Nonexistence of Even Fibonacci Pseudoprimes of The

With regard to this problem, Di Porto and Filipponi, in [4], conjectured that there are no even-Fibonacci pseudoprimes of the 1 kind, providing some constraints are placed on their existence, and Somer, in [12], extends these constraints by stating some very interesting theorems. Moreover, in [1], a solution has been found for a similar problem, that is, for the sequence {Vn(2,1)}, defined by F0(2,l) = 2, ^(2,1) = 3, F„(2,1) = 3F^1(2,1)-2FW_2(2,1) = 2 + 1 Actu-ally Beeger, in [1], shows the existence of infinitely many even pseudoprimes n, that is, even n such that 2" s=2 (modn) <=>F„(2,1) = 2 + 1 = ̂ (2,1) (modw). After defining (in this section) the generalized Lucas numbers, Vn(m), governed by the positive integral parameter m, and after giving some properties of the period of the sequences {Vn(m}} reduced modulo a positive integer t, we define in section 2 the Fibonacci pseudoprimes of the mkind(m-F.'Psps.) and we give some propositions. Finally, in section 3, we demonstrate the above theorem. Throughout this paper, p will denote an odd prime and Vn(m) will denote the generalized Lucas numbers (see [2], [7]), defined by the second-order linear recurrence relation