Incorrect guess based on limited data: Number of Primes 6k-1 > Number of Primes 6k+1
 

[COLOR="RoyalBlue"][/COLOR][B]Guess based on limited data[/B]: [U]The number of primes of type 6[I]k[/I]-1 is greater than the number of primes of type 6[I]k[/I]+1.[/U]

Range, [I]π[/I](Range), [COLOR="Red"]Number of Primes 6[I]k[/I]-1[/COLOR], [COLOR="SeaGreen"]Number of Primes 6[I]k[/I]+1[/COLOR], [B][COLOR="royalblue"]Difference[/COLOR][/B]
1,000,000,000, 50,847,534, [COLOR="Red"]25,424,819[/COLOR], [COLOR="SeaGreen"]25,422,713[/COLOR], 25,424,819 - 25,422,713 = [B][COLOR="royalblue"]2,106[/COLOR][/B]
2,000,000,000, 98,222,287, [COLOR="red"]49,112,582[/COLOR], [COLOR="Seagreen"]49,109,703[/COLOR], 49,112,582 - 49,109,703 = [B][COLOR="royalblue"]2,879[/COLOR][/B]
3,000,000,000, 144,449,537, [COLOR="red"]72,226,055[/COLOR], [COLOR="Seagreen"]72,223,480[/COLOR], 72,226,055 - 72,223,480 = [B][COLOR="royalblue"]2,575[/COLOR][/B]
...

[I]Note: Primes 2 and 3 are counted in [I]π[/I](Range).[/I]

There is a consistent slight difference of a few thousand primes.
Is there a simple way to explain said difference?
Also, please share if there are existing references related to this matter.

(* Wolfram code *)
nrange = 3000000000; nmax = PrimePi[nrange]; tpc = 0; tp5 = 0; tp1 = 0; tpother = 0;
n = 1; While[(n <= nmax), p = Prime[n]; tpc++; If[Mod[p, 6] == 5, tp5++;, If[Mod[p, 6] == 1, tp1++;, tpother++;];]; n++];
Print[tpc, ", ", tp5, ", ", tp1, ", ", tpother];

You've rediscovered a very well-known phenomenon called [URL="https://en.wikipedia.org/wiki/Chebyshev%27s_bias"]Chebyshev's bias[/URL]. Assuming some strong versions of the Riemann hypothesis, it is known that for most N, there are more primes of the form 2 mod 3 than 1 mod 3 up to N. 1 mod 3 does sometimes take the lead; this first happens at N = 608981813029. The effect relates to the fact that numbers of the form 1 mod 3 can be squares while those of the form 2 mod 3 cannot. In some sense, it's actually the numbers of *primes and prime powers* (edit: with a suitable scaling on the prime powers) that we should expect to be balanced between the two classes, but it's not easy to explain why. See [URL="https://arxiv.org/pdf/math/0408319v1.pdf"]here[/URL] for an overview of the field.

The ratio (primes 1 mod 3)/(primes 2 mod 3) tends to 1, by de la Vallee Poussin's theorem that for any k the primes are evenly distributed among the residue classes mod k that are coprime to k.

Occasionally, 6k+1 will be exposed to possible factoring by one higher prime than 6k-1 is, as least factor > 1. For example 167 is prime and [TEX]sqrt(167)[/TEX] = 12.9228...; 169=13[SUP]2[/SUP].

In the dim past, I posted [url=https://mersenneforum.org/showpost.php?p=492715&postcount=34]here[/url], and recommended reading [url=http://www.dms.umontreal.ca/~andrew/PDF/PrimeRace.pdf]this paper[/url], which is the same as the one linked to in the preceding post to this thread, though at a different URL.

[QUOTE=charybdis;586526]You've rediscovered a very well-known phenomenon called [URL="https://en.wikipedia.org/wiki/Chebyshev%27s_bias"]Chebyshev's bias[/URL]. Assuming some strong versions of the Riemann hypothesis, it is known that for most N, there are more primes of the form 2 mod 3 than 1 mod 3 up to N. 1 mod 3 does sometimes take the lead; this first happens at N = 608981813029. The effect relates to the fact that numbers of the form 1 mod 3 can be squares while those of the form 2 mod 3 cannot. In some sense, it's actually the numbers of *primes and prime powers* (edit: with a suitable scaling on the prime powers) that we should expect to be balanced between the two classes, but it's not easy to explain why. See [URL="https://arxiv.org/pdf/math/0408319v1.pdf"]here[/URL] for an overview of the field.

The ratio (primes 1 mod 3)/(primes 2 mod 3) tends to 1, by de la Vallee Poussin's theorem that for any k the primes are evenly distributed among the residue classes mod k that are coprime to k.[/QUOTE]
Thanks, [B]charybdis[/B], this was useful. It would take several days to reach 608981813029 on a Raspberry Pi 4B device with a compiled Wolfram script. Out of curiosity, I may continue further for some time to observe the tendency after the reversal point.

