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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,160

Hi Stangerzv,

These are the digital-root pairs found up to Pₓ=280,000:

{1,2}, {1,5}, {1,8}, {2,4}, {2,7}, {4,5}, {4,8}, {5,7}, {7,8}.

Still no {1,7}, but I'll let my program keep looking for it overnight.

72 Perfect Twin Primes have been found so far, for Pₓ just <4million.

*Last edited by phrontister (2013-05-30 13:22:35)*

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 218

Ok, so far we got most of them but do you notice or not that digital root pair (1,4) occurs only once at Px=3 & Pt=3. By adding this digital root we get 1+4=5, yet the rest of digital roots above when added would give you a multiple of 3 digital roots examples, 1+2=3, 1+5=2x3, 1+8=3x3, 2+4=2x3, 2+7=3x3, 4+5=3x3, 5+7=12=>3, 7+8=15=2x3 but 1+7=8. I think it would be odd to get perfect prime pairs with the digital root {1,7} or if it does exist, it would be a special one. On the other hands, maybe there is only one pair of digital root of {1,4}.

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,160

Yes, it is interesting that all of the DR pairs I've got so far are a multiple of 3...except for that one case of {1,4} in Pₜ=3. There are a dozen or more solutions that I haven't posted, but they too are a multiple of 3.

I have to reboot my computer soon and so have terminated the search for a {1,7} pair in Pₜ=2. The last solution I found was at Pₓ=313219. Anyway, we now know that it's unlikely to exist...as is probably also the case with {1,4}.

I may continue the search for the two missing pairs in the first layer of other groups that I haven't tested yet, as I'm only up to Pₜ=19. But higher values will take much longer.

*Last edited by phrontister (2013-05-30 23:19:42)*

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,160

I may continue the search for the two missing pairs in the first layer of other groups that I haven't tested yet, as I'm only up to Pₜ=19. But higher values will take much longer.

I think I'll give up on that idea, as the evaluations take faaaaar too long at the higher values and would take forever per result.

My program found just one solution overnight, which was Pₜ=23 with DR={1,2} for approx Pₓ=3.5million (I forgot to note the exact result ).

When I aborted the program it was busy solving for Pₜ=29, as I'd changed my code so that after each first solution for the Pₜ it was evaluating it would automatically advance the search to the next Pₜ. For Pₜ=29 it was up to nearly Pₓ=2million, and so I could see it was time to pull the plug...which was disappointing.

Solving for these higher values is a job for someone with a much more powerful computer set up than mine, and should preferably be tackled by leaving their program running while away on long service leave...unless they have a spare computer.

Btw, none of my solutions have the DR of Pₓ as a multiple of 3, other than that one in Pₜ=3 for which the DR of Pₓ=3...which also is the only one with a single-digit Pₓ.

*Last edited by phrontister (2013-05-30 23:19:11)*

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 218

Hi phrontister

I think I know how to find the proof. You see, other than Px=3 & Pt=3 we will always having multiple of 3 digital roots for

. So, now consider equations as follows:------------(1)and

------------(2)

(1)+(2), yields,

Since Digital Root of

is multiple of 3, so as with the digital roots of .Since none of the values of

is multiple of 3 we can prove that Px is not a multiple of 3. Lets consider this equation,(1)-(2)

Taking digital root both sides yields

If RHS is a negative DG, add 9. Since DG of RHS is never a multiple of 3 through prove by exhaustion. Therefore, digital root of LHS must also not a multiple of 3.

*Last edited by Stangerzv (2013-05-30 14:07:44)*

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,160

Hi Stangerzv,

Isn't the proof based on an assumption, though? We've tested a few numbers from several squillion, and even with those the results are not conclusive because they're not exhaustive.

I agree that our testing gives certain very strong indications, but I don't think we can actually call something that is based on those findings a 'proof' as such.

I suppose we'd have to construct some sort of logical mathematical argument to support the proof, but I don't know how to do that in this case.

*Last edited by phrontister (2013-05-30 15:43:58)*

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 218

Hi phrontister

I agree with you, what to make it conclusive is that when you can prove there is no pair of {1,7} and do you notice or not that

is always either 9 or 6 for perfect prime twin pair other than when Px=3. Basically the proof is not complete otherwise people could at least prove one of the conditions above is true. If we could prove that would always be either 6 or 9, then we can get the conclusive proof.It is becoming more interesting I guess.

*Last edited by Stangerzv (2013-05-30 18:13:11)*

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,160

Hi Stangerzv,

...do you notice or not that

is always either 9 or 6 for perfect prime twin pair other than when Px=3...

...If we could prove that would always be either 6 or 9, then we can get the conclusive proof.

There are many cases with Pₓ<300,000 (that's all I've tested to) where DR=3 for Pₜ=2.

These are the first three:

*Last edited by phrontister (2013-05-31 22:28:35)*

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 218

So, it means the sums of prime power for prime would be a multiple of 3 then. But have you tried prime power bigger than 2? I think there is no digital root of 3 for power greater than 2.

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,160

Stangerzv wrote:

So, it means the sums of prime power for prime would be a multiple of 3 then. But have you tried prime power bigger than 2? I think there is no digital root of 3 for power greater than 2.

Yes, I had tried powers higher than 2, but not many. I found these:

Pₜ=3

Pₜ=5

*Last edited by phrontister (2013-05-30 23:17:45)*

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 218

Ok, 2 and 3 always behave that way. I missed Pt=3 but I think more than Pt=3 the DG=3 would diminish unless you can find a counter example.

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,160

Sorry, my edit to post #135 crossed with your latest post. I had added a DR=3 for Pₜ=5.

Here's the next DR=3 for Pₜ=5:

*Last edited by phrontister (2013-05-30 23:29:24)*

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 218

So, it seems that the sums of prime-power would be a multiple of 3. Well, do you think is it possible for it to have other than multiple of 3? for Pt>3. Maybe I need to learn programming again and run it on my computer. Mine is just core i5. Maybe it is just enough for smaller numbers Anyway, phrontister, thanks for the input, it is really a big help.

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,160

So, it seems that the sums of prime-power would be a multiple of 3.

Yes, that would be highly likely, other than for the first (smallest) result of Pₜ=3.

Well, do you think is it possible for it to have other than multiple of 3? for Pt>3.

I lack experience and knowledge and can't think of any way of proving that it is not *possible*, but from the fact that I've tested quite a few Pₜ and all of them other than the first solution for Pₜ=3 resulted in DR=a multiple of 3, it certainly seems *very improbable*.

Maybe I need to learn programming again and run it on my computer. Mine is just core i5.

I don't know how yours compares with mine, which is an AMD Athlon Dual Core 5200+ 2.71GHz with 3Gb RAM, but larger Pₜ certainly slow things down for mine.

Could be time for an upgrade, maybe?

Thanks for the thanks...I never do this sort of testing and it was a nice opportunity. Glad to help.

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 218

It is easier to put something than making sure it is true for all. I think it would take like forever to proof that, why these prime always working with 3, 6 & 9 through computation.

phrontister, how often DR 3, 6 & 9 of

with prime Px resulted in non-prime Ps?Offline

**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,160

I think it would take like forever to proof that, why these prime always working with 3, 6 & 9 through computation.

It might even take longer than forever(!), as the way we're doing it means we can only ever test a tiny sample from an infinite range. I guess a conclusive proof would have to be by algebraic logic or some other means of argument that I wouldn't have any idea about as my maths level is too low. Someone else with more knowledge would have a better idea about how to construct a proof...or know whether or not it's even feasible.

how often DR 3, 6 & 9 of

with prime Px resulted in non-prime Ps?

I'm not sure I understand you correctly.

Pₛ must always be even, otherwise Pₛ+Pₓ and Pₛ-Pₓ could not be prime.

*Last edited by phrontister (2013-05-31 22:50:47)*

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 218

Well, DR for even number not always an even but also can be a prime or odd number. DR(12)=3. What did I mean is that, would sums of prime-power with digital root 3, 6 and 9 always resulting in a prime Ps when Px also a prime? This is because, I think there are not that many of sums of prime-power with digital root 3, 6 & 9 could exist at the same time with prime Px, if they do, would all of them resulted in prime Ps or not necessarily to be prime.

