Survey of Cