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The generalized twin prime can be formulated as follows:
or
Where all Pn are consecutive primes and Ps is the resulting primes.
Example:
For P1=5, there would be no twin prime existed
For P1=7
and as P1>7, there is no twin prime numbers could be formed (A conjecture).
I love to play with the numbers and usually I don't claim it is mine until it is proven a novel idea. Because it is kinda frustration to know someone else has found it. I think the generalized formulation could be written as P1P2..Pn+-(Pn-1)=Ps, where all of them are prime.
I see you went up to 7919 and found none. I thought you have found one, I think it would be impossible to find the others as the largest twin primes which differ by 2 so far is having 200,700 digits
. So, finding a twin primes which differ by two and also a summation of consecutive multiplied primes +- 1 could be impossible, I guess.Hi bobbym
Kool, can you paste the number pairs?
Where P1, P2,...,Pn are the consecutive primes and Ps is the resulting Prime
Example:
What is the next prime?
Where P1, P2,...,Pn are the consecutive primes and Ps is the resulting Prime
Example:
Maybe this is the only prime of this form.
There is nothing mystery though. Here is why:
Let A=V
Rearranging the equation above yields,
So, r=0 or 3
"Then what triggered the bang?" Most of us would stop here because none of us was there when it happened and it is beyond our mind to understand singularity. How could everything we had on earth were made from a tiny mass of zero volume and to make things even worse the stars, galaxies, etc also came from this same thing. We were compressed in a tiny space of zero volume in the beginning but this is what the science tells us. Ref: http://www.universeadventure.org/eras/era1-plankepoch.htm and for every birth there must be a death and this universe would have its death too. Maybe our world would be dead at the same time this universe died.
"The energy is not going anywhere, the energy can't be destroyed" Energy is conserved according to classical physics but not in the outer space because many strange things out there not to our comprehension. At t=0s before the big bang the universe had zero volume. Zero volume means nothing, so, no energy.
"And exactly where is all this energy going?" It is like a man lifting a 50kg weight and when he is running out of energy he would release it and the weight would hit the ground. Same with the universe, the energy is expanding the universe and once the energy is lacking the system has no resistance to hold them (Galaxies, stars etc) and they would collapse and hit each others. Or when the energy runs out we would have heat death of universe where the temperature reaches zero kelvin. Even if there is no big crunch with zero kelvin or -273C no living things would exist. So the end of the world is inevitable but when? But for sure when you die..it is the end of the world for you:) Does it really matter to know it when?
If you could measure the deceleration of the expansion of universe then you could predict the end of the world (Universe). Deceleration means the energy is exhausting and when the energy is not enough to sustain the expansion, the big crunch would occur. Maybe we couldn't even measure the retardation speed if the difference is so small or negligible to us but enough to roll back the universe. We are just tiny speck in this entire universe lets say there is imbalance in the universe and we got hit by VY Canis Majoris, this would be million times the end of the world. This is how big VY Canis Majoris compared to our earth http://www.youtube.com/watch?v=Ov5AHcCQtd8
Hi cmowla
I happened to read the article regarding the sums of power of integer using Bernoulli & Pascal long time ago. Could look similar but you can read it here www.sanjosemathcircle.org/handouts/2008-2009/20081112.pdf
Alternative proof for Fermat's Last Theorem Using Sums of Power Formulation for p=3.
Now, let consider n=3,
When n=3, the sums of power for p=3 reduces into:
Let
Then
Or
Now consider this equation
Where
and
Assuming w is an even, thus
and
Therefore
Solving the equation yields
Let s=1, then
w=12
Solving the equation yields,
, andand z=2a=6
Therefore, there is a solution for this equation, which is given as follows
Now consider when n=2 and using the same procedure.
When n=2, the sums of power for p=3 reduces into:
Let
Then
Assuming w is an even, thus
and
Solving the equation yields:
since w=2a=2(s/2)=s,
This is a trivial solution,
or
Consider
and
Solving the equations yields:
Imaginary Solution.Other way to get sums of power for smaller p.
Let the expansion of (x+y) as follows:
Now let the T-th term of arithmetic progression as (a+bi).
Thus,
Summing the terms above yields:
Example:
=>
=>
As the p is getting larger, the calculation would be becoming tedious.
Thanks anonimystefy, yes you are right, basically, I need to rewrite the equation but got no time to edit the whole paper. Anyway, Ej is the Euler number. There are two coefficients, Oj and Qj and both are using Euler or Zig/Secant number. The sums of power for arithmetic progression is using Bernoulli's number but the alternating sums of power is using Euler's number instead.
