# On Generalized 2-primes Numbers

## Main Article Content

## Abstract

In this paper, we introduce the generalized 2-primes sequences and we deal with, in detail, three special cases which we call them 2-primes, Lucas 2-primes and modified 2-primes sequences. We present Binet’s formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.

Keywords:

2-primes numbers, Lucas 2-primes numbers, generalized Fibonacci numbers.

## Article Details

How to Cite

*Asian Journal of Advanced Research and Reports*,

*9*(2), 34-53. https://doi.org/10.9734/ajarr/2020/v9i230217

Section

Original Research Article

## References

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Horadam AF. Special properties of the sequence wn(a; b; p; q). Fibonacci Quarterly. 1967;5(5):424-434.

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Article no.AJARR.56064

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Available:http://oeis.org/

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Fibonacci Quarterly. 1988;26:77-83.

Kocer GE. Circulant, negacyclic and semicirculant matrices with the modified Pell, Jacobsthal and Jacobsthal-Lucas numbers. Hacettepe Journal of Mathematics and Statistics.

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Dasdemir A. On the Pell, Pell-Lucas and modified Pell numbers by matrix method. Applied Mathematical Sciences.

;5(64):3173-3181.

Ercolano J. Matrix generator of Pell sequence. Fibonacci Quarterly.

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Mech. 2017;4(4):1-4.

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Taiwanese Journal of Mathematics.

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Hendel RJ. Proof and generalization of the cassini-catalan-tagiuri-gould identities.

Fibonacci Quarterly. 2017;55(5):76-85.

Koshy T. Gelin-cesaro identity for the gibonacci family. Math. Scientist.

;40:59-61.

Lang CL, Lang ML. Fibonacci numbers and identities; 2013. preprint,

arXiv:1303.5162v2 [math.NT].

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arXiv:1304.3388v4 [math.NT].

Melham RS. A Fibonacci identity in the spirit of simson and gelin-cesaro. Fibonacci

Quarterly. 2003:142-143.

Melham RS. On product difference fibonacci identities. Integers. 2011;11:8.

Soykan Y. Simson identity of generalized mstep fibonacci numbers. Int. J. Adv. Appl.

Math. and Mech. 2019;7(2):45-56.Soykan Y. generalized fibonacci numbers: Sum formulas. Journal of Advances in

Mathematics and Computer Science.

;35(1):89-104.

DOI: 10.9734/JAMCS/2020/v35i130241

Kalman D. Generalized fibonacci numbers by matrix methods. Fibonacci Quarterly.

;20(1):73-76.

Fibonacci Quarterly. 1965;3(3):161-176.

Horadam AF. A generalized fibonacci sequence. American Mathematical Monthly.

;68:455-459.

Horadam AF. Special properties of the sequence wn(a; b; p; q). Fibonacci Quarterly. 1967;5(5):424-434.

Horadam AF. Generating functions for powers of a certain generalized sequence of numbers. Duke Math. J. 1965;32:437-446.

Article no.AJARR.56064

Sloane NJA. The on-line encyclopedia of integer sequences.

Available:http://oeis.org/

Akbulak M, O¨ teles¸ A. On the sum of Pell and Jacobsthal numbers by matrix method. Bull. Iranian Mathematical Society.

;40(4):1017-1025.

Aydın FT. On generalizations of the Jacobsthal sequence. Notes on Number Theory and Discrete Mathematics.

;24(1):120-135.

Catarino P, Vasco P, Campos APA, Borges A. New families of Jacobsthal and Jacobsthal-Lucas numbers. Algebra and Discrete Mathematics. 2015;20(1):40-54.

Cˇ erin Z. Formulae for sums of Jacobsthal– Lucas numbers. Int. Math. Forum.

;2(40):1969-1984.

Cˇ erin Z. Sums of squares and products of jacobsthal numbers. Journal of Integer Sequences. 2007;10. Article 07.2.5, 2007 Dasdemir A. On the jacobsthal numbers by matrix method. SDU Journal of Science.

;7(1):6976.

Das¸demir A. A study on the Jacobsthal and Jacobsthal-Lucas numbers by matrix method. DUFED Journal of Sciences.

;3(1):13-18.

Gnanam A, Anitha B. Sums of squares Jacobsthal numbers. IOSR Journal of Mathematics. 2015;11(6):62-64.

Horadam AF. Jacobsthal representation numbers. Fibonacci Quarterly. 1996;34:40- Horadam AF. Jacobsthal and Pell curves.

Fibonacci Quarterly. 1988;26:77-83.