The material is nicely presented, however I tend to be careful when I see Granvilles' name (I don't like him much, after many gaffes he did, and after the story with Beal).

Yeah, the two number lines cross infinitely times, however, one "team" is leading "almost" the whole time. Which reminds me of the old joke with two old guys talking on a bench in the park, "how's your sex life these days", "oh, no problem I have sex almost every day", "I don't believe you, I only have it two or three times per year, you are not serious", "of course I am serious, I had sex almost every day", "how come?", "well, Monday I almost had sex, Tuesday I almost had sex, Wednesday I almost had sex, Thursday..."

100+ years old talk to Doctor

-friend of mine tell me, that he can have sex in row at the one night, and I can not, Help me!
-Ok, its very easy to Help! You just can [B]tell[/B] him that You can do the same too!

The original paper by Bays and Hudson (1978, available in JSTOR, see [url]https://www.jstor.org/stable/2006165?refreqid=excelsior%3Ac83e4bdacade0e9fd265734d170a779e[/url]) has no mention of several cases for small primes when the numbers of primes of the two types [I]π[/I][SUB]3,2[/SUB]([I]x[/I]) and [I]π[/I][SUB]3,1[/SUB]([I]x[/I]) are equal.

Actually, [I]π[/I][SUB]3,2[/SUB]([I]x[/I]) = [I]π[/I][SUB]3,1[/SUB]([I]x[/I]) for [I]x[/I] = 2, 3, 7, 13, 19, 37, 43, 79, 163, 223 and 229:
[I]π[/I][SUB]3,2[/SUB](2) = [I]π[/I][SUB]3,1[/SUB](2) = 0 (trivial case)
[I]π[/I][SUB]3,2[/SUB](3) = [I]π[/I][SUB]3,1[/SUB](3) = 0 (trivial case)
[I]π[/I][SUB]3,2[/SUB](7) = [I]π[/I][SUB]3,1[/SUB](7) = 1
[I]π[/I][SUB]3,2[/SUB](13) = [I]π[/I][SUB]3,1[/SUB](13) = 2
[I]π[/I][SUB]3,2[/SUB](19) = [I]π[/I][SUB]3,1[/SUB](19) = 3
[I]π[/I][SUB]3,2[/SUB](37) = [I]π[/I][SUB]3,1[/SUB](37) = 5
[I]π[/I][SUB]3,2[/SUB](43) = [I]π[/I][SUB]3,1[/SUB](43) = 6
[I]π[/I][SUB]3,2[/SUB](79) = [I]π[/I][SUB]3,1[/SUB](79) = 10
[I]π[/I][SUB]3,2[/SUB](163) = [I]π[/I][SUB]3,1[/SUB](163) = 18
[I]π[/I][SUB]3,2[/SUB](223) = [I]π[/I][SUB]3,1[/SUB](223) = 23
[I]π[/I][SUB]3,2[/SUB](229) = [I]π[/I][SUB]3,1[/SUB](229) = 24

It seems that the problem is still open for [I]x[/I] -> Infinity as there is no strict proof.

[QUOTE=LaurV;586540]Yeah, the two number lines cross infinitely times, however, one "team" is leading "almost" the whole time.[/QUOTE]
Indeed, it is unclear (at least to me) what happens at [I]x[/I] -> Infinity.
It could be [I]π[/I][SUB]3,2[/SUB]([I]x[/I]) = [I]π[/I][SUB]3,1[/SUB]([I]x[/I]) + [I]C[/I] for some small non-zero constant [I]C[/I]. Or [I]C[/I]=0?

[QUOTE=Dr Sardonicus;586528]In the dim past, I posted [url=https://mersenneforum.org/showpost.php?p=492715&postcount=34]here[/url], and recommended reading [url=http://www.dms.umontreal.ca/~andrew/PDF/PrimeRace.pdf]this paper[/url], which is the same as the one linked to in the preceding post to this thread, though at a different URL.[/QUOTE]
[QUOTE=Dr Sardonicus;492715]According to [url=http://www.dms.umontreal.ca/~andrew/PDF/PrimeRace.pdf]Prime Races[/url] [which I recommend reading], 1 mod 6 doesn't take the lead until p = 608,981,813,029.[/QUOTE]

The limit of the ratio [I]π[/I][SUB]3,2[/SUB]([I]x[/I]) / [I]π[/I][SUB]3,1[/SUB]([I]x[/I]) would tend to 1 as [I]x[/I] -> Infinity for the infinite number of reversals.
What I am seeking an answer to is if [I]π[/I][SUB]3,2[/SUB]([I]x[/I]) = [I]π[/I][SUB]3,1[/SUB]([I]x[/I]) + [I]C[/I] for some small non-zero constant [I]C[/I].

