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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

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;

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

Would there be primes at n=48?

From calculation n=48 would only give one prime

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

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

*Last edited by Stangerzv (2015-06-24 01:21:16)*

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

Smallest solution for Pt=4, when n=3

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

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

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**danaj****Member**- Registered: 2014-03-03
- Posts: 29

Isn't that a solution for Pt=5? Similarly n=3 Pt=6 => +/-9, n=3 Pt=7 => +/- 63

*Last edited by danaj (2015-06-27 17:07:29)*

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

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.

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**danaj****Member**- Registered: 2014-03-03
- Posts: 29

Oh, I see. You're restricting *n* to both the number of terms and the +/- unit.

I believe for Pt=5, n=1281; Pt=6, n=609; Pt=7, n=711; Pt=8, n=957; Pt=9, n=4035. Double check them.

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

Dear Danaj, thanks for the calculation. Have you checked the prime with prime provable algorithm like in the mathematica?

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**danaj****Member**- Registered: 2014-03-03
- Posts: 29

I didn't before (just using something similar to Mathematica's PrimeQ), but these numbers are small so I changed to is_provable_prime. They're provable primes (if they weren't we'd have found something that was a BPSW + extra M-R tests counterexample, which would be astonishing).

*Last edited by danaj (2015-06-30 01:48:41)*

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**danaj****Member**- Registered: 2014-03-03
- Posts: 29

By the way, here is what I used. Just brute force.

`perl -Mntheory=:all -MMath::GMPz -E 'for my $pt (1..11) { for my $n (1 .. 50000) { my($sign,$s)=(1,0); forprimes { $s += $sign* Math::GMPz->new($_)**$pt; $sign=-$sign } nth_prime($n); if ($n < $s && is_provable_prime($s-$n) && is_provable_prime($s+$n)) { say "$pt: $s +/- $n"; last; } } }'`

Takes about 1.5 minutes. Generating the proofs vs. plain BPSW makes no noticeable timing difference.

```
1: 8 +/- 5
2: 20 +/- 3
3: 106 +/- 3
4: 560 +/- 3
5: 59779076418421204640 +/- 1281
6: 4255809704937643329920 +/- 609
7: 63835705531104194020747282 +/- 711
8: 5199099913899300539585745185120 +/- 957
9: 86486076326025141392980473635379192344452 +/- 4035
10: 1287183127776263297776662121192361760476608760 +/- 3723
11: 101678827728094895209692286827415375382 +/- 435
```

I imagine this could be translated into Pari/GP with little effort, and its isprime is APR-CL with modern versions. Mathematica would probably not translate directly but should be easy enough. ProvablePrimeQ according to documentation uses various methods including ECPP, but I have no idea how fast it is.

*Last edited by danaj (2015-07-02 19:17:17)*

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

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:)

*Last edited by Stangerzv (2015-07-04 02:28:33)*

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

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.

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

For larger value of primes, it seems that the value of n is always divisible by 3. Unless a counterexample is found.

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

There are so far 3 groups of primes for consecutive power when n=3.

The list is given as follows:

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

Perfect Primes:

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

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).

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

The largest solution for

could be only whenOffline

**googol****Member**- From: Delft, The Netherlands
- Registered: 2016-04-22
- Posts: 13

First results for Pt=23: (no perfect twins)

-4475 : 5889137212050938735368472902508869483835853899657458332272001992354134048771473543186122962291841184947527129

+4475 : 5889137212050938735368472902508869483835853899657458332272001992354134048771473543186122962291841184947536079

-11737- : 763535938306412512047674015563031733445522730069155715016873757788272794603779526321867531591867496118778511572255374563

+11737+ : 763535938306412512047674015563031733445522730069155715016873757788272794603779526321867531591867496118778511572255398037

-11848 : 977088126028151808796565733124095583631954744365488527199737265166828731869558855032529267557676836542668809798898003537

+11848 : 977088126028151808796565733124095583631954744365488527199737265166828731869558855032529267557676836542668809798898027233

**10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000**

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**googol****Member**- From: Delft, The Netherlands
- Registered: 2016-04-22
- Posts: 13

**10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000**

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**Primenumbers****Member**- Registered: 2013-01-22
- Posts: 149

Can be written as....

And we know

So,

569,673=(1x2)+(2x4)+(4x6)+(6x8)+(10x12)+(12x14)+(16x18).

**"Time not important. Only life important."*** - The Fifth Element 1997*

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

Dear Primenumbers, the idea of the equations is to get the largest primes or perhaps they simply non-existence as n goes into the infinity. There are several types of equations formed from the generalize equation (i.e. n=integers, n=primes, n=square numbers).

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

Extension into negative Primes

Example

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

Extension into odd numbers instead of using primes

Where

Example

*Last edited by Stangerzv (2016-06-17 18:35:05)*

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 265

There are lesser resultant primes using odd number inputs. The next prime for Pt=2 is given as follows

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