A Study On Generalized Mersenne Numbers
Keywords:
Mersenne numbers, Mersenne-Lucas numbers, generalized Fibonacci numbers.
Abstract
In this paper, we introduce the generalized Mersenne sequence and we deal with, in detail, two special cases, namely, Mersenne and Mersenne-Lucas 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.
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References
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Brillhart, J., On the Factors of Certain Mersenne Numbers, Math. Comp. 14(72), 365-369, 1960.
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Catarino, P., Campos, H., Vasco, P., On the Mersenne Sequence. Annales Mathematicae et Informaticae, 46, 37--53, 2016.
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Daşdemir, A., Mersene, Jacobsthal, and Jacobsthal-Lucas Numbers with Negative Subscripts, Acta Mathematica Universitatis Comenianae, 88(1), 145--156, 2019.
Ehrman, J. R.,The Number of Prime Divisors of Certain Mersenne Numbers, Math. Comp. 21(100), 700--704, 1967.
Ford, K., Luca, F., Shparlinski, I. E., On the Largest Prime Factor of the Mersenne Numbers, Bull. Austr. Math. Soc. 79(3), 455--463, 2009.
Goy, T., On New Identities for Mersenne Numbers, Applied Mathematics E-Notes, 18, 100--105, 2018.
Granger, R., Moss, A., Generalized Mersenne Numbers Revisited, Mathematics of Computation, 82(284), 2389--2420, 2013.
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Horadam, A.F., Special Properties of The Sequence w_{n}(a,b;p,q), Fibonacci Quarterly, 5(5), 424-434, 1967.
Horadam, A.F., Generating functions for powers of a certain generalized sequence of numbers. Duke Math. J 32, 437-446, 1965.
Jaroma, J. H., Reddy, K. N., Classical and Alternative Approaches to the Mersenne and Fermat Numbers, American Mathematical Monthly, 114(8), 677--687, 2007. DOI: 10.1080/00029890.2007.11920459
Murata, L., Pomerance, C., On the Largest Prime Factor of a Mersenne Number, Number Theory 36, 209--218, 2004.
Ochalik, P., Włoch, A., On Generalized Mersenne Numbers, Their Interpretations and Matrix Generators, Ann. Univ. Mariae Curie-Skłodowska Sect. A., 72(1), 69--76, 2018. doi: 10.17951/a.2018.72.1.69
Pomerance, C., On Primitive Divisors of Mersenne Numbers, Acta Arith. 46, 355--367, 1986.
Samuel, S., Wagstaff, Jr., Divisors of Mersenne Numbers, Math. Comp. 40, 385--397, 1983.
Schinzel, A., On Primitive Prime Factors of aⁿ-bⁿ, Proc. Cambridge Philos. Soc. 58(4), 555--562, 1962.
Sloane, N.J.A., The on-line encyclopedia of integer sequences, http://oeis.org/
Šolcová, A., Křižek, M., Fermat and Mersenne Numbers in Pepin's Test, Demonstratio Mathematica, 39(4), 2006.
Solinas. J.A., Generalized Mersenne Numbers. Technical report CORR-39, Dept. of C&O, University of Waterloo, 1999. Available from http://www.cacr.math.uwaterloo.ca
Soykan, Y., Simson Identity of Generalized m-step Fibonacci Numbers, Int. J. Adv. Appl. Math. and Mech. 7(2), 45-56, 2019.
Soykan, Y., On Generalized (r,s)-numbers, International Journal of Advances in Applied Mathematics and Mechanics, 8(1), 1-14, 2020.
Soykan, Y., Some Properties of Generalized Fibonacci Numbers: Identities, Recurrence Properties and Closed Forms of the Sum Formulas ∑_{k=0}ⁿx^{k}W_{mk+j}, Archives of Current Research International, 21(3), 11-38, 2021. DOI: 10.9734/ACRI/2021/v21i330235
Stewart, C. L., The Greatest Prime Factor of aⁿ-bⁿ, Acta Arith. 26, 427--433, 1975.
Zatorsky, R., Goy, T., Parapermanents of Triangular Matrices and Some General Theorems on Number Sequences, Journal of Integer Sequences, 19, Article 16.2.2, 2016.
Brillhart, J., On the Factors of Certain Mersenne Numbers, Math. Comp. 14(72), 365-369, 1960.
Brillhart, J., On the Factors of Certain Mersenne Numbers, II, Math. Comp. 18, 87--92, 1964.
