Tree of Fermat-Pramanik Series and Solution of <i>A^M </i>+<i>B</i>² =<i>C</i>² with Integers Produces a New Series of (<i>C</i>₁²- <i>B</i>₁²)=(<i>C</i>₂²- <i>B</i>₂²)=(<i>C</i>₃²- <i>B</i>₃²)=Others

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 A M + B 2 = C 2 which produces series of the type . It has been shown that this equation is solvable with N { A, B, C, M } . where , , M = M 1 + M 2 and M 1 > M 2 . Subsequently, it has been shown that using M = M 1 + M 2 + M 3 +.. . The combinations of M s should be taken so that the values of both the parts ( C n + B n ) and ( C n - B n ) 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.