Some of the results:
Anybody got an idea how to submit this code "O_{m,j}=n^{2m}+\sum_{j=1}^{m}\left \left [ \left ( -1 \right )^{j}\binom{m}{j} \left ( 2j+1 \right )E_{j}n^{2(m-j)}\frac{\prod_{k=0}^{j-1}(1+2(m-k))}{\prod_{k=0}^{j}(1+2(j-k))}\right]" I couldn't display it.
I have also formulated the formulation for alternating sums of power for arithmetic progression.
For odd power:
For even power:
This is part of my paper, hope nobody ever thought of it yet.
Let
Where P(x,y) is the non-trivial factor of Fermat's Last Theorem and could be prime.
Example,
P(x,y)=331 when x=1, y=2
Another Example,
So far I haven't found any prime yet but there are plenty of primes of the form as below within the calculation.
Or in the generalize form:
Example,
Ps=146677501881935178590642884931590473174737559368161423241221 when x=18 and y=19
It is ok then! This is how mathematics flows! We think about something and other people think too! I have developed many mathematics formulations since I was 12 years old, the first was binomial expansion but of course someone had found it. Then I developed unidigit or digital root in characterizing equation in quest for the Fermat's Last theorem proof and I found out after sometimes that the hindus had using this digital roots for thousands years before. I had formulated sums of power for integers more than 10 years ago without even knowing that 300 years ago someone had found it but I never give up. Few new formulations that I think people had not finding it yet like sums of power for arithmetic progression, alternating sums of power for arithmetic progression, symmetric prime numbers. Symmetric prime is my conjecture and it explains how Mersenne's Prime, Wagstaff's Prime, Fermat's Prime etc could be derived. Maybe people had found it but so far 300 years back none of literatures on sums of power for arithmetic progression did exist otherwise it could be used in the Fermat's Last Theorem long time ago. In few months time I would collaborate with one of the universities back here to find this prime using grid computing. This Primes could be bigger than Mersenne's primes anytime because of the bigger inputs. Hope none had found this type of Primes yet, otherwise I had to look for a way of finding something new:)
I agree with anonimnystefy. You should put the condition at the end and not at the beginning because you know at the end the condition is not valid, implying it is not valid through out the derivation. In other words, it is wrong to put a=b in the first place. You should refer to the flaw proof of 2=1 when it is assumed at the beginning that a=b.
I think nobody could shake the fundamental of mathematics. The reason why all seem working is because of the division by zero is ignored. The concept of proving 2=1 also ignores the division by zero. This is why when you have statement like a=b and the derivation of the equation consist of expression that leads to the division by zero then the statement a=b must be not true in the first step. Consider this, a=b, multiplying both sides by a we get a^2=ab and minus both sides by b^2 yields, a^2-b^2=ab-b^2=> (a+b)(a-b)=b(a-b)=>a+b=b and a=b, thus 2b=b=>2=1. Anything leads to the division by zero through a-b or ln(a/b) must not be true in the beginning when it is stated a=b.
Thanks for the formula bobbym! Using mathcad, you would get the value of
. Using method of area integration, , where "a" is the first term and "b" the final term . My method is simple and giving better accuracy. I would write it down later on.Thanks bobbym for the info. I have developed two ways of finding the sums, first method is using Area Integration which gives quite high error. Another one is using unbounded sums of power of arithmetic progression. After using 4 internal coefficients I managed to get 0.004% Error. As the internal coefficients reach infinity the sum will approach 100% accuracy. Can you show me the closed form formulation? As to my knowledge, if the closed form for this series existed then the sums of power for arithmetic progression would be technically found ages ago. There is a paper on the sums of power formulation by Chen et al dated 2008 and accepted by 2010, you can read their paper by the title Faulhaber's Theorem on Power Sum at arxiv, he actually managed to formulate for odd power.
I have developed new method for Numerical Analysis for sums of power for arithmetic progression of non-integer power and still working on it to improve its accuracy. However, before I could present it, has anybody got an idea how to sum this series
. Is there any formulation that could sum it?Hi cmowla
Basically, there are many Bernoulli's formulations and the finding of new Bernoulli's formulation not that significant. In my paper, the development of new bernoulli's formulation is only a small portion and without this formulation I can get others formulation to get the numbers. Since, Sums of power got bernoulli's numbers in it, I managed to manipulate it to get new forms of bernoulli's formulation but the main purpose is to develop sums of power for arithmetic progression. I do believe finding new sums of power for arithmetic progression is a big thing before people get to know it. It can be used for numerical analysis, Riemman's zeta function, Fermat's Last Theorem, generating function for finding prime numbers etc. I had demonstrated few examples of the use of this formulation.
Fulhaber is known as one of the greatest mathematicians because he developed sums of power for integers. This encourage me to work on bringing this formulation to the world as it is the umbrella for all sums of power because it can do integers, non-integers, integer power, complex power and many more.
Here the bernoulli's formulation: