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Re: Goldbach : Two new conjectures inside the first ?

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  • sopadeajo2001
    ... 43 ... or ... (r) ... (g) ... (g) ... (r) ... (g) ... The answer is that we do not need any experimental checking. Every even number is of the form 4k or
    Message 1 of 2 , Feb 23, 2005
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      --- In primenumbers@yahoogroups.com, "Jean-Pierre Pratali"
      <jpierre.pratali@w...> wrote:
      >
      > According to the Goldbach conjecture every even integer >2 is, by
      > one or several manners, the sum of two primes.
      > I suppose it exists a large table of such decompositions for a wide
      > range of even numbers. Personally I could not find where.
      > So I built, "by hand", a small table for even numbers :
      > ...
      > 14 = 3 + 11 = 7 + 7
      > 16 = 3 + 13 = 5 + 11
      > 18 = 5 + 13 = 7 + 11
      > ...
      > 46 = 3 + 43 = 5 + 41 = 17 + 29 = 23 +23
      > 48 = 5 + 43 = 7 + 41 = 11 + 37 = 17 + 31 = 19 + 29
      > 50 = 3 + 47 = 7 + 43 = 13 + 37 = 19 + 31
      > ...
      > 90 = 3 + 87 = 7 + 83 = 11 + 79 = 17 + 73 = 19 + 71 =
      > 23 + 67 = 29 + 61 = 31 + 59 = 37 + 53 = 43 + 47
      > 92 = 3 + 89 = 13 + 79 = 19 + 73 = 31 + 61
      > 94 = 5 + 89 = 11 + 83 = 23 + 71 = 41 + 53 = 47 + 47
      > 96 = 7 + 89 = 13 + 83 = 17 + 79 = 23 + 73 = 37 + 59 =
      43
      > + 53
      > 98 = 19 + 79 = 31 + 67 = 37 + 61
      >
      > Now let us examine each set of decomposition for each even.
      > We remember that a prime is either a "gaussian" one (-1 mod 4) or
      > a "not gaussian" one (1 mod 4) (they also call it "rational"; it is
      > also the sum of 2 squares )
      >
      > So let us look at the pairs of primes for each even : We can see
      > that
      >
      > 1. the corresponding pairs are either all "homogeneous" (gaussian-
      > gaussian or rational-rational) or all "mixed" (gaussian-rational
      or
      > rational-gaussian).
      >
      > Next, if we consider the set of all the even numbers >4, we can see
      > that
      >
      > 2. a "mixed" (m) always succeeds to a "homogeneous" (h) and
      > inversely.
      >
      > For example (with (g) for "gaussian", (r) for "rational", (h) for
      > homogeneous" and (m) for "mixed" :
      >
      > 46(h) = 3(g) + 43(g) = 5(r) + 41(r) = 17(r) + 29(r) = 23g)
      > + 23(g)
      > 48(m) = 5(r) + 43(g) = 7(g) + 41(r) = 11(g) + 37(r) = 17
      (r)
      > + 31(g) = 19(g) + 29(r)
      > 50(h) = 3(g) + 47(g) = 7(g) + 43(g) = 13(r) + 37(r) = 19
      (g)
      > + 31(g)
      > 52(m) = 5(r) + 47(g) = 11(g) + 41(r) = 23(g) + 29(r)
      > ...
      > 90(h) = 3(g) + 87(g) = 7(g) + 83(g) = 11(g) + 79(g) = 17
      > (r) + 73(r) = 19(g) + 71(g) = 23(g) + 67(g) = 29(r) + 61(r)
      > = 31(g) + 59(g) = 37(r) + 53(r) = 43(g) + 47(g)
      > 92(m) = 3(g) + 89(r) = 13(r) + 79(g) = 19(g) + 73(r) = 31
      (g)
      > + 61(r)
      > 94(h) = 5(r) + 89(r) = 11(g) + 83(g) = 23(g) + 71(g) = 41
      (r)
      > + 53(r) = 47(g) + 47(g)
      > 96(m) = 7(g) + 89(r) = 13(r) + 83(g) = 17(r) + 79(g) = 23
      (g)
      > + 73(r) = 37(r) + 59(g) = 43(g) + 53(r)
      >
      > Does a large table confirm (1) and (2) indefinitely ?



      The answer is that we do not need any experimental checking.

      Every even number is of the form 4k or 4k+2
      If the even number is 4k=4(k1+k2+1)=(4k1+1)+(4k2+3)
      If the even number is 4k+2=4(k3+k4)+2=(4k3+1)+(4k4+1)
      4k+2=4(k5+k6+1)+2=(4k5+3)+(4k6+3)

      So even 4k+2 are always "homogeneous" and even 4k are always "mixed"
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