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Forum: Dobri 2022-06-09, 10:15
Replies: 29
Views: 3,279
Posted By Dobri
This is a continuation of the previous post #29...

This is a continuation of the previous post #29 at https://www.mersenneforum.org/showpost.php?p=606844&postcount=29.

Number of 2kp+1 primes for p within the range 2 ≤ p ≤ 999,999,937 which...
Forum: Dobri 2022-05-30, 22:54
Replies: 29
Views: 3,279
Posted By Dobri
Number of 2kp+1 primes for p within tne range 2 ≤...

Number of 2kp+1 primes for p within tne range 2 ≤ p ≤ 999,999,937 which divide the composite Mersenne numbers Mp = 2p-1 so that Mp mod (2kp+1) = 0:

k=1: 1655600 (for which p mod 4 = 3 and p mod 6...
Forum: Homework Help 2022-05-30, 22:52
Replies: 4
Views: 1,063
Posted By Dobri
See also the web page on "Euler and Lagrange on...

See also the web page on "Euler and Lagrange on Mersenne Divisors" by Chris K. Caldwell at <https://primes.utm.edu/notes/proofs/MerDiv2.html>.
Forum: Homework Help 2022-05-25, 09:17
Replies: 4
Views: 1,063
Posted By Dobri
For the first known proof, see the following...

For the first known proof, see the following manuscript:
Joseph Louis de Lagrange, Recherches d'arithmétique (1775), pp. 695-795 (in French).
Available:...
Forum: Homework Help 2022-05-24, 23:11
Replies: 4
Views: 1,063
Posted By Dobri
See the following manuscript: Édouard Lucas,...

See the following manuscript:
Édouard Lucas, Théorie des Fonctions Numériques Simplement Périodiques, American Journal of Mathematics, Vol. 1, No. 4 (1878), pp. 184-240 and 289-321 (in French). ...
Forum: Dobri 2022-05-22, 21:18
Replies: 29
Views: 3,279
Posted By Dobri
For k = 5, out of the total of 50,847,534 prime...

For k = 5, out of the total of 50,847,534 prime numbers 2 ≤ p ≤ 999,999,937, there are 408,660 prime exponents (0.8%) with composite Mersenne numbers Mp mod (10p + 1) = 0 for which p mod 4 = 3 and p...
Forum: Dobri 2022-05-18, 14:56
Replies: 53
Views: 6,270
Posted By Dobri
In the attached image, the lowest smooth orange...

In the attached image, the lowest smooth orange curve is the plot of x1/2ln(ln(ln(x)))/(2*ln(x)).
The gray smooth curve above it is the plot of Kx1/2ln(ln(ln(x)))/(2*ln(x)) with an empirically...
Forum: Dobri 2022-05-14, 14:34
Replies: 53
Views: 6,270
Posted By Dobri
This update contains images of the values of the...

This update contains images of the values of the absolute minima and maxima (up to a prime x on the horizontal axis) of the prime race ∆(x) = 𝜋6,5(x) - 𝜋6,1(x) for the first 185,000,000,000 primes....
Forum: Dobri 2022-05-13, 05:43
Replies: 29
Views: 3,279
Posted By Dobri
Excluding the 3rd Mersenne prime number M5 (for...

Excluding the 3rd Mersenne prime number M5 (for which p = 5, p mod 4 = 1, 2p - 1 = 6p + 1 = 27×12 + 22 = 31 is a prime, and also 2p + 1 = 11 is a prime):

1) Out of the total of 50,847,534 prime...
Forum: Lounge 2022-05-12, 14:01
Replies: 50
Views: 2,889
Posted By Dobri
Alexander Grothendieck ...

Alexander Grothendieck
https://en.wikipedia.org/wiki/Alexander_Grothendieck
Forum: Number Theory Discussion Group 2022-05-06, 16:01
Replies: 18
Views: 1,859
Posted By Dobri
The Wolfram code below represents the prime...

The Wolfram code below represents the prime exponent p of each known Mersenne prime 2p-1 as a unique sum of two squares p = a2 + b2 with the exception of the prime exponents p = 4k + 3 for which...
Forum: MattcAnderson 2022-05-02, 21:54
Replies: 6
Views: 446
Posted By Dobri
Marin Mersenne wrote about the birds that ...

Marin Mersenne wrote about the birds that

"...qu'ils se seruent de leurs voix pour exprimer leurs passions naturelles, & non des paroles..."
("...they use their voices to express their...
Forum: Information & Answers 2022-05-02, 15:39
Replies: 11
Views: 1,846
Posted By Dobri
Below is a link to one external thread related to...