Catarino, P., Campos, H., Vasco, P., On the Mersenne Sequence. Annales Mathematicae et Informaticae, 46, 37--53, 2016.
Chelgham, M., Boussayoud, A., On the k-Mersenne--Lucas Numbers, Notes on Number Theory and Discrete Mathematics, 27(1), 7--13, 2021. DOI: 10.7546/nntdm.2021.27.1.7-13
Daşdemir, A., Mersene, Jacobsthal, and Jacobsthal-Lucas Numbers with Negative Subscripts, Acta Mathematica Universitatis Comenianae, 88(1), 145--156, 2019.
Ehrman, J. R.,The Number of Prime Divisors of Certain Mersenne Numbers, Math. Comp. 21(100), 700--704, 1967.
Ford, K., Luca, F., Shparlinski, I. E., On the Largest Prime Factor of the Mersenne Numbers, Bull. Austr. Math. Soc. 79(3), 455--463, 2009.
Goy, T., On New Identities for Mersenne Numbers, Applied Mathematics E-Notes, 18, 100--105, 2018.
Granger, R., Moss, A., Generalized Mersenne Numbers Revisited, Mathematics of Computation, 82(284), 2389--2420, 2013.
Horadam, A.F., A Generalized Fibonacci Sequence, American Mathematical Monthly, 68, 455-459, 1961.
Horadam, A.F., Basic Properties of a Certain Generalized Sequence of Numbers, Fibonacci Quarterly, 3.3, 161-176, 1965.
Horadam, A.F., Special Properties of The Sequence w_{n}(a,b;p,q), Fibonacci Quarterly, 5(5), 424-434, 1967.
Horadam, A.F., Generating functions for powers of a certain generalized sequence of numbers. Duke Math. J 32, 437-446, 1965.
Jaroma, J. H., Reddy, K. N., Classical and Alternative Approaches to the Mersenne and Fermat Numbers, American Mathematical Monthly, 114(8), 677--687, 2007. DOI: 10.1080/00029890.2007.11920459
Murata, L., Pomerance, C., On the Largest Prime Factor of a Mersenne Number, Number Theory 36, 209--218, 2004.
Ochalik, P., Włoch, A., On Generalized Mersenne Numbers, Their Interpretations and Matrix Generators, Ann. Univ. Mariae Curie-Skłodowska Sect. A., 72(1), 69--76, 2018. doi: 10.17951/a.2018.72.1.69
Pomerance, C., On Primitive Divisors of Mersenne Numbers, Acta Arith. 46, 355--367, 1986.
Samuel, S., Wagstaff, Jr., Divisors of Mersenne Numbers, Math. Comp. 40, 385--397, 1983.
Schinzel, A., On Primitive Prime Factors of aⁿ-bⁿ, Proc. Cambridge Philos. Soc. 58(4), 555--562, 1962.
Sloane, N.J.A., The on-line encyclopedia of integer sequences, http://oeis.org/
Šolcová, A., Křižek, M., Fermat and Mersenne Numbers in Pepin's Test, Demonstratio Mathematica, 39(4), 2006.
Solinas. J.A., Generalized Mersenne Numbers. Technical report CORR-39, Dept. of C&O, University of Waterloo, 1999. Available from http://www.cacr.math.uwaterloo.ca
Soykan, Y., Simson Identity of Generalized m-step Fibonacci Numbers, Int. J. Adv. Appl. Math. and Mech. 7(2), 45-56, 2019.
Soykan, Y., On Generalized (r,s)-numbers, International Journal of Advances in Applied Mathematics and Mechanics, 8(1), 1-14, 2020.
Soykan, Y., Some Properties of Generalized Fibonacci Numbers: Identities, Recurrence Properties and Closed Forms of the Sum Formulas ∑_{k=0}ⁿx^{k}W_{mk+j}, Archives of Current Research International, 21(3), 11-38, 2021. DOI: 10.9734/ACRI/2021/v21i330235
Stewart, C. L., The Greatest Prime Factor of aⁿ-bⁿ, Acta Arith. 26, 427--433, 1975.
Zatorsky, R., Goy, T., Parapermanents of Triangular Matrices and Some General Theorems on Number Sequences, Journal of Integer Sequences, 19, Article 16.2.2, 2016.
Published
2021-09-20
How to Cite
Soykan, Y. (2021). A Study On Generalized Mersenne Numbers. Journal of Progressive Research in Mathematics, 18(3), 90-108. Retrieved from http://scitecresearch.com/journals/index.php/jprm/article/view/2088
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