On Generalized Lucas Pseudoprimality of Level k

Abstract

We investigate the Fibonacci pseudoprimes of level k, and we disprove a statement concerning the relationship between the sets of different levels, and also discuss a counterpart of this result for the Lucas pseudoprimes of level k. We then use some recent arithmetic properties of the generalized Lucas, and generalized Pell–Lucas sequences, to define some new types of pseudoprimes of levels k+ and k− and parameter a. For these novel pseudoprime sequences we investigate some basic properties and calculate numerous associated integer sequences which we have added to the Online Encyclopedia of Integer Sequences.

Comparative Study Between a Novel Deterministic Test for Mersenne Primes and the Well-Known Primality Tests

“…This test is considered a generalization for the SST and it is based on a special recursive sequence called Luca's sequence 3,11 . If P and Q are any integers and if 𝛼 = (𝑃 + √𝐷)/2 and 𝛽 = (𝑃 − √𝐷)/2 are the roots of the quadratic equation 𝑥 2 − 𝑃𝑥 + 𝑄 = 0 whose discriminant 𝐷 = 𝑃 2 − 4𝑄 is positive, then the following relations are obtained: 𝑃 = 𝛼 + 𝛽, 𝑄 = 𝛼𝛽, and 𝐷 = (𝛼 − 𝛽) 2 .…”

“…This study suggests a new approach that can be used to solve this problem by doing some modifications on SLT to present a new modified test called Hindi Selfridge Lucas test (HLT) with the help of FLT using base 3. Also, the selection of the Lucas sequence criteria and 𝐷 are based on Selfridge's method 11,12 .…”

“…The following lemma can be found in 11 . Lemma 1: If n is odd and {𝑈 𝑛 } 𝑛≥0 is a Lucas sequence so that ( 𝐷 𝑛 ) = −1 and 𝑛|𝑈 𝑛+1 , then 𝑔𝑐𝑑(𝑛, 𝑄𝐷) = 1.…”

“…The overall time complexity of the HLT approach is twice the time needed for the computation of 𝑎 𝑛 (𝑚𝑜𝑑 𝑚). So, by using matrix representation 11,12 , it is acquired that the time complexity is of order 𝑂(𝑛 4 log 3 𝑛). This result is tested on all numbers until 800,000 digits and none of them satisfies Theorem 7.…”

Carlitz’s Equations on Generalized Fibonacci Numbers

“…Their fascinated properties lead to abundant applications in totally surprising and unrelated fields (cf. [1][2][3][4][5][6]).…”