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Andrica conjecture, Beal’s conjecture, degree n polynomial equation, Diophantine analysis, Fermat’s last theorem, Identity for solving equations, Legendre conjecture, proof, Proof of the abc conjecture, quintic equation, rational points on elliptic curve Cramer’s conjecture, Riemann hypothesis., Strong Goldbach conjecture

*In this research an identity is proposed for the proof of Beal’s conjecture. The identity is demonstrated to have the capacity to solve the congruent number problem and a host of other unsolved problems including the Pillai conjecture and so forth. Additionally the identity provides for an algebraic relationship connecting two or more terms being added to the ﬁnal sum. Thus the identity provides a completely new and simple approach that can be used solve the polynomial equation. Thus in this paper the identity used to prove Beal’s conjecture is extended to provide an algebraic solution of the degree n polynomial equation. In this research propositions are also made on how the above proposed identity is can be used to prove the Andrica, Legendre and the strong Goldbach conjecture. Other more basic considerations are brought considered in this paper to come up with a direct proof of the strong Goldbach conjecture. In the paper we examine the problem of determining the number of rational points on an elliptic curve through the aid of the prosed identity. Additionally the paper deals with proof of Cramer’s conjecture, proof of the abc conjecture and a strong case for disproof of the Riemannhypothesis.*

**View Book: **http://bp.bookpi.org/index.php/bpi/catalog/book/126

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