We study new primality tests based on linear recurrent sequences of degree two exploiting a matrix approach. The classical Lucas test arises as a particular case and we see how it can be easily improved. Moreover, this approach shows clearly how the Lucas pseudoprimes are connected to the Pell equation and the Brahamagupta product. We also introduce two new specific primality tests, which we will call generalized Lucas test and generalized Pell test. We perform some numerical computations on the new primality tests and we do not find any pseudoprime up to 238. Moreover, we combined the generalized Lucas test with the Fermat test up to 264 and we did not find any composite number that passes the test. We get the same result using the generalized Pell test.

Primality tests, linear recurrent sequences and the Pell equation / Bazzanella, Danilo; Di Scala, Antonio; Dutto, Simone; Murru, Nadir. - In: RAMANUJAN JOURNAL. - ISSN 1382-4090. - 57:2(2022), pp. 755-768. [10.1007/s11139-020-00373-9]

Primality tests, linear recurrent sequences and the Pell equation

Murru, Nadir
2022-01-01

Abstract

We study new primality tests based on linear recurrent sequences of degree two exploiting a matrix approach. The classical Lucas test arises as a particular case and we see how it can be easily improved. Moreover, this approach shows clearly how the Lucas pseudoprimes are connected to the Pell equation and the Brahamagupta product. We also introduce two new specific primality tests, which we will call generalized Lucas test and generalized Pell test. We perform some numerical computations on the new primality tests and we do not find any pseudoprime up to 238. Moreover, we combined the generalized Lucas test with the Fermat test up to 264 and we did not find any composite number that passes the test. We get the same result using the generalized Pell test.
2022
2
Bazzanella, Danilo; Di Scala, Antonio; Dutto, Simone; Murru, Nadir
Primality tests, linear recurrent sequences and the Pell equation / Bazzanella, Danilo; Di Scala, Antonio; Dutto, Simone; Murru, Nadir. - In: RAMANUJAN JOURNAL. - ISSN 1382-4090. - 57:2(2022), pp. 755-768. [10.1007/s11139-020-00373-9]
File in questo prodotto:
File Dimensione Formato  
pseudo-exp-rev.pdf

Solo gestori archivio

Tipologia: Pre-print non referato (Non-refereed preprint)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 314.17 kB
Formato Adobe PDF
314.17 kB Adobe PDF   Visualizza/Apri
Bazzanella2021_Article_PrimalityTestsLinearRecurrentS.pdf

Solo gestori archivio

Tipologia: Versione editoriale (Publisher’s layout)
Licenza: Tutti i diritti riservati (All rights reserved)
Dimensione 378.44 kB
Formato Adobe PDF
378.44 kB Adobe PDF   Visualizza/Apri

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/312536
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 1
  • ???jsp.display-item.citation.isi??? 1
  • OpenAlex ND
social impact