Primeness of bases

[gd_barnes]

OK, I just got done playing around with the spreadsheet. Using the aforemention method, it comes in pretty close. Here is what I did:

1. Divided your # of candidates of 4236 by 10 to get 424.
2. Plugged in n=50, 150, 250, etc. up to n=950 into the n-value for all 10 ranges.
3. For each of the n-values in #2, I entered the # of expected primes into a separate column.

Adding up the separate column gave a total expected primes of ~605. Dividing by 4236 gives 14.28%. I'm surprised it's that close to your ~16% for 2 reasons:
1. The spreadsheet doesn't work well at low n-values.
2. The % decerease in primes is very high at this lowest of all possible ranges.

I'll try one more thing: For n=50 and 424 candidates, it came up with 270.492 expected primes. Becauase the ratio of top to bottom n is still very high for this smallest of ranges, I'll subdivide n=1 to 100 into 10 ranges of n=1 to 10, 11 to 20, etc. (using n=5, 15, 25, etc.) and leave the rest of the ranges as is. Now, what is strange about this is that for n=1 to 40, ALL candidates should be prime! Why? Because 2^40 ~= 1M^2. That means for candidates <= 2^40, their largest possible factor is ~1M.

In checking this on the spreadsheet, there is actually a much smaller error than I would have expected. Up to n=~30, it shows the same # of primes as candidates but starts to drop below the # of candidates for n>30. Technically it should stay the same as the # of candidates up to n=40. But that's close enough for our reference. Here is what I come up with:

287.562 expected primes for n=1 to 100. Adding it to the rest of the n=100 ranges expected primes gives a total expected primes of ~622.

622 is 14.68% of 4236.

The ~1-1.5% error from what you are getting is now virtually solely as a result of the spreadsheet not quite being able to handle extremely low n-ranges perfectly as demonstarted by the 2^30 and 2^40 example above.

I did some more tweaking of the ranges and was able to get it to > 15%; far closer to actual than I would have expected.

If you were to do this same test for 2^1024 thru 2^2048 instead of 1 thru 2^1024, I think this would come in quite accurately.

Edit: One more thing: It's fairly clear that there is no statistical significance in the differences in the percentages amongst the bases. To demonstrate it, I would suggest an "out-of-sample" test. That is, try the exact same test for 2^1024 thru 2^2048. Then see what the mathematical correlation is between the 2 sets of percentages. If that correlation is near zero or is negative, then there is nothing that can be gleaned by this.


Gary

[henryzz]

I will do some testing from n=1000-2000.
I have found a huge bug when i start from an n!=0

[henryzz]

Originally Posted by henryzz

I will do some testing from n=1000-2000.
I have found a huge bug when i start from an n!=0

I think this bug is now fixed(along with some others but the previous results won't be more that 1 out at any point and that will be candidates not primes).
Testing has started. I will post once several bases are done.

[henryzz]

n=1001-2000 is a bit all over the place because there is only 10% of the primes we had last time.
I am going to search ks from 1-500 for this range instead of 1-100 to lose some of the randomness.

[gd_barnes]

BTW, the reference that I frequently make about a base being "more prime" than another refers to the average weight of k's in the base. That is the # of candidates remaining after sieving to a set depth. It does not refer to the k's magically being found prime at a higher rate after doing sieving to a usual depth.

Therefore I feel this excercise will not bare any fruit in the same way that the first twin k base 2 for each n excercise at TPS was pointless other than entertainment for those involved.

Some bases are much "more prime" than others because their k's have far fewer small factors than others. It doesn't mean that the remaining candidates after sieving are any more likely to be prime.

Some known facts:

1. Bases that are 2^q-1 are among the heaviest weight bases.

2. Bases where b+1 has many small factors > 2 will usually have more difficulty finding primes for many of their k's. For the 2^q-1 bases, since b+1 only has a factor of 2, then they have the heaviest weight.

3. Find all of the factors > 2 of b+1 and call the factors f(x). Include in such factorization b+1 itself. The lowest weight k's base b are k's where k==[1 mod f(x)] and k==[(f(x)-1) mod f(x)].

A good example on #3 is base 19 on either side. The factorization of b+1, i.e. 20, is 2*2*5 so the only factor > 2 is 5. If you look at base 19, you'll see that almost all k's remaining end in a 4 of 6. Why is that? It's certainly possible to have k's remaining ending in a 0, 2, or 8 and there are but they are extremely rare. This is because k's ending in a 4 or 6 are (1 mod 5) or (4 mod 5). The others are more easily found prime.

What happens where k==[1 mod f(x)] or k==[(f(x)-1) mod f(x)] where f(x) > 2 is that every other n-value has a factor of f(x). This effectively halves the weight and causes them to be much more difficult to find a prime for.

What's amazing about this is the percentage of k's that are remaining where f(x) is quite high or "medium" sized. Refer to base 10 on both sides. There are only 3 final k's combined remaining. All 3 of them are k==(1 mod 11) or (10 mod 11) but that's not the amazing part. If you consider the top 10 primes on both sides, then a phenominal 22 out of 23 k's are either k==(1 mod 11) or (10 mod 11). It was base 10 that alerted me to this fact about 2 years ago when I compiled a web page of k*10^n-1 primes.

To further the point: I've also seen factors of b+1 that are 30<f(x)<50 cause over half of all remaining k's base b.

Doing an analysis of the phenomina in #3 would make for much more useful information.


Gary