[QUOTE=paulunderwood;588860]Congrats to James Winskill for the mega primorial prime: [URL="https://primes.utm.edu/primes/page.php?id=132758"]3267113# - 1[/URL] (1,418,398 decimal digits).

:banana:[/QUOTE]

Yes, in the last few days we have had two new categories of primes entering the megaprime territory. A [URL="https://en.wikipedia.org/wiki/Palindromic_prime"]Palindromic[/URL] with 1,234,567 digits and this [URL="https://en.wikipedia.org/wiki/Primorial_prime"]Primorial[/URL] with 1,418,398 digits. The next one coming is probably the 3[SUP]rd[/SUP] term of a [URL="https://en.wikipedia.org/wiki/Primes_in_arithmetic_progression"]Prime in A.P.
[/URL]
We now have close to 1,125 megaprimes

[QUOTE=rudy235;588876]...A [URL="https://en.wikipedia.org/wiki/Palindromic_prime"]Palindromic[/URL] with 1,234,567 digits and ...[/QUOTE]How about [I]two [/I]of them? :rolleyes:

[QUOTE=Batalov;588934]How about [I]two [/I]of them? :rolleyes:[/QUOTE]

You mean "two more"?

[url]https://primes.utm.edu/primes/page.php?id=132766[/url]
[url]https://primes.utm.edu/primes/page.php?id=132767[/url]

:banana: :banana:

How difficult is to prove a primorial Prime?


[URL="https://primes.utm.edu/primes/page.php?id=132758"]3267113# - 1 [/URL]

[B][COLOR="Navy"]Verification status (*): InProcess[/COLOR][/B]

Is still unproven. I would think that having the primorial +1 100% factored would make proving it a matter of a couple of says. A week in the worse case.

[QUOTE=rudy235;590778]How difficult is to prove a primorial Prime?


[URL="https://primes.utm.edu/primes/page.php?id=132758"]3267113# - 1 [/URL]

[B][COLOR="Navy"]Verification status (*): InProcess[/COLOR][/B]

Is still unproven. I would think that having the primorial +1 100% factored would make proving it a matter of a couple of says. A week in the worse case.[/QUOTE]

Some numbers require proof attempts at increasing sizes of FFT.

Two birds with one stone
 

Congrats tp Ryan and Serge for the record Near-rep Digit / Palindrome prime [URL="https://primes.utm.edu/primes/page.php?id=132851"]10^1888529 - 10^944264 - 1[/URL]

[QUOTE=paulunderwood;591030]Congrats tp Ryan and Serge for the record Near-rep Digit / Palindrome prime [URL="https://primes.utm.edu/primes/page.php?id=132851"]10[SUP]1888529[/SUP] - 10[SUP]944264[/SUP] - 1[/URL][/QUOTE]
Yet another custom sieve for such hybrid beasts:
quick sketch:

We are searching for NRP(K,n) = 10[SUP]2n+1[/SUP]-K*10[SUP]n[/SUP]-1. K can only be 1,2,4,5,7,8. (K=3 has algebraic factorization, which is not needed ...because the whole expression is divisible by 3 when 3|K).

Step 1. Let x=10^n, then NRP(K,n) = 10x[SUP]2[/SUP]-Kx-1 . I solve this quadratic equation just like in school but x is some Mod(x,p) then sieve by p

Step 2. If quadratic equation has solution (nearly half the time; if it doesn't , nothing to sieve out), then --

Step 3. Solve 10^n = x[SUB]1[/SUB] and 10^n = x[SUB]2[/SUB]. This is called znlog() and these values will periodically repeat with period znorder().

Step 4. Sieve out and repeat for 7<= p <= 10^11 or 10^12.

Step 5: remove special cases for p={7,11,13} (this actually removes a huge fraction of candidates with K=2, that's why [URL="https://stdkmd.net/nrr/9/99799.htm"]it is the "thinnest" K[/URL])

The trick is to code steps 1, 2 and 3, and to know how.

Step 6. Test. (we test all six number forms in order of size. The fact that K=1 produced the first hit is accidental. With K=1, the number looks a bit more elegant.)

[QUOTE=rudy235;590778]How difficult is to prove a primorial Prime?


[URL="https://primes.utm.edu/primes/page.php?id=132758"]3267113# - 1 [/URL]

[B][COLOR="Navy"]Verification status (*): InProcess[/COLOR][/B]

Is still unproven. I would think that having the primorial +1 100% factored would make proving it a matter of a couple of days. A week in the worse case.[/QUOTE]

After 20 days (10/17/21) it was proven prime.

[QUOTE=Batalov;591031]Step 5: remove special cases for p={7,11,13} (this actually removes a huge fraction of candidates with K=2, that's why [URL="https://stdkmd.net/nrr/9/99799.htm"]it is the "thinnest" K[/URL])[/QUOTE]

You mean that the [URL="https://www.rieselprime.de/ziki/Nash_weight"]Nash weight[/URL] (or [URL="https://stdkmd.net/nrr/prime/primedifficulty.txt"]difficulty[/URL]) for K=2 (999...9997999...999) is very low?

[QUOTE=sweety439;591120]You mean that the [URL="https://www.rieselprime.de/ziki/Nash_weight"]Nash weight[/URL] ...[/QUOTE]
Dare you to define it (for these six sequences), but yes.
[QUOTE=sweety439;591120]... (or [URL="https://stdkmd.net/nrr/prime/primedifficulty.txt"]difficulty[/URL]) for K=2 (999...9997999...999) is very low?[/QUOTE]
Dare you to define it (for these six sequences), but yes.

If I may ask how many candidates remain after that ?