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+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 (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.