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.