[QUOTE=Dobri;586573]
[I]π[/I][SUB]3,2[/SUB](2) = [I]π[/I][SUB]3,1[/SUB](2) = 0 (trivial case)
[I]π[/I][SUB]3,2[/SUB](3) = [I]π[/I][SUB]3,1[/SUB](3) = 0 (trivial case)
[I]π[/I][SUB]3,2[/SUB](7) = [I]π[/I][SUB]3,1[/SUB](7) = 1
...[/QUOTE]

Don't forget 2 is a prime of the form 2 mod 3.

[QUOTE=Dobri;586575]Indeed, it is unclear (at least to me) what happens at [I]x[/I] -> Infinity.
It could be [I]π[/I][SUB]3,2[/SUB]([I]x[/I]) = [I]π[/I][SUB]3,1[/SUB]([I]x[/I]) + [I]C[/I] for some small non-zero constant [I]C[/I]. Or [I]C[/I]=0?[/QUOTE]

Littlewood's result from 1914, quoted in Granville and Martin's survey, shows that the difference oscillates from positive to negative infinitely many times, and also takes arbitrarily large positive and negative values.

[QUOTE=charybdis;586577]Don't forget 2 is a prime of the form 2 mod 3.[/QUOTE]
My bad, the Wolfram script is actually testing the [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) and [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) cases.

Therefore, the following correction applies: [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) = [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) for [I]x[/I] = 2, 3, 7, 13, 19, 37, 43, 79, 163, 223 and 229.
[I]π[/I][SUB]6,5[/SUB](2) = [I]π[/I][SUB]6,1[/SUB](2) = 0 (trivial case)
[I]π[/I][SUB]6,5[/SUB](3) = [I]π[/I][SUB]6,1[/SUB](3) = 0 (trivial case)
[I]π[/I][SUB]6,5[/SUB](7) = [I]π[/I][SUB]6,1[/SUB](7) = 1
[I]π[/I][SUB]6,5[/SUB](13) = [I]π[/I][SUB]6,1[/SUB](13) = 2
[I]π[/I][SUB]6,5[/SUB](19) = [I]π[/I][SUB]6,1[/SUB](19) = 3
[I]π[/I][SUB]6,5[/SUB](37) = [I]π[/I][SUB]6,1[/SUB](37) = 5
[I]π[/I][SUB]6,5[/SUB](43) = [I]π[/I][SUB]6,1[/SUB](43) = 6
[I]π[/I][SUB]6,5[/SUB](79) = [I]π[/I][SUB]6,1[/SUB](79) = 10
[I]π[/I][SUB]6,5[/SUB](163) = [I]π[/I][SUB]6,1[/SUB](163) = 18
[I]π[/I][SUB]6,5[/SUB](223) = [I]π[/I][SUB]6,1[/SUB](223) = 23
[I]π[/I][SUB]6,5[/SUB](229) = [I]π[/I][SUB]6,1[/SUB](229) = 24

Perhaps this is helpful for you:

p 1 3 5 mod 6
------------------
10 1 1 1
10^2 11 1 12
10^3 80 1 86
10^4 611 1 616
10^5 4784 1 4806
10^6 39231 1 39265
10^7 332194 1 332383
10^8 2880517 1 2880936
10^9 25422713 1 25424819
10^10 227523123 1 227529386
10^11 2059018668 1 2059036143
10^12 18803933520 1 18803978496
10^13 173032692013 1 173032844824
10^14 1602470745574 1 1602471005226

[QUOTE=bhelmes;586579]Perhaps this is helpful for you:

p 1 3 5 mod 6
...[/QUOTE]
Thanks, this is of limited use as the task is to study the exact locations of the reversal points for which the sign of [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) - [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) flips to the opposite, or eventually [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) - [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) = 0 with or without a subsequent sign reversal for larger primes. There is no need of considering the only prime [I]π[/I][SUB]6,3[/SUB]([I]x[/I]) = 3 for said task.

[QUOTE=charybdis;586577]Littlewood's result from 1914, quoted in Granville and Martin's survey, shows that the difference oscillates from positive to negative infinitely many times, and also takes arbitrarily large positive and negative values.[/QUOTE]
Indeed, the question is what is happening [B]on the average[/B] (rather than at distinct sampling points or limited sampling intervals) when the number of reversal points approaches infinity.

It should be noted that the prime number race [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) vs. [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) is not the same with the race [I]π[/I][SUB]3,2[/SUB]([I]x[/I]) vs. [I]π[/I][SUB]3,1[/SUB]([I]x[/I]) and their respective reversal points might differ.
Therefore, unless a reference is found to prove that the reversal points of the race [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) vs. [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) have been studied in the past, the said race is a new topic.
Thus I do not understand why an anonymous mod had to change the thread icon from 'question' sign to 'minus' sign.

[QUOTE=Dobri;586593]Indeed, the question is what is happening [B]on the average[/B] (rather than at distinct sampling points or limited sampling intervals) when the number of reversal points approaches infinity.[/QUOTE]

It's likely unknown what happens to the average value of C in the long term, as we don't even know that the Chebyshev bias exists without assuming GRH. However, I would be surprised if it tended to any actual limit, as the fluctuations are too irregular. It might even be possible to prove this from known results, I'm not an expert here.

