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I have tabulated digital roots for equations system and from these tabulation we can deduce whether a set of data could be represented by a set of equation or not. From this plot we could see why it is not possible to find an equation that could represent the prime numbers, if this equation existed, it would be super complex with infinite variables, I think. Hope someone could show me how to publish/upload image/pic to this forum.
There is no such thing biggest number but bigger number because of the fact 1/x is too big when x=0. If there is such thing biggest number, it could always be 1 because the infinite universe was confined in a single unit when t=0s during the big bang:)
I am writing a paper on this subject as I have developed Sums of Power for arithmetic progression. Setting n=2, the generalized equation reduces to polynomials of stepping down 2nd power. It looks like Kummer's cyclotomic expression but a slightly different form. This form had been used by Euler for p=3 and Sophie Germain for p=5, they got it by substitution of variables, it was not known during their time that there is a generalized equation that can describe the same form for any p.
There are two simplified forms for these equations:
Consider Fermat's Last Theorem Equation as follows:
Therefore:
For odd p
For even p
Where:
andSome of the equations:
p=2
p=3
p=4
p=5
Maybe it could offer an alternative proof for Fermat's Last Theorem. I had tried rational root theorem and substitution of solution of these forms:
orStill got stumbled upon few steps. Maybe Galois theory might be used to explain why
can never be rational in the form of root w and variable s.Sum of Power of Integers using Generalized Equation for Sums of Power for the Arithmetic Progression.
Where:
and
Hi Bobbym
Thanks for your comment. For your information, I have emailed the link to John Coates, Andrew Wiles advisor when he was in Cambridge. He said it could be new since it could do sums of power for arithmetic progression but he hasn't got time to look into the 17-19th century papers and p-adic articles. I need to change the format of the paper to latex before I could submit the paper to a mathematical journal. Thanks again with the latex info, I thought it was difficult to use it.
To others, infinity is a concept not a number but to me there is even and odd infinity. To them 1+infinity=infinity but to me 1+infinity>infinity. They got all those values like 1/12 or etc because they play with the infinity as they like. If you have a function of 1-1+1-1+1..infinity, if you use their concept you would get sometimes S=1/2 yet you know when it is even, S=0 and when it is odd, S=1 and this function alternates 0 & 1 to the infinity. I think people need to respect the infinity, otherwise we would be hay-wired. I do sometimes play with the infinity and I can proof that Zeta function
is not always true and converge to value 4.Hi Wintersolstice
It was started more than 17 years ago when I was a teenager, my brother introduced me Fermat's Last Theorem. While thinking about solving Fermat's Last Theorem I got an Idea, why not expressing
into function. That was how it started, not knowing that Jakob Bernoulli and few other guys already developed sums of power for integers few hundred years back I managed to formulated the formulation. The good things about this generalize equation is that it uses the most elementary symmetric function and making it possible to describe almost all series either real or complex numbers. I think this is the first ever formulation that could do sums of power for complex numbers, example as follows:Let's say we got 1000 series, the first term is 3+2i and the common difference between successive terms is s=i+1, find the 3rd power of this series
Substituting the arithmetic sums into the 3rd power equation yields the result as follow:
Hi Bobbym
I haven't try AKS method yet, if you could provide me the link of any program using that method let me know. I know, ECM is not for proving and disproving but the guy had incorporated Rabin-Miller probabilistic test in the program and it would tell you whether the number is prime or not at the end of factorization. Anyway, it would be a great help if you could provide me a link where to download or use AKS method. For your information I haven't got into programming for quite long time and to write a program for checking primality on my own would take some times.
By the way, thanks for your suggestion and idea it is a great help.
Hi Bobbym
I am running quad core i5 with 8gb memory and it took me around a day to check the primality of that 2241 digits prime. I am using ECM program written by an Argentinian's programmer. You can get his program at alpertron dot com dot ar. His program only could test up to 20,000 digits. I think if someone could write a primality checker for larger prime using the formulation I got, it would be kool. I am working to get first 1 million digits prime using the formulation but I think it is going to take for ages. Got a friend who tried to write the program but he quit. Can you suggest me the better way to check the primality accurately and fast? From the equation, theoretically we could get bigger prime than the Mersenne because it allows bigger numbers in the equation.
Hi Bobbym
You are welcome and thanks for your latex link.
