.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
References
[1]
Vinogradov, I.M. (2003) Elements of Number Theory. Dover Publications, Mineola.
[2]
Long, C.T. (1972) Elementary Introduction to Number Theory. 2nd Edition, D.C. Heath and Company, Lexington.
[3]
Hardy, G.H. and Wright, E.M. (2008) An Introduction to the Theory of Numbers. Oxford University Press, Oxford.
[4]
Niven, I.M., Zuckerman, H.S. and Montgomery, H.L. (2008) An Introduction to the Theory of Numbers. John Wiley & Sons, Hoboken.
Granville, A. (2008) Analytic Number Theory. In: Gowers, T., Barrow-Green, J., Leader, I., Eds., The Princeton Companion to Mathematics. Princeton University Press, Princeton.
[7]
Singh, S. (1997) Fermat’s Last Theorem: The Story of a Riddle that Confounded the World’s Greatest Minds for 358 Years. Fourth Estate.
[8]
Weil, A. (1984) Number Theory: An Approach through History—from Hammurapi to Legendre. Birkhäuser, Boston.
[9]
Wiles, A. (1995) Modular Elliptic Curves and Fermat’s Last Theorem. Annals of Mathematics, 141, 443-551. https://doi.org/10.2307/2118559
[10]
Wünsche, A. (2024) Three-and Four-Dimensional Generalized Pythagorean Numbers. Advances in Pure Mathematics, 14, 1-15. https://doi.org/10.4236/apm.2024.141001
[11]
Beji, S. (2021) A Variant of Fermat’s Diophantine Equation. Advances in Pure Mathematics, 11, 929-936. https://doi.org/10.4236/apm.2021.1112059
[12]
Pramanik, S., Das, D.K. and Pramanik, P. (2023) Products of Odd Numbers or Prime Number Can Generate the Three Members’ Families of Fermat Last Theorem and the Theorem Is Valid for Summation of Squares of More Than Two Natural Numbers. Advances in Pure Mathematics, 13, 635-641. https://doi.org/10.4236/apm.2023.1310043
[13]
Kraft, J.S. and Washington, L.C. (2018) An Introduction to Number Theory with Cryptography. 2nd Edition. Chapman and Hall/CRC Press, Boca Raton.