[QUOTE=Dobri;586594]Therefore, unless a reference is found to prove that the reversal points of the race [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) vs. [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) have been studied in the past, the said race is a new topic.[/QUOTE]

This has been investigated. See OEIS [URL="https://oeis.org/A096449"]A096449[/URL].

[QUOTE=charybdis;586595]This has been investigated. See OEIS [URL="https://oeis.org/A096449"]A096449[/URL].[/QUOTE]
The link to OEIS only shows how the mod 3 terms can be rearranged to list the mod 6 terms, it is trivial.
There is no investigation of any kind in OEIS.

What sort of investigation do you want? From a mathematical point of view, there's no real difference between the mod 3 and mod 6 races. Results like Littlewood's and Rubinstein and Sarnak's aren't going to be changed by the omission of the single prime 2.

[QUOTE=charybdis;586597]What sort of investigation do you want? From a mathematical point of view, there's no real difference between the mod 3 and mod 6 races. Results like Littlewood's and Rubinstein and Sarnak's aren't going to be changed by the omission of the single prime 2.[/QUOTE]
There appears to be a significant difference. For instance, the list of [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) = [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) in my corrected post is invalid for the mod 3 case. Changing the sequence interval from 3 to 6 changes the behavior of the prime-counting function.

You get a few extra crossover points by going from mod 3 to mod 6. That's it. Number theorists are not particularly concerned with the exact positions of the crossover points - especially once the first one has been found - but about the long-term trends. Changing the difference by 1 does not affect these. The difference is usually much larger than 1.

[QUOTE=charybdis;586600]You get a few extra crossover points by going from mod 3 to mod 6. That's it. Number theorists are not particularly concerned with the exact positions of the crossover points - especially once the first one has been found - but about the long-term trends. Changing the difference by 1 does not affect these. The difference is usually much larger than 1.[/QUOTE]
This is my humble task, to find the first crossover point for the mod 6 prime number race. Let the number theorists ponder about the long-term trends after that indeed.

[QUOTE=Dobri;586594]Thus I do not understand why an anonymous mod had to change the thread icon from 'question' sign to 'minus' sign.[/QUOTE]Conjecture is too strong of a term for what is really just a basic observation over small numbers, so perhaps that was the impetus for the change. Anyhow, I didn't change it, it is merely guess.

[QUOTE=retina;586602]Conjecture is too strong of a term for what is really just a basic observation over small numbers, so perhaps that was the impetus for the change. Anyhow, I didn't change it, it is merely guess.[/QUOTE]
Please kindly delete "Conjecture" from the title if this would remove also the 'minus' icon.

[QUOTE=Dobri;586601]This is my humble task, to find the first crossover point for the mod 6 prime number race. Let the number theorists ponder about the long-term trends after that indeed.[/QUOTE]

It's at [URL="https://oeis.org/A275939"]608981813017[/URL], immediately before the first crossover in the mod 3 race which occurs at the following prime.

[QUOTE=charybdis;586605]It's at [URL="https://oeis.org/A275939"]608981813017[/URL], immediately before the first crossover in the mod 3 race which occurs at the following prime.[/QUOTE]
It is not explicitly said that [I]x[/I] in [I]π[/I]([I]x[/I]) has to be a prime at a crossover point.

[QUOTE=Dobri;586604]Please kindly delete "Conjecture" from the title if this would remove also the 'minus' icon.[/QUOTE]It isn't just the title, the entire post reinforces the claim.

[QUOTE=Dobri;586607]It is not explicitly said that [I]x[/I] in [I]π[/I]([I]x[/I]) has to be a prime at a crossover point.[/QUOTE]

Leaving aside that the OEIS sequence I linked has an error in the definition (the terms given are q*k+1, not k), how do you expect a crossover to occur at a number that isn't prime?

[QUOTE=retina;586611]It isn't just the title, the entire post reinforces the claim.[/QUOTE]
Apparently, I am unable to edit the initial post anymore after the elapsed time.

[QUOTE=charybdis;586613]Leaving aside that the OEIS sequence I linked has an error in the definition (the terms given are q*k+1, not k), how do you expect a crossover to occur at a number that isn't prime?[/QUOTE]
I meant that after the occurrence of a crossover, it still could be observed for some [I]x[/I] afterwards as the prime count would remain intact.

What is of importance is a relative term in mathematics. One could state that a shift by 1 is irrelevant. But a shift down by one for Mersenneries could mean getting a new prime instead of dealing just with powers of 2.

[QUOTE=Dobri;586619]What is of importance is a relative term in mathematics. One could state that a shift by 1 is irrelevant. But a shift down by one for Mersenneries could mean getting a new prime instead of dealing just with powers of 2.[/QUOTE]

Making this sort of "point" to defend your arguments is likely related to why the thread got the red mark of "no new math in this here conjecture."