I am not a mathematician and my maths could be more to classical approach rather than contemporary one. If I were to find a proof of something, I rather find something which is understandable to all people including the laymen. The good about maths is that, it was/is also developed by amateurs through history and that is why, we shouldn't feel bad about not having proper education in the mathematics.

*Last edited by Stangerzv (2013-06-01 07:06:57)*

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,160

Well, DR for even number not always an even but also can be a prime or odd number. DR(12)=3.

Sorry, I'd overlooked the fact that "+/-" (ie, +Px and -Px) isn't included this time, otherwise Ps would have to be even for Ps+Px and Ps-Px to be prime.

...would sums of prime-power with digital root 3, 6 and 9 always resulting in a prime Ps when Px also a prime?

I can only find even numbers for Ps with DR=3, 6 or 9 when Px is prime.

I tested quite a range of Pt and found the following Ps with DR=3, 6 or 9 when Px is prime:

1. For Pt>2 the DR occurs at Px>=97.

(a) The DR occurs at the same Px for all Pt.

(b) The DR is not necessarily the same at each Px for all Pt (see image).

2. Pt=2 gives many more results than for the other Pt, with some Px being the same as the others and some not.

(a) DR is 9 for all Px<=97.

...we shouldn't feel bad about not having proper education in the mathematics.

It's not that I actually feel bad about it. I enjoy doing puzzles and I know enough to be able to solve a reasonable range of them with logic, pen and paper, my calculator, Excel, Mathematica and BASIC programming. That keeps me pretty happy, but sometimes I wish I had more knowledge so that I could understand more of the posts here on MIF. I don't have the desire, dedication or time to broaden my maths knowledge much, but have been picking up a few things here and there from bobbym and others.

*Last edited by phrontister (2013-06-02 20:21:23)*

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,160

Here's a table like the one in my previous post, but this one also includes Ps and the range of Pt.

I did some more testing with higher Pt and still found no prime Ps with DR=3, 6 or 9 when Px is prime.

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 218

Hi phrontister, thanks for the input. It would be wonderful to have a supercomputer indeed. I used to work with OSCAR Cray-SGI supercomputer when I was in the UK last time. I wish I was doing mathematics those times and the calculation would be lightening speed fast for sure. It is a sure thing that next Ps would be a rare thing and finding the bigger one is something kool because the probability to get the twin prime at higher Px is very small. It would be more challenging than finding Mersenne's prime because these primes exist in pair and usually bigger primes more than few millions digits rarely occurred next to each other.

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 218

Dear Bobbym and Phrontister

I am writing up a journal for this prime number, can you both give me your details so I can include both of you in the paper for helping me with the calculation. Please message me with the detail if you guys don't mind.

Thanks

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 98,926

Hi;

It appears the bulk of the computing and work was done by phrontister so mention him. Instead of mentioning me you might mention the Math Is Fun Forum as where you were helped too.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.** **Thinking is cheating.**

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 4,160

Hi Stangerzv,

I'd be happy just to have the Math Is Fun Forum mentioned as the source for obtaining help with your calculation.

If you named me, then someone might get the mistaken impression that I actually knew what I was talking about!

I turn to bobbym for help when I'm stuck, and learnt some things from him in that thread...in which he was involved way before I joined in.

Thanks again, though, for the opportunity to work on your puzzle.

*Last edited by phrontister (2013-10-20 22:24:11)*

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**Stangerzv****Member**- Registered: 2012-01-30
- Posts: 218

Bobbym & Phrontister. Thanks, I would mention this forum then:)

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 98,926

I turn to bobbym for help when I'm stuck, and learnt some things from him in that thread...in which he was involved way before I joined in.

Oh boy, if only some of that were true. My cousin is being kind.

Hi Stangerzv;

Tell me when you get it published.

**In mathematics, you don't understand things. You just get used to them.****If it ain't broke, fix it until it is.** **Thinking is cheating.**

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