TITLE:
Tree of Fermat-Pramanik Series and Solution of AM +B2 =C2 with Integers Produces a New Series of (C12- B12)=(C22- B22)=(C32- B32)=Others
AUTHORS:
Panchanan Pramanik, Susmita Pramanik, Sabyasachi Sen
KEYWORDS:
Fermat Theorem, Fermat-Pramanik Tree, Solution of AM +B2 =C2, Deductive Series, Generation of Fermat’s Triode, Generation of Fermat Series
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.14 No.3,
March
21,
2024
ABSTRACT: The Fermat–Pramanik series are like below:
.The mathematical principle has been established by factorization principle. The Fermat-Pramanik tree can be grown. It produces branched Fermat-Pramanik series using same principle making Fermat-Pramanik chain. Branched chain can be propagated at any point of the main chain with indefinite length using factorization principle as follows:
Same principle is applicable for integer solutions of AM+B2=C2which produces series of the type . It has been shown that this equation is solvable with N{A, B, C, M}. where , , M=M1+M2 and M1>M2. Subsequently, it has been shown that using M= M1+M2+M3+... The combinations of Ms should be taken so that the values of both the parts (Cn+Bn)and (Cn-Bn) should be even or odd for obtaining Z{B,C}. Hence, it has been shown that the Fermat triple can generate a) Fermat-Pramanik multiplate, b) Fermat-Pramanik Branched multiplate and c) Fermat-Pramanik deductive series. All these formalisms are useful for development of new principle of cryptography.