On the properties of r-circulant matrices involving Mersenne and Fermat numbers

Document Type : Research Paper


1 Department of Mathematics, Central University of Jharkhand, Ranchi, India

2 Department of Mathematics, Babasaheb Bhimrao Ambedkar University, India

3 Department of Mathematics, Erzincan Binali Yıldırım University, Turkey


The aim of this study is to investigate r-circulant matrices containing Mersenne and Fermat numbers with arithmetic indices. We obtain the eigenvalues and determinants of these matrices implicitly. In addition, limits for matrix norms and spectral norms of these matrices are obtained. Thus, the results for right and skew-right circulant matrices appear immediately.


[1] A.C.F. Bueno, On the eigenvalues and the determinant of the right circulant matrices with Pell and Pell–Lucas numbers, Int. J. Math. Sci. Comput. 4 (2014), no. 1, 19–20.
[2] P. Catarino, H. Campos, and P. Vasco, On the Mersenne sequence, Ann. Math. Inf. 46 (2016), 37–53.
[3] M. Chelgham and A. Boussayoud, On the k-Mersenne–Lucas numbers, Notes Number Theory Discrete Math. 1 (2021), no. 27, 7–13.
[4] R.E. Cline, R.J. Plemmons, and G. Worm, Generalized inverses of certain Toeplitz matrices, Linear Algebra Appl. 8 (1974), no. 1, 25–33.
[5] B. Fischer and J. Modersitzki, Fast inversion of matrices arising in image processing, Number Algorithms 22 (1999), no. 1, 1–11.
[6] R. Frontczak and T. Goy, Mersenne-horadam identities using generating functions, Carpathian Math. Publ. 12 (2020), no. 1, 34–45.
[7] S. Georgiou and C. Kravvaritis, New good quasi-cyclic codes over GF(3), Int. J. Algebra 1 (2007), no. 1, 11–24.
[8] S. Kounias, C. Koukouvinos, N. Nikolaou, and A. Kakos, The non-equivalent circulant d-optimal designs for n ≡ 2 mod 4, n ≤ 54, n = 66, J. Comb. Theory Ser. A. 65 (1994), no. 1, 26–38.
[9] I. Kra and S.R. Simanca, On circulant matrices, Notices AMS 59 (2012), no. 3, 368–377.
[10] M. Kumari, J. Tanti, and K. Prasad, On some new families of k-Mersenne and generalized k-gaussian Mersenne numbers and their polynomials, Contrib. Discrete Math. (2023), (to appear).
[11] E. Ozkan, A. G¨o¸cer, and I. Altun, ¨ A new sequence realizing Lucas numbers, and the Lucas bound, Electronic J. Math. Anal. Appl. 5 (2017), no. 1.
[12] E. Ozkan and M. Uysal, ¨ Mersenne-Lucas hybrid number, Math. Montisnigri 52 (2021), 17–29.
[13] S.-Q. Shen and J.-M. Cen, On the spectral norms of r-circulant matrices with the k-Fibonacci and k-Lucas numbers, Int. J. Contemp. Math. Sci. 5 (2010), no. 12, 569–578.
[14] S.-Q. Shen and J.-M. Cen, On the bounds for the norms of r-circulant matrices with the Fibonacci and Lucas numbers, Appl. Math. Comput. 216 (2010), no. 10, 2891–2897.
[15] S. Solak, On the norms of circulant matrices with the Fibonacci and Lucas numbers, Appl. Math. Comput. 160 (2005), no. 1, 125–132.
[16] Y. Soykan, A study on generalized Mersenne numbers, J. Prog. Res. Math. 18 (2021), no. 3, 90–108.
[17] R. T¨urkmen and H. G¨okba¸s, On the spectral norm of r-circulant matrices with the Pell and Pell-Lucas numbers, J. Inequal. Appl. 2016 (2016), Article no. 65.
[18] Y. Yu and Z. Jiang, On the norms and spreads of Fermat, Mersenne and Gaussian Fibonacci RFMLR-circulant matrices, WSEAS Trans. Math. 15 (2016), Art. #4, 34–43.
[19] Y. Zheng and S. Shon, Exact inverse matrices of Fermat and Mersenne circulant matrix, Abstr. Appl. Anal. 2015 (2015), Article ID 760823.
Volume 14, Issue 5
May 2023
Pages 121-131
  • Receive Date: 21 July 2022
  • Revise Date: 22 February 2023
  • Accept Date: 03 March 2023
  • First Publish Date: 03 March 2023