Well, at least there is no strict proof about what happens with the prime race at infinity and it seems that since 1978 there was no new attempt to find a second reversal point or even verify the validity of the first one.

With a good eps viewer (I use acdSee) this pari liner will generate in just 4 seconds a very beautiful plot which can be scrolled, panned, and zoomed in. To have an idea how this function looks. It will "go under" an infinite number of times, but how much it will stay under when the primes get really HUGE, nobody knows. In the ranges it was tested, it is mostly positive, as already mentioned.

[CODE]gp > default(realprecision,3); maxprime=5*10^6; v=primepi(maxprime); x=vector(v); y=vector(v); for(i=1,v,x[i]=i); i=0; c1=0; c5=0; forprime(p=5,maxprime,if(p%6==1,c1++,c5++);i++; y[i]=c5-c1); plothraw(x,y)[/CODE]Starting point for "study" if you (non general you) are serious about that.

Edit: of course the "implementation" is far away of being efficient, a sieve would do wonders, hihi, but pari stores a large number of small primes in a vector. You may need "allocatemem()" few times if you want to go larger. But yet, this is only didactic, it is too slow for "large study".

[QUOTE=LaurV;586626]But yet, this is only didactic, it is too slow for "large study".[/QUOTE]
Indeed, the computational task is to reach
maxprime=5*10^(2*6)
and beyond which, interestingly, could be parallelized.

[QUOTE=Dobri;586625]Well, at least there is no strict proof about what happens with the prime race at infinity and it seems that since 1978 there was no new attempt to find a second reversal point or even verify the validity of the first one.[/QUOTE]

A general theme of this thread is that you haven't looked hard enough to see whether something has been investigated before. I can't speak for the moderators but I would guess the :minus: may have something to do with this.

In this case, OEIS is a good place to look: an OEIS search for the first crossover point, 608981813029, might find the sequence of crossover points. Indeed, it gives [URL="https://oeis.org/A007352"]A007352[/URL] as the first result. The b-file tells us that there are over 9000 further crossovers up to 610968213803, at which point 3n+2 takes the lead until 6148171711663 (note this has one more digit than the previous two numbers!).

A Google search for 6148171711663 turns up [URL="http://math101.guru/wp-content/uploads/2018/09/01-A3-Presentation-v7.3EN-no.pdf"]this presentation[/URL]. They verify this value, and presumably the first crossover too, although they claim that there were errors in the search that found it. I assume this means some of the terms in A007352 beyond 6148171711663 are wrong, unless the b-file has been corrected.

I will reach the crossover point in days with two Raspberry Pi 4B devices running the same task. Of course, I could do that much faster with my gaming PC, but I am also testing a custom solar-powered configuration independent of the power grid. One Raspberry Pi 4B device is connected to the power grid and another one to a battery charged by a solar panel. After the testing phase is over and hopefully both devices produce the same result, I will be able to arrange an array of Edge computing devices for prime testing for which there will be no need to pay electricity bills.

The computational experiment was able to reach the first [U]crossover[/U] point [I]x[/I] = 608,981,813,017 ([I]π[/I]([I]x[/I]) = 23,338,590,791) where [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) - [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) = -1.
This is preceded by three [U]equilibrium[/U] points [I]x[/I] = 608,981,812,891, 608,981,812,951, and 608,981,812,993 ([I]π[/I]([I]x[/I]) = 23,338,590,786, 23,338,590,788, and 23,338,590,790) for which [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) - [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) = 0.
It appears that there is an existing OEIS A096629 ([URL]https://oeis.org/A096449[/URL]) description of the [U]equilibrium[/U] points (and a list of 85,508 [U]equilibrium[/U] points up to [I]x[/I] = 6,156,051,951,809 ([I]π[/I]([I]x[/I]) = 216,682,882,516)) with the following description: "Values of [I]n[/I] for which {p_3, p_4, ..., p_n} (mod 3) contains equal numbers of 1's and 2's."
I was able to find it after a Google search with the [I]π[/I]([I]x[/I]) values of equilibrium points obtained with the Raspberry Pi devices.
The current version of the Wolfram code generates consecutive lists of 10,000,000 consecutive primes for which a prime count of [I]π[/I][SUB]6,5[/SUB]([I]x[/I]) and [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) is performed.

The prime number races draw an analogy with the Big Bang of the universe and the existence of more matter than antimatter in the current universe.
What if the unknown fabric of the universe creates new universes with Big Bangs after consecutive equilibrium zero points and in some universes the antimatter dominates over matter?

The maximum difference observed in the entire interval before the first crossover point, [I]π[/I]([I]x[/I]) < 23,338,590,791, is Max[[I]π[/I][SUB]6,5[/SUB]([I]x[/I]) - [I]π[/I][SUB]6,1[/SUB]([I]x[/I])] = [B]47,050[/B] = 2[FONT=&quot]×5[/FONT][FONT=&quot][SUP]2[/SUP]×941.[/FONT]
Correction to post #38: The link to A[OEIS]096629[/OEIS] is [URL]https://oeis.org/A096629[/URL].

