<![CDATA[Math Is Fun Forum / New Math Formula: Sums of Power for Arithmetic Series]]> 2020-01-18T00:36:40Z FluxBB https://www.mathisfunforum.com/viewtopic.php?id=17387 <![CDATA[Re: New Math Formula: Sums of Power for Arithmetic Series]]> An approach to proof Fermat's Last Theorem.

Consider when n=3, and i=3, yields:

Let

and

then;

let

Then

this equation is solvable.

Now let n=2

Let

and

then

There is no whole number solution according to Euler.

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https://www.mathisfunforum.com/profile.php?id=184422 2020-01-18T00:36:40Z https://www.mathisfunforum.com/viewtopic.php?pid=412183#p412183
<![CDATA[Re: New Math Formula: Sums of Power for Arithmetic Series]]> New Formulation for alternating sums of power

Let the p-th power of an alternating arithmetic series as follows

The General Equations are given as follows:

For odd power:

For even power:

Where

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https://www.mathisfunforum.com/profile.php?id=184422 2019-11-21T23:51:20Z https://www.mathisfunforum.com/viewtopic.php?pid=411413#p411413
<![CDATA[Re: New Math Formula: Sums of Power for Arithmetic Series]]> The published work can be read here https://www.tandfonline.com/doi/abs/10.1080/09720529.2015.1102945 & https://www.researchgate.net/publication/264158476_A_TREATY_OF_SYMMETRIC_FUNCTION_AN_APPROACH_IN_DERIVING_GENERAL_FORMULATION_FOR_SUMS_OF_POWER_FOR_AN_ARBITRARY_ARITHMETIC_PROGRESSION

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https://www.mathisfunforum.com/profile.php?id=184422 2019-11-04T01:34:52Z https://www.mathisfunforum.com/viewtopic.php?pid=411205#p411205
<![CDATA[Re: New Math Formula: Sums of Power for Arithmetic Series]]> The paper had been accepted and would be published in Journal of Discrete Mathematical Sciences & Cryptography ISSN: 0972-0529 (Print)  ISSN: 2169-0065 (Online). I would update the new formulation for sums of power for an arithmetic progression on Wikipedia.

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https://www.mathisfunforum.com/profile.php?id=184422 2019-10-16T23:46:12Z https://www.mathisfunforum.com/viewtopic.php?pid=411061#p411061
<![CDATA[Re: New Math Formula: Sums of Power for Arithmetic Series]]> The journal has been accepted for publication in the Journal of Discrete Mathematical Sciences & Cryptography

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https://www.mathisfunforum.com/profile.php?id=184422 2015-11-19T22:43:50Z https://www.mathisfunforum.com/viewtopic.php?pid=371079#p371079
<![CDATA[Re: New Math Formula: Sums of Power for Arithmetic Series]]> This paper is about to be accepted. Then, I can update on the Wikipedia for a new formulation for sums of power of an arbitrary arithmetic progression.

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https://www.mathisfunforum.com/profile.php?id=184422 2015-09-27T00:08:23Z https://www.mathisfunforum.com/viewtopic.php?pid=368513#p368513
<![CDATA[Re: New Math Formula: Sums of Power for Arithmetic Series]]> 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,

,
and

and 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.

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https://www.mathisfunforum.com/profile.php?id=184422 2013-02-22T16:50:13Z https://www.mathisfunforum.com/viewtopic.php?pid=254537#p254537
<![CDATA[Re: New Math Formula: Sums of Power for Arithmetic Series]]> 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.

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https://www.mathisfunforum.com/profile.php?id=184422 2012-11-10T18:55:37Z https://www.mathisfunforum.com/viewtopic.php?pid=239498#p239498
<![CDATA[Re: New Math Formula: Sums of Power for Arithmetic Series]]> 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.

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https://www.mathisfunforum.com/profile.php?id=184422 2012-10-12T20:03:55Z https://www.mathisfunforum.com/viewtopic.php?pid=234987#p234987
<![CDATA[Re: New Math Formula: Sums of Power for Arithmetic Series]]> Have you seen post #16?

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https://www.mathisfunforum.com/profile.php?id=118786 2012-10-12T19:54:58Z https://www.mathisfunforum.com/viewtopic.php?pid=234986#p234986
<![CDATA[Re: New Math Formula: Sums of Power for Arithmetic Series]]> Some of the results:

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https://www.mathisfunforum.com/profile.php?id=184422 2012-10-12T19:51:49Z https://www.mathisfunforum.com/viewtopic.php?pid=234984#p234984
<![CDATA[Re: New Math Formula: Sums of Power for Arithmetic Series]]>

You had an extra \left in there.

Btw, I would imagine it to be O_{m,k}, because the value of j in the expression isn't really constant...

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https://www.mathisfunforum.com/profile.php?id=118786 2012-10-12T19:49:57Z https://www.mathisfunforum.com/viewtopic.php?pid=234983#p234983
<![CDATA[Re: New Math Formula: Sums of Power for Arithmetic Series]]> 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.

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https://www.mathisfunforum.com/profile.php?id=184422 2012-10-12T19:38:27Z https://www.mathisfunforum.com/viewtopic.php?pid=234982#p234982
<![CDATA[Re: New Math Formula: Sums of Power for Arithmetic Series]]> I have also formulated the formulation for alternating sums of power for arithmetic progression.

For odd power:

For even power:

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https://www.mathisfunforum.com/profile.php?id=184422 2012-10-12T19:03:03Z https://www.mathisfunforum.com/viewtopic.php?pid=234980#p234980
<![CDATA[Re: New Math Formula: Sums of Power for Arithmetic Series]]> 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:

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https://www.mathisfunforum.com/profile.php?id=184422 2012-07-05T05:19:31Z https://www.mathisfunforum.com/viewtopic.php?pid=224977#p224977