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Consider the equation below:
=Prime SequenceExample:
Prime Sequence
Value of a
1
1
2
2
3
3
4
4
4
Value of b
1
2
1
3
2
4
1
3
7
New Prime
Another Larger Sequence
Prime Sequence
93703
93719
93739
93761
93763
93787
93809
Value of a
306
306
306
306
306
306
306
Value of b
67
83
103
125
127
151
173
New Prime
373
389
409
431
433
457
479
Perhaps there are lengthy new prime listing could be generated from the normal prime sequence.
The largest prime value of n would always be 3. As the value of Pt and n becoming larger, n would always be divisible by 3 (a conjecture).
Perfect Primes:
There are so far 3 groups of primes for consecutive power when n=3.
The list is given as follows:
For larger value of primes, it seems that the value of n is always divisible by 3. Unless a counterexample is found.
It is quite amazing to find out that the plus-minus n is always having a cycle of digital roots of 2,4,3,6 or 9.
Thanks for the calculation. I do believe in the future the calculation time for ProvablePrimeQ would be smaller as the computing power is getting more powerful. I was using a supercomputer with a cluster of 2000 CPUs in the late 90s and it took sometimes up to 4 months to complete a finite element analysis of a small section of turbine jet engine component. Now, a single GPU like NVDIA tesla comprises of thousand cores in a single unit of processor would make as if you are having a supercomputer at home. However, as the computing power getting faster the prime number is also getting larger and limitless. So back to the square one:)
Dear Danaj, thanks for the calculation. Have you checked the prime with prime provable algorithm like in the mathematica?
Dear danaj, that's not a solution because n is the number of terms used. Your equation uses n=3 and shouldn't be 57.
For n<25, no apparent primes for Pt=5
For n<25, no apparent primes for Pt=6
For n<25, no apparent primes for Pt=7
Smallest solution for Pt=4, when n=3
Smallest solution for Pt=3, when n=2
Next prime when n=4
Next prime when n=6
Next prime when n=9
Next prime when n=15
Would there be primes at n=48?
From calculation n=48 would only give one prime
Smallest solution for Pt=2, when n=2;
Next solution, when Pt=2 and n=3;
Next solution, when Pt=2 and n=6;
Next solution, when Pt=2 and n=12;
Next solution, when Pt=2 and n=24;
Twin Prime of Alternate Components, consider this equation:
Where
Dear danaj, is it possible to run a program that you wrote using gpu processors with thousands of core doing the calculation? I am upgrading my computing power to mini-supercomputer using nvidia processors. It would be a great thing to find any series of in tens or perhaps near hundreds long series. If they don't exist then we can conjecture there would be no series longer than max number found so far.
I can do it even faster:) With a calculator!
Thanks danaj for the input. Can you list the 10 consecutive primes. Thanks again for the script.
Thanks danaj..the idea is to get all the Pr as prime with the sequential prime-th (i.e. s). Mathematica uses PrimeQ, a good primality algorithm for small primes, for larger prime you can use provableprimeQ which verifies the number to be prime. This is why it takes so long and if your computer is slow it could take months. I am trying to get longer series, perhaps up to 40 consecutive primes. So far, 8 is the largest.
Dear Danaj can you list down the primes of 8 consecutive from your calculation? I am using mathematica and it is a bit slow. Mathematica got primality algorithm to verify the numbers, perhaps that makes it so slow. Anyway, thanks for the script, I am looking for a longer prime consecutive with the inputs.
Just found another 8 consecutive prime for range 23,000,000<prime-th <24,000,000
No apparent result bigger than 8 consecutive primes for Prime-th up to 20,000,000 for Pr=34+3s
The max number of consecutive primes for Pr=34+3s is 8 (s<13,500,000)
For Pr=34+3s
Prime-th{13155307, 13155308, 13155309, 13155310, 13155311, 13155312, 13155313, 13155314}=s={239878543, 239878571, 239878579, 239878603, 239878621, 239878649, 239878663,
239878673}
Pr={719635663, 719635747, 719635771, 719635843, 719635897, 719635981, 719636023, 719636053} Consecutive Primes
Another Primes for For Pr=34+3s (7 consecutive)
Prime-th{10654019, 10654020, 10654021, 10654022, 10654023, 10654024, 10654025}=s={191885429, 191885471, 191885483, 191885509, 191885539, 191885543, 191885563}
Pr={575656321, 575656447, 575656483, 575656561, 575656651, 575656663, 575656723} Consecutive Primes
The max number of consecutive primes for s<10,000,000 still 7.