The maximum difference Max[[I]π[/I][SUB]6,5[/SUB]([I]x[/I]) - [I]π[/I][SUB]6,1[/SUB]([I]x[/I])] = [B]47,050[/B] = [B]2[/B][FONT=&quot]×[B]5[/B][SUP]2[/SUP][/FONT][FONT=&quot]×[B]941[/B] occurs at [I]x[/I] = [B]457,861,654,499[/B] ([/FONT][FONT=&quot][I]π[/I]([I]x[/I]) = 17,741,340,390).[/FONT]
[FONT=&quot]The primes [B]2[/B], [B]5[/B], [B]941[/B], and [/FONT][FONT=&quot][FONT=&quot][B]457,861,654,499 [/B][/FONT]are [/FONT][FONT=&quot]congruent to [B]2 (mod 3)[/B] thus [/FONT][FONT=&quot][FONT=&quot]5, 941, and [/FONT][/FONT][FONT=&quot][FONT=&quot][FONT=&quot]457,861,654,499[/FONT] are of the type [/FONT][FONT=&quot][B]6[I]k[/I]-1 = [/B][B]3(2[I]k[/I]-1)+2[/B][/FONT].[/FONT]
[FONT=&quot]Therefore, said primes are also [/FONT][FONT=&quot]Eisenstein primes with zero imaginary part (see [URL]https://mathworld.wolfram.com/EisensteinPrime.html[/URL]).
[/FONT]
[FONT=&quot]
[/FONT]

As the computational experiment continues beyond the first crossover point (and multiple subsequent crossovers are encountered), the observed minimum Min[[I]π[/I][SUB]6,5[/SUB]([I]x[/I]) - [I]π[/I][SUB]6,1[/SUB]([I]x[/I])] is equal to -1,539 = -3[SUP]4[/SUP]×19 so far but its value may change with the future updates on this prime number race.

The following local maxima were observed:
[I]π[SUB]6,5[/SUB][/I]([I]x[/I]) - [I]π[SUB]6,1[/SUB][/I]([I]x[/I]) = 47,716 = [FONT=&quot]2[SUP]2[/SUP]×79×151[/FONT][FONT=&quot], [I]x[/I] = 683,008,329,317, [/FONT][FONT=&quot][I]π[/I]([I]x[/I]) = 26,060,805,816;[/FONT]
[I]π[/I][SUB]6,5[/SUB]([I]x[/I]) - [I]π[SUB]6,1[/SUB][/I]([I]x[/I]) = 48,542 = [FONT=&quot]2×13×1,867[/FONT][FONT=&quot], [I]x[/I] = 684,347,039,021, [/FONT][FONT=&quot][I]π[/I]([I]x[/I]) = 26,109,930,072;[/FONT]
[I]π[/I][SUB]6,5[/SUB]([I]x[/I]) - [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) = 48,591 = [FONT=&quot]3[SUP]2[/SUP]×5,399[/FONT][FONT=&quot], [I]x[/I] = 684,349,485,899, [/FONT][FONT=&quot][I]π[/I]([I]x[/I]) = 26,110,020,031; and
[/FONT][I]π[/I][SUB]6,5[/SUB]([I]x[/I]) - [I]π[/I][SUB]6,1[/SUB]([I]x[/I]) = [B]48,910[/B] = [FONT=&quot]2×5×67×73[/FONT][FONT=&quot], [I]x[/I] = [B]684,706,312,967[/B], [/FONT][FONT=&quot][I]π[/I]([I]x[/I]) = [B]26,123,111,894[/B].[/FONT]

[FONT=&quot][FONT=&quot]The primes 2, 5, and 5,399 [/FONT][FONT=&quot]are [/FONT][FONT=&quot]congruent to 2 (mod 3), then 13, 67, 73, 79, 151, and 1, 867 are [/FONT][/FONT][FONT=&quot][FONT=&quot][FONT=&quot][FONT=&quot]congruent to 1 (mod 3), and 3 is [/FONT][/FONT][/FONT][/FONT][FONT=&quot][FONT=&quot][FONT=&quot][FONT=&quot]congruent to 0 (mod 3).[/FONT][/FONT][/FONT][/FONT]
[FONT=&quot]
[/FONT]The observed difference [B]48,910[/B] is the largest one since the computational experiment was initiated.

Well, isn't it fun racing the race that already finished in [URL="https://www.ams.org/journals/mcom/1978-32-142/S0025-5718-1978-0476616-X/S0025-5718-1978-0476616-X.pdf"]1976[/URL] and reporting results as if they were happening right now, with great urgency, with minute-by-minute updates and all?

Hunter S. Thompson you are not, Sir. [SPOILER]This is no Las Vegas, ...only fear and loathing.[/SPOILER]

P.S. Your "blogging" belongs in [URL="https://mersenneforum.org/forumdisplay.php?f=136"]Blogorrhea[/URL] section (where the blogger is the only one reading). Blog as much as you want.