Basically the formulation is a conjecture and I need some time to prove it. In the old days, Mersenne, Wagstaff and Fermat did derive their equations by playing with the equations and numbers. The generalize equation for prime was found during my research on symmetric function involving Fermat's Polynomials which is in stepping down of 2nd power. This function was actually used by Euler and Sophie Germaine to prove p=3&5 for Fermat's Last theorem using infinite descent method. I have developed a function using sums of power for arithmetic progression and found out there is a factorization pattern in the Fermat's last theorem when we set n=2. I named this function rule of division of symmetric function and from this generalize equation we can get infinite sub-formulations for primes like Mersenne, Wagstaff and Fermat. The thing that amazes me is that the equation is so simple but it can do a lots of things. Using this equation, I can even generated bigger primes than Mersenne at smaller prime input. The prime above was obtained by using p=281 and symmetric function (10002 and 10003). My limitation is that I need bigger computing power to check for the primality test. Here the generalize equation for prime:
and this one gives you 849 digits prime
Other primes
I have formulated a generalize equation for Prime numbers and most of prime can be described using this equation. Few of the primes that lie under this formulation are Mersenne, Wagstaff and Fermat prime.
Here one of the primes using this formulation:
This prime is 2241 digits [11507067905299776611167663020
41590820051565995672304288311
24854137822480173849330291461
66087367460386669118580647249
30052681875635965617124940310
62326938026349750640865076202
03736318769140528216656033596
44722489898447864802153907872
27300511462471040094207321270
90554844559843700179014387094
18711097783152486213088339385
71160237500018069648323469545
68456524256016347658045443319
53262052348945357102129180166
13434453629744837793476024582
94674804197353103153987874654
59943642151201364331007751892
96425124029747391445436875529
28501871772617302046028319930
01243378185688089736375529835
79093892581749382816469541124
35425111220333572540153219290
48997955954591971369247628415
93681545530044116107712656841
34326600658063029626751434962
24123539953325905994586049538
94293795422174439212170655862
61629910962908281206699254406
40894865787684694841534937813
51183142744391230210968312832
97125852159307941535125376738
31691036872766879337859307696
89249971644131434793031660037
13686942780437566554379784876
39523010057682582534115066202
27471586245145046138132408605
29725343923440292133829627878
86192693980231385671987258056
19719084243186856232253678301
58320066502319045978542194300
68163673355668883889104829136
80636692160116791042330612047
67590855543616358502660982725
02499776086860852743643061075
42872410233457793232918649292
96194473399825261215374706935
09107041251706156044897741900
54577787182575412514440471591
17459515308966495842580774699
25448653647411653670217926543
38228436679228994630246306543
30382349143519853968166423962
34925730484716753057321805154
44191564580373361653042665048
76415258148510464374012060422
82647018701310201324806753679
28127886170683068570890544080
56336771271953474474032329183
69246979601097391109845011856
95642127480440609912594627485
36426506457693532707595612327
17594273764876973834356898572
32631247788528229678127928732
21469394190199916742560269984
83523768730518439732639512064
50553817892930225740351817482
69367973377232288579420198280
87658206805864897478365851624
50915566680066251071210443389
18934327827719202572717325462
08235698011349885492159938160
15222469285995577561649810793
70501655784930752373089650499
78224714356131112735263185117
99669421636956973444063380527
20122798020869243613385645059
85692174307382376534038923542
29745901]
Hi Bobbym
You go to vixra dot org/numth/ and type sums of power and all five papers are mine. Select "A Treaty of Symmetric Function Part 1 Sums of Power" and read version 2 (v2).
This is part of my works. I have formulated a new formulation for sums of power and it works for any numbers (i.e. real & complex arithmetic progression). The generalize equation can generate any power p (it works fine also with complex p).
Let the p-th power of an arithmetic series as follows
The general equation for the sum is given as follows
Below are the equations for p=2-7
The value s is the common difference of successive terms in arithmetic progression and
is the sum of arithmetic terms. The paper is pending for publication but you can read on vixra. The beauty of this equation is that when you set n=2, it describes the Fermat's Last Theorem in a polynomial forms and if you set p to be negative, you can get new form of Riemmann's Zeta Function. I would be happy if anyone could point me the references if someone else has found it before me.Here, you can see how the coefficients are repetitive:
Dear Bobbym
Thanks a lot for your help
Hi guys, kinda new here. A brief about myself, I am a mechanical+ aerospace engineer, microbiologist and having a passion in mathematics. I have formulated new formulation for sums of power for arithmetic progression, still pending for publication. Could be new until someone proves otherwise it was discovered before. By the way, anybody knows how to type the math formulation on this forum?
Cheers all:)