_{1}

A Simple Mathematical Solutions to Beal’s Conjecture and Fermat’s Marginal Conjecture in his diary notes, Group Theoretical and Calculus Solutions to Fermat’s Last theorem & Integral Solution to Riemann Hypothesis are discussed.

Beal’s conjecture states if

Since it is a conjecture it should either be proved or disproved so that we have to find simple way to handle it. In the search of a solution to the open problem after many trial and errors we decided to tackle it in the following way. Take

positive integer the square coefficient factors appearing in the left hand side of

Further

There are bigger five set of number identities given by Don Zagier but non of them fit with Beal’s Conjecture due to Beal’s restriction all

^{th} powers of divisors of n”, Ramanujan had begun. If n = 6 for example, its divisors are 6, 3, 2 and 1. So that if, say 3, “the sum of the s^{th} powers of the divisors”,

Now let us get back to our answering based on what is said by Robert Kanigel [

satisfying Beal’s Conjecture could be derived from sum of 1th powers of divisors of 2^{n}^{−1} including 1 and itself, based on the mathematics of Ramanujan’s paper on certain arithmetical functions appeared in the transactions of the Cambridge Philosophical Society, XX11, No. 9, 1916, 159 - 184. If other number identities exist they all might be derivable by the same technique introduced here at last a genuine approach to finalize the story. Further

n = 4,15 is not a prime factor, n = 6,63 is not a prime factor are very simple counter examples to Beal’s Conjecture, So it is very clear factor need not be prime always but can be prime or not prime. Very simple mathematics founded by W. Nadun, Kanigel, S. Ramanujan at last but not least defeated Beal’s Claim of a prime factor always for exponents greater than 2 for Beal-Fermat Equation a generalization of Fermat’s Theorem at Last and Beal’s Theorem at First. Andrew Beal was very sincere if at least few counter examples are found for his famous conjecture now he is provided with two such cases. If you computerize it with a programming language it may give many counter examples as well as examples satisfying Beal. In our discussion, we end this story on Beal’s Conjecture. Wikipedia on Beal can provide more information. We have provided sufficient ground in support of Beal’s Conjecture as well as against it.

Now assume there are such positive integer

since

Fermat’s Last Theorem states

Few technical points can be noted as follows. The resulting polynomial equation is

be the elementary symmetric functions in the in-determinates

For quartic equation the Galois group

Therefore

Alternating Group

Therefore for

As

Start with

when

gers, consider the case with

leads to complex number

Therefore by the method called Reductio ad Absurdum that is reduction to absurdity Fermat’s Criteria get satisfied that there are no positive integer solutions when

this point but does not exist due tonon availability of real positive integers for zero discriminant that must be what Frey has stated as forbidden of existence.

Further

A proof is given for the Riemann Hypothesis in the strictly open interval

is the Riemann Zeta function,

ber. What Hypothesis says is that

In theory of Complex Numbers

and with the substitution at relevant places in I & J integrals we get for

what values I and J integrals take?

trigonometric functions.

This completes the proof which has sufficient ground unnoticed before for the long awaited Riemann Hypothesis strictly in the open interval

to come up with their sketch of proofs following similar line of thought the existence of two more zero lines and beyond this closed range no more such lines exists.

These problems were newly proposed by A. C. Ranjith De Alwis at E-Mail: charles@hansexportaustralia.com.au, due to the announcement of new awards by Clay Institute up to seven million US Dollars appearing in the internet web can be found under Google search where Edward Witten, Andrew Wile … are Founding Members and Solved by A. C. Wimal Lalith De Alwis present author assisted by W. Nadun, Susitha S. Present Author [another E-Mail: dealwis_a@yahoo.com, Home Address: No. 299, Galle Road, Goraka, Moratuwa] the author of new principia is staying in Lanka the Golden Lanka of Ramayan Epicand National Aeronautical Space Agency of America (NASA) found high rock peaks & their height up to 20,000 feets in Adams Bridge path from Talaimannar of Lanka to Danushkodi of India by Geographical Earth satellite over length of 35 km strip that India is going to cover the sea gap by Rama Sethu Bridge project with assistance of NASA if not by Stones by Cables.

de Alwis, A.C.W.L. (2016) Solutions to Beal’s Conjecture, Fermat’s Last Theorem and Riemann Hypothesis. Advances in Pure Mathematics, 6, 638- 646. http://dx.doi.org/10.4236/apm.2016.610053