Please share if knowing about a reference to a previous study investigating [U]computationally[/U] the exact local [U]maxima[/U] and [U]minima[/U] of the Δ[I]π[/I] differences of prime number races.
I am aware about a few papers and several OEIS sequences concerned with Δ[I]π[/I] = -1, 0, and 1 differences only.

[U]Lemma 1[/U]: The prime-count distance[I] π[/I]([I]x[/I][SUB]2[/SUB]) [I]- π[/I]([I]x[/I][SUB]1[/SUB]) between two primes [I]x[/I][SUB]1[/SUB] and [I]x[/I][SUB]2[/SUB], [I]x[/I][SUB]1[/SUB] < [I]x[/I][SUB]2[/SUB], for which Δ[I]π[/I]([I]x[/I][SUB]1[/SUB]) = Δ[I]π[/I]([I]x[/I][SUB]2[/SUB]) = 0 (two consecutive zeros without other zeros in between) in a binary prime number race of two prime types is an even number, [I]π[/I]([I]x[/I][SUB]2[/SUB]) [I]- π[/I]([I]x[/I][SUB]1[/SUB]) = 2[I]k[/I].
Proof: The number of prime-count steps in one direction must be equal to the number of prime-count steps in the opposite direction of the binary prime number race so that starting from Δ[I]π[/I]([I]x[/I][SUB]1[/SUB]) = 0 the staircase approximation of distinct prime-count steps has to end up in Δ[I]π[/I]([I]x[/I][SUB]2[/SUB]) = 0 for which an even number of steps is required.

[U]Lemma 2[/U]: The absolute value of an extremum Δ[I]π[/I]([I]x[SUB]m[/SUB][/I]), [I]x[/I][SUB]1[/SUB] < [I]x[SUB]m[/SUB][/I] < [I]x[/I][SUB]2[/SUB], between two primes [I]x[/I][SUB]1[/SUB] and [I]x[/I][SUB]2[/SUB], [I]x[/I][SUB]1[/SUB] < [I]x[/I][SUB]2[/SUB], for which Δ[I]π[/I]([I]x[/I][SUB]1[/SUB]) = Δ[I]π[/I]([I]x[/I][SUB]2[/SUB]) = 0 (two consecutive zeros without other zeros in between) in a binary prime number race is less than or equal to ([I]π[/I]([I]x[/I][SUB]2[/SUB]) [I]- π[/I]([I]x[/I][SUB]1[/SUB]))/2.
Proof: The absolute value of the prime count of the extremum cannot exceed [I]k[/I] in Lemma 1.
In the [U]ideal[/U] case with only one extremum, the absolute value of said extremum (minimum or maximum) = [I]k[/I].

Therefore, the knowledge of the distribution of consecutive zeros Δ[I]π[/I]([I]x[/I]) = 0 can be used to estimate in first approximation the Max[Δ[I]π[/I]([I]x[/I])] distribution and plot an [U]ideal[/U] bar graph (assuming a single extremum between consecutive zero Δ[I]π[/I]([I]x[/I]) points).

The Littlewood theorem states that Δ[I]π[/I]([I]x[/I]) can take arbitrarily large values toward infinity but does not explicitly specify how frequently this could happen in comparison with other Δ[I]π[/I]([I]x[/I]) values. Therefore, analyzing the prime number race results from the past, and also performing recounts to obtain the exact local minima and maxima distributions, does matter.

An [U]ideal[/U] bar graph (assuming a single extremum between two consecutive zero points) on the basis of the 85,508 zero Δ[I]π[/I]([I]x[/I]) points in A[URL="https://oeis.org/A096629"]096629[/URL] can be plotted to show in first approximation the shape of the distribution of the extrema of Δ[I]π[/I]([I]x[/I]).

Whoever modified the title of this thread, how do you know that the guess is incorrect?

[QUOTE=Batalov;588371]P.S. Your "blogging" belongs in [URL="https://mersenneforum.org/forumdisplay.php?f=136"]Blogorrhea[/URL] section (where the blogger is the only one reading). Blog as much as you want.[/QUOTE]


You abuse your power systematically. Attempting to intimidate and belittle ordinary users, you just reveal your true self.

[quote=Dobri;586524][u]The number of primes of type 6k-1 is greater than the number of primes of type 6k+1.[/u][/quote]
[quote=Dobri;588384]Whoever modified the title of this thread, how do you know that the guess is incorrect?[/quote]
[color=red]2. 5 points for every statement that is clearly vacuous, logically inconsistent, or widely agreed on to be false.
3. 10 points for each such statement that is adhered to despite careful correction.[/color]
[quote=Dobri;588380]The Littlewood theorem states that Δπ(x) can take arbitrarily large values toward infinity but does not explicitly specify how frequently this could happen in comparison with other Δπ(x) values. Therefore, analyzing the prime number race results from the past, and also performing recounts to obtain the exact local minima and maxima distributions, does matter.[/quote]
[color=red]7. 10 points for stating that your ideas are of great financial, theoretical and/or spiritual value.[/color]
[quote=Dobri;588387]You abuse your power systematically. Attempting to intimidate and belittle ordinary users, you just reveal your true self.[/quote]
[color=red]26. 20 points for each complaint that the list moderators are out to get you or are blocking your substantial and useful posts.[/color]

That's 45 points on The PrimePages' Crackpot index right there.