Kocer GE. Circulant, negacyclic and semicirculant matrices with the modified Pell, Jacobsthal and Jacobsthal-Lucas numbers. Hacettepe Journal of Mathematics and Statistics.

;36(2):133-142.

K¨oken F, Bozkurt D. On the Jacobsthal numbers by matrix methods. Int. J. Contemp Math. Sciences. 2008;3(13):605-614.

Mazorchuk V. New families of Jacobsthal and Jacobsthal-Lucas numbers. Algebra and Discrete Mathematics. 2015;20(1):40-Uygun S¸ . Some sum formulas of (s; t)- Jacobsthal and (s; t)-Jacobsthal lucas matrix sequences. Applied Mathematics.;7:61-69.

Bicknell N. A primer on the Pell sequence and related sequence. Fibonacci Quarterly.

;13(4):345-349.

Dasdemir A. On the Pell, Pell-Lucas and modified Pell numbers by matrix method. Applied Mathematical Sciences.

;5(64):3173-3181.

Ercolano J. Matrix generator of Pell sequence. Fibonacci Quarterly.

;17(1):71-77.

G¨okbas H, K¨ose H. Some sum formulas for products of Pell and Pell-Lucas numbers. Int. J. Adv. Appl. Math. and

Mech. 2017;4(4):1-4.

Horadam AF. Pell Identities. Fibonacci Quarterly. 1971;9(3):245-263.

Kilic¸ E, Tas¸c¸i D. The linear algebra of the pell matrix. Bolet´ın de la Sociedad Matem´ atica Mexicana. 2005;3(11).

Koshy T. Pell and Pell-Lucas numbers with applications. Springer. New York; 2014.

Melham R. Sums involving fibonacci and pell numbers. Portugaliae Mathematica.

;56(3):309-317.

Kilic¸ E, Tas¸c¸i D. The generalized Binet formula, representation and sums of the generalized order-k pell numbers.

Taiwanese Journal of Mathematics.

;10(6):1661-1670.

Kilic¸ E, Stanica P. A matrix approach for general higher order linear Recurrences.

Bulletin of the Malaysian Mathematical Sciences Society. 2011;34(1):51-67.

Soykan Y. On generalized third-order pell numbers. Asian Journal of Advanced Research and Reports. 2019;6(1):1-18.

Soykan Y. A study of generalized fourthorder pell sequences. Journal of Scientific Research and Reports. 2019;25(1-2):1-18.

Soykan Y. Properties of generalized fifthorder pell numbers. Asian Research Journal of Mathematics. 2019;15(3):1-18.

Soykan Y. On generalized sixth-order pell sequence. Journal of Scientific Perspectives. 2020;4(1):49-70.

Koshy T. Fibonacci and Lucas numbers with applications, a wiley-interscience publication. New York; 2001.

Vajda S. Fibonacci and Lucas numbers and the golden section. Theory and Applications, John Wiley & Sons, New York; Vorobiev NN. Fibonacci numbers,birkh¨auser basel; 2002. Originally published

under the title of ”Chisla Fibonacci” byNauka, Moscow; 1992 (6th Edition).

Cooper C. Some identities involving differences of products of generalized fibonacci numbers. Colloquium

Mathematicae. 2015;141(1):45-49.

Fairgrieve S. H.W. gould, product difference fibonacci identities of simson, gelin-cesaro, tagiuri and generalizations. Fibonacci Quarterly. 2005;137-141.

Hendel RJ. Proof and generalization of the cassini-catalan-tagiuri-gould identities.

Fibonacci Quarterly. 2017;55(5):76-85.

Koshy T. Gelin-cesaro identity for the gibonacci family. Math. Scientist.

;40:59-61.

Lang CL, Lang ML. Fibonacci numbers and identities; 2013. preprint,

arXiv:1303.5162v2 [math.NT].

Lang CL, Lang ML. Fibonacci numbers and identities II; 2013. preprint,

arXiv:1304.3388v4 [math.NT].

Melham RS. A Fibonacci identity in the spirit of simson and gelin-cesaro. Fibonacci

Quarterly. 2003:142-143.

Melham RS. On product difference fibonacci identities. Integers. 2011;11:8.

Soykan Y. Simson identity of generalized mstep fibonacci numbers. Int. J. Adv. Appl.

Math. and Mech. 2019;7(2):45-56.Soykan Y. generalized fibonacci numbers: Sum formulas. Journal of Advances in

Mathematics and Computer Science.

;35(1):89-104.

DOI: 10.9734/JAMCS/2020/v35i130241

Kalman D. Generalized fibonacci numbers by matrix methods. Fibonacci Quarterly.

;20(1):73-76.