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Topic: Vindication of Goldbach's conjecture
Replies: 74   Last Post: Aug 9, 2012 6:50 PM

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 Richard Tobin Posts: 1,427 Registered: 12/6/04
Re: Vindication of Goldbach's conjecture
Posted: Jun 22, 2012 4:41 PM

Call the number of ways that even N can be expressed as the sum of two
(unordered) primes g(N). g(N) tends to be much higher when N is a
multiple of 3, because only a third of the "potential pairs" a+b=N are
ruled out by a or b being a multiple of 3, while for other numbers two
thirds are ruled out.

A consequence of that is the it's very rare to find a sequence of more
than three consecutive even numbers whose g(N) is non-increasing.
Here are all of them up to 10^7:

run of 4 starting at 14
run of 4 starting at 22
run of 4 starting at 910
run of 4 starting at 620620
run of 4 starting at 920920
run of 4 starting at 1521520
run of 4 starting at 1701700
run of 4 starting at 1851850
run of 4 starting at 2212210
run of 4 starting at 2662660
run of 4 starting at 2992990
run of 4 starting at 3233230
run of 4 starting at 3803800
run of 4 starting at 4344340
run of 4 starting at 4644640
run of 4 starting at 4944940
run of 4 starting at 5275270
run of 4 starting at 5515510
run of 4 starting at 5785780
run of 4 starting at 6446440
run of 4 starting at 6760390
run of 4 starting at 6806800
run of 4 starting at 7257250
run of 4 starting at 7710010
run of 4 starting at 7797790
run of 4 starting at 8128120
run of 4 starting at 8338330
run of 4 starting at 8518510
run of 4 starting at 8938930
run of 4 starting at 9202270

A suprising set of starting points! Well, not really. The repeated
digits are because they are multiples of 10010. And 10010 is
2x5x7x11x13, so being a multiple of 10010 is (roughly speaking) the
next best thing to being a multiple of 3.

-- Richard

Date Subject Author
6/6/12 mluttgens
6/6/12 Brian Q. Hutchings
6/6/12 GEIvey
6/7/12 Richard Tobin
6/8/12 mluttgens
6/8/12 Count Dracula
6/9/12 mluttgens
6/9/12 Brian Q. Hutchings
6/9/12 mluttgens
6/25/12 GEIvey
6/9/12 Richard Tobin
6/9/12 mluttgens
6/9/12 Richard Tobin
6/9/12 Brian Q. Hutchings
6/9/12 Brian Q. Hutchings
6/14/12 mluttgens
6/16/12 mluttgens
6/16/12 Frederick Williams
6/20/12 mluttgens
6/20/12 Rick Decker
6/21/12 mluttgens
6/21/12 Frederick Williams
6/21/12 mluttgens
6/22/12 mluttgens
6/22/12 mluttgens
6/22/12 Brian Q. Hutchings
6/25/12 Michael Stemper
6/26/12 mluttgens
6/26/12 Frederick Williams
6/28/12 Michael Stemper
7/19/12 mluttgens
7/19/12 Timothy Murphy
7/19/12 mluttgens
7/19/12 Gus Gassmann
7/20/12 mluttgens
8/1/12 Tim Little
8/4/12 mluttgens
8/4/12 Frederick Williams
8/6/12 mluttgens
8/6/12 gus gassmann
8/6/12 Brian Q. Hutchings
8/9/12 Pubkeybreaker
7/19/12 J. Antonio Perez M.
7/20/12 mluttgens
6/25/12 Michael Stemper
6/25/12 Thomas Nordhaus
6/17/12 mluttgens
6/17/12 quasi
6/18/12 Count Dracula
6/18/12 quasi
6/19/12 Count Dracula
6/19/12 quasi
6/20/12 mluttgens
6/22/12 Michael Stemper
6/22/12 mluttgens
6/22/12 Robin Chapman
6/22/12 Michael Stemper
6/23/12 mluttgens
6/22/12 Richard Tobin
6/22/12 Richard Tobin
6/25/12 Richard Tobin
6/25/12 Michael Stemper
6/14/12 Count Dracula
6/21/12 Luis A. Rodriguez
6/21/12 Brian Q. Hutchings
6/21/12 mluttgens
6/25/12 GEIvey
6/20/12 J. Antonio Perez M.