[QUOTE=Dr Sardonicus;588392][COLOR=red]2. 5 points for every statement that is clearly vacuous, logically inconsistent, or widely agreed on to be false.
3. 10 points for each such statement that is adhered to despite careful correction.[/COLOR]
[COLOR=red]7. 10 points for stating that your ideas are of great financial, theoretical and/or spiritual value.[/COLOR]
[COLOR=red]26. 20 points for each complaint that the list moderators are out to get you or are blocking your substantial and useful posts.[/COLOR]
That's 45 points on The PrimePages' Crackpot index right there.[/QUOTE]

Do you imply that the the initial guess is false, and you arrived at this conclusion after a careful consideration?

Then you must be able to provide tangible evidence, preferably works published by others, to support your careful consideration, and explain yourself clearly without speaking in riddles or giving 'homework'.

Please do not put words in my mouth as it was never stated in this thread that the "ideas are of great financial, theoretical and/or spiritual value."

If you are wrong about points [COLOR=Red]#2[/COLOR], [COLOR=red]#3[/COLOR], and [COLOR=red]#7[/COLOR], then adding point [COLOR=red]#26[COLOR=Black] does not make sense.[/COLOR][/COLOR]

[QUOTE=Dobri;588400]Do you imply that the the initial guess is false, and you arrived at this conclusion after a careful consideration?
<snip>[/QUOTE][i]Objection! Already asked and answered![/i]

You have already publicly acknowledged references proving your guess is wrong that were offered, e.g. in [url=https://mersenneforum.org/showpost.php?p=586526&postcount=2]this post[/url] and [url=https://mersenneforum.org/showpost.php?p=586577&postcount=11]this post[/url] to this thread.

Objection sustained. I find you guilty of trolling, and will see about imposing a 1-month timeout. :judge:

[FONT=&quot]This update contains images of the values of the absolute [COLOR=Red]minima[/COLOR] and [COLOR=Red]maxima[/COLOR] [/FONT][FONT=&quot][FONT=&quot](up to a prime [I]x[/I] on the horizontal axis) [/FONT]of the prime race ∆([I]x[/I]) = [/FONT][I][FONT=&quot]𝜋[/FONT][/I][FONT=&quot][SUB]6,5([/SUB][I]x[/I]) - [/FONT][I][FONT=&quot]𝜋[/FONT][/I][FONT=&quot][SUB]6,1[/SUB]([I]x[/I]) for the first 185,000,000,000 primes.[/FONT]
[FONT=&quot]The ongoing exhaustive computation is getting closer to the next equilibrium point [/FONT][FONT=&quot]∆([I]x[/I]) = 0 listed in OEIS A[/FONT][OEIS]096449[/OEIS][FONT=&quot] (see post #38, [URL]https://mersenneforum.org/showpost.php?p=588100&postcount=38[/URL]).[/FONT]

OEIS sequences references of your problem:

[URL="https://oeis.org/A007350"]https://oeis.org/A007350[/URL]
[URL="https://oeis.org/A007352"]https://oeis.org/A007352[/URL]
[URL="https://oeis.org/A199547"]https://oeis.org/A199547[/URL]
[URL="https://oeis.org/A306891"]https://oeis.org/A306891[/URL]
[URL="https://oeis.org/A038698"]https://oeis.org/A038698[/URL]
[URL="https://oeis.org/A112632"]https://oeis.org/A112632[/URL]
[URL="https://oeis.org/A066520"]https://oeis.org/A066520[/URL]
[URL="https://oeis.org/A321856"]https://oeis.org/A321856[/URL]
[URL="https://oeis.org/A275939"]https://oeis.org/A275939[/URL]
[URL="https://oeis.org/A326615"]https://oeis.org/A326615[/URL]
[URL="https://oeis.org/A306499"]https://oeis.org/A306499[/URL]
[URL="https://oeis.org/A306500"]https://oeis.org/A306500[/URL]

In the attached image, the lowest smooth orange curve is the plot of [I]x[/I][SUP]1/2[/SUP]ln(ln(ln([I]x[/I])))/(2*ln([I]x[/I])).
The gray smooth curve above it is the plot of [I]Kx[/I][SUP]1/2[/SUP]ln(ln(ln([I]x[/I])))/(2*ln([I]x[/I])) with an empirically chosen coefficient [I]K[/I] = 28.
Correction to my previous post #52, [URL]https://mersenneforum.org/showpost.php?p=605876&postcount=52[/URL]: The OEIS sequence is A[URL="https://oeis.org/A096629"]096629[/URL].