (47,80)
      (36,91)
      > > > (6,121)
      > > > k = 128: (110,18) (106,22) (98,30) (68,60) (36,92) (32,96)
      > > > k = 129: (125,4) (77,52)
      > > > k = 130: No primes!
      > > > k = 131: (115,16) (103,28) (96,35) (53,78) (36,95) (24,107)
      > > > k = 132: No primes!
      > > > k = 133: (66,67) (6,127) (3,130) (1,132)
      > > > k = 134: (69,65)
      > > > k = 135: (97,38) (44,91) (23,112)
      > > > k = 136: (134,2) (88,48) (56,80) (40,96) (26,110) (10,126)
      > > > k = 137: (109,28) (80,57) (48,89) (30,107) (12,125)
      > > > k = 138: (95,43) (49,89)
      > > > k = 139: (100,39) (93,46) (87,52) (67,72) (40,99) (37,102)
      > (35,104)
      > > > k = 140: (116,24)
      > > > k = 141: No primes!
      > > > k = 142: No primes!
      > > > k = 143: (123,20) (103,40) (86,57) (81,62) (73,70) (72,71)
      > (24,119)
      > > > k = 144: No primes!
      > > > k = 145: (127,18) (126,19) (106,39) (82,63) (77,68) (28,117)
      > > > k = 146: (65,81) (3,143)
      > > > k = 147: No primes!
      > > > k = 148: (108,40) (18,130)
      > > > k = 149: (144,5) (133,16) (123,26) (120,29) (107,42) (85,64)
      > > (67,82)
      > > > k = 150: (127,23)
      > > > k = 151: (132,19) (123,28) (111,40) (108,43) (81,70) (66,85)
      > > (63,88)
      > > > k = 152: (148,4)
      > > > k = 153: (148,5) (133,20) (61,92) (29,124)
      > > > k = 154: (15,139)
      > > > k = 155: (104,51) (101,54) (73,82) (56,99) (44,111) (29,126)
      > > > k = 156: (152,4) (68,88)
      > > > k = 157: (147,10) (114,43) (108,49) (81,76) (70,87) (48,109)
      > > (25,132)
      > > > k = 158: (83,75)
      > > > k = 159: No primes!
      > > > k = 160: (154,6) (44,116)
      > > > k = 161: (151,10) (137,24) (85,76) (25,136) (24,137)
      > > > k = 162: No primes!
      > > > k = 163: (149,14) (115,48) (70,93) (6,157)
      > > > k = 164: (162,2)
      > > > k = 165: No primes!
      > > > k = 166: No primes!
      > > > k = 167: (152,15) (138,29) (131,36) (107,60) (32,135)
      > > > k = 168: No primes!
      > > > k = 169: (162,7) (114,55) (76,93) (47,122)
      > > > k = 170: (149,21) (143,27) (131,39) (63,107)
      > > > k = 171: (128,43) (92,79) (13,158)
      > > > k = 172: (116,56) (38,134)
      > > > k = 173: (109,64) (89,84) (84,89) (12,161) (6,167)
      > > > k = 174: (35,139) (19,155)
      > > > k = 175: (148,27) (131,44) (124,51) (103,72) (66,109)
      > > > k = 176: (162,14) (80,96) (50,126) (18,158)
      > > > k = 177: (121,56) (76,101) (16,161) (7,170)
      > > > k = 178: (175,3) (55,123)
      > > > k = 179: (116,63)
      > > > k = 180: (8,172)
      > > > k = 181: (135,46) (132,49) (70,111) (30,151) (1,180)
      > > > k = 182: (129,53)
      > > > k = 183: (83,100)
      > > > k = 184: (158,26) (40,144) (26,158)
      > > > k = 185: (177,8) (87,98)
      > > > k = 186: No primes!
      > > > k = 187: (179,8) (135,52) (36,151) (35,152) (12,175) (6,181)
      > > > k = 188: (164,24) (158,30) (152,36)
      > > > k = 189: (145,44)
      > > > k = 190: (189,1) (183,7)
      > > > k = 191: (55,136) (5,186)
      > > > k = 192: (176,16) (160,32)
      > > > k = 193: (174,19) (106,87) (59,134) (24,169) (9,184)
      > > > k = 194: (165,29) (143,51) (119,75) (69,125)
      > > > k = 195: (136,59) (49,146)
      > > > k = 196: (142,54) (122,74) (110,86) (104,92) (92,104)
      > > > k = 197: (161,36) (44,153) (29,168) (21,176)
      > > > k = 198: (175,23) (119,79) (73,125) (23,175)
      > > > k = 199: (160,39) (147,52) (132,67) (4,195)
      > > > k = 200: No primes!
      > > >
      > > >
      > > >
      > > > --- In primenumbers@yahoogroups.com, "eharsh82" <harsh@u...>
      > wrote:
      > > > > Hi,
      > > > >
      > > > > I came across a new series that produces lots of primes
      > > > >
      > > > > Here is the format of the series
      > > > > (2^x)*(3^y)+-1
      > > > >
      > > > > Where x+y is a constant. (x+y=c)
      > > > >
      > > > > I had a few conjectures concerning this series, please let me
      > > know
      > > > if
      > > > > you can prove or disprove any. These conjectures are valid
      for
      > > the –
      > > > 1
      > > > > and the +1 considered separately and not together.
      > > > >
      > > > > 1)For every c there is at least one prime in the + range.
      > > > >
      > > > > 2)If 1) is not true then there is at least one prime for
      every
      > > even
      > > > c
      > > > > or there is at least one prime for every odd c for the + range
      > > > >
      > > > > 3)If 2) is not true then there is at least one prime for
      every
      > > > prime c
      > > > >
      > > > > These are the conjectures for the + side, similar conjectures
      > can
      > > > be
      > > > > made for the – side also. These conjectures can be extended
      to
      > > the
      > > > +
      > > > > and – side combined.
      > > > >
      > > > > Leaving this these series have a similarity with the
      Goldbach's
      > > > > conjecture. As the value of c rises so does the number of
      > > > candidates
      > > > > and hence the expectancy to find a prime remains the same.
      > > > >
      > > > >
      > > > >
      > > > > Please let me know if you can help me.
      > > > > The first and second case for the + side can be disproved
      using
      > > > > c=12,33
      > > > > The first, second and the third case for the + side can be
      > > > disproved
      > > > > using
      > > > > c=6,149
      > > > > The first and second case for the combined range side can be
      > > > > disproved using
      > > > > c=46,165
      > > > > I don't know what values disprove the third case for the +
      side
      > > and
      > > > > the combined range. I have checked for all primes under 1000.
      (c
      > > is
      > > > a
      > > > > prime)
      > > > >
      > > > >
      > > > > An example ABC file to find c's is
      > > > > ABC2 (2^$a)*(3^(43-$a))-1
      > > > > a: from 1 to 43
      > > > >
      > > > > for c=43
      > > > >
      > > > > Then look in the pfgw.out file for primes or PRP's
      > > > >
      > > > >
      > > > > Thanks,
      > > > > Harsh Aggarwal
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