Re: Goldbach fine line
- Of course if one of your "close" to failing cases is the first of a
prime number pair, your next even number will be a close or better
A real failure will have as a necessary condition the lack of the
above property in the previous pair. A very shallow observation I
- --- In email@example.com, "Mark Underwood"
> I was having some fun putting (rather arbitrary!) restrictions on the
> goldbach conjecture to make it come *close* to failing.
> (Goldbach conjecture: Every even number n > 4 is the sum of two
>It turns out that for GP Pari to work on my mac, I needed to get a C
> For instance what if the smallest prime is restricted to be less than
> (log(n))^2.5 ?
> I checked up to n = 10^7, and here are all the cases where the number
> of solutions did not exceed 2 (Format: n, solution count) :
> 6,1 8,2 12,2 38,2 68,2 128,2 332,2 346,2 548,2 992,2 2642,2
> 29816,2 37952,2 66284,2 133802,2 134972,2 168568,2 194464,2
> 250658,2 256864,2 279362,2 311932,2 395936,2 420218,2 455138,2
> 492368,2 503222,2 643898,2 676892,2 853592,2 880778,2 1077422,2
> 1542776,2 1613174,2 2857214,2 3064088,2 3507104,2 3507344,2
> 4358656,2 6206272,2 8549774,2 9002584,2 9190322,2 9867986,2
> When does it fail? My Windows 98 computer is just too slow for me to
> go very far with this.
> PS If anybody has experience getting GP Pari to work on a new mac, I
> would appreciate knowing how to do so. My attempts thus far have been
> in vain. Thanks!
compiler which will produce programs which could run on the mac. Apple
provides such for free in their huge Xcode programming tools. (One
just has to become a member first, again free.) So I downloaded the
almost 1 gigabyte of Xcode stuff.
I had already downloaded Darwin Ports and Fink, two ways to download
software. Darwin and Fink say I have now Pari 'installed', and I
naively thought it would compile GP Pari automatically. But since I
don't see any Pari program, it appears that I will have to compile it
myself. I'll figure out how to do that in the near future. Thanks to
those who privately offered some assistance.
Back to the subject of making arbitrary restrictions on Goldbach's
conjecture, to make it come close to failing.
Here's a better restriction which walks a tighter line. We restrict
Goldbach such that the lowest prime p1 of
p1 + p2 = n (n even)
is less than (log(n)/log(2.28))^2.28
Up to 10^8, there are 318 cases where there are exactly 2 solutions,
37 cases where there is exactly one solution, and 0 cases where there
are no solutions. But I'm almost sure it will fail at some point. But
it does make me wonder if there is a formula that will yield indefinitely
many very close calls, but yet will not fail. (Not that we can prove
it wouldn't fail: we can't yet even prove Goldbach's conjecture!)
Here are the 37 cases of n where there is only one solution:
6, 8, 12, 168568, 485326, 492368, 503222, 676892, 734648, 1077422,
1542776, 3807404, 6206272, 6352972, 9867986, 10105742, 12033188,
16456228, 17561876, 20578742, 20970616, 21991748, 24106882, 27789878,
33720656, 35211998, 37053182, 37998938, 40622522, 44689544, 47600414,
48071014, 59347322, 60119912, 71445742, 78042182, 98754346.
For instance, (log(98754346)/log(2.28))^2.28 =~ 1190
There is one prime p1 less than 1190 such that p1 + p2 = 98754346.
That is, 149 + 98754197 = 98754346.