Below is a link to one external thread related to the discussion.
https://forums.tomshardware.com/threads/prime95-worker-failing.3457965
Forum: Lounge 2022-04-24, 17:57
Replies: 1,936
RIP
Views: 226,908
Posted By Dobri
My condolences as you grieve, xilman.

My condolences as you grieve, xilman.
Forum: Dobri 2022-04-24, 08:25
Replies: 12
Views: 1,962
Posted By Dobri
For quality assurance (QA) purposes, a stage-1...

For quality assurance (QA) purposes, a stage-1 P-1 test was performed on M726064763 without excluding in quotation marks the 12 known factors.
The result is a 145-digit 481-bit composite factor...
Forum: No Prime Left Behind 2022-04-23, 10:42
Replies: 24
Views: 823
Posted By Dobri
All three pages are accessible with Chromium for...

All three pages are accessible with Chromium for Raspberry Pi 4B.
Forum: Math 2022-04-22, 11:41
Replies: 2
Views: 1,045
Posted By Dobri
Here is a Wolfram code to test the quoted...

Here is a Wolfram code to test the quoted heuristic recursion. :smile:

p = 1277; Mn = 2^p - 1; xn = 3;
While[(GCD[xn - 1, Mn] == 1) && (xn != 0), xn = Mod[(xn^(12*p*5) - 1)*(xn^(12*p*7) - 1) + 1,...
Forum: Dobri 2022-04-20, 12:58
Replies: 29
Views: 3,279
Posted By Dobri
This forms a sequence of composite p values...

This forms a sequence of composite p values indeed,
p = 21, 121, 153, 221, 237, 245, 305, 333, 357, 381, 445, 465, 545, 565, 605, 637, 657, 737, 753, 777, 793, 861, 917,...


kc = 3; ic = 2;...
Forum: Dobri 2022-04-20, 10:19
Replies: 29
Views: 3,279
Posted By Dobri
It should be noted that it is assumed that 6p + 1...

It should be noted that it is assumed that 6p + 1 is a prime number.
If 6p + 1 is not a prime number, there are instances when Mp mod (6p + 1) ≠ 0 and 6p + 1 = 27a2 + b2, for example:
p = 41, (241...
Forum: Dobri 2022-04-20, 09:29
Replies: 29
Views: 3,279
Posted By Dobri
I just connect a Raspberry Pi 4B device to the...

I just connect a Raspberry Pi 4B device to the HDMI port of my 4K TV set and use Wolfram Mathematica for free.
Here is the Wolfram code for the Mersenne primes

MPData = {2, 3, 5, 7, 13, 17, 19,...
Forum: Dobri 2022-04-20, 07:58
Replies: 29
Views: 3,279
Posted By Dobri
There is a typo, it has to be "... for p = 101,...

There is a typo, it has to be "... for p = 101, 137, 173, 181, 241, 257, 277, 293, 313,..."
Forum: Dobri 2022-04-20, 06:48
Replies: 29
Views: 3,279
Posted By Dobri
Excluding p = 5 for which 25 - 1 = 6×5 + 1 =...

Excluding p = 5 for which 25 - 1 = 6×5 + 1 = 27×12 + 22, there are no other counterexamples for p mod 4 = 1 and k = 3 within the limited sample of known Mersenne primes.

Starting from the...
Forum: Dobri 2022-04-20, 05:14
Replies: 29
Views: 3,279
Posted By Dobri
Here is a counterexample: p = 5, p mod 4 = 1, and...

Here is a counterexample: p = 5, p mod 4 = 1, and 2p - 1 = 6p + 1 = 27×12 + 22 is the 3rd Mersenne prime number M5.
Forum: Dobri 2022-04-19, 16:25
Replies: 29
Views: 3,279
Posted By Dobri
Concerning k = 5, there are 10 known Mersenne...

Concerning k = 5, there are 10 known Mersenne primes, with prime exponents p = 3, 7, 13, 19, 31, 1279, 2203, 2281, 23209, and 44497, for which 2*5*p+1 is a prime number. Here p mod 6 = 1 for p = 7,...
Forum: Dobri 2022-04-19, 15:52
Replies: 29
Views: 3,279
Posted By Dobri
Within the limited sample of known Mersenne...

Within the limited sample of known Mersenne primes, there are 7 Mersenne primes, with prime exponents p = 2, 5, 89, 9689, 21701, 859433, and 43112609, for which 2p + 1 is a prime number. Obviously, p...
Showing results 1 to 25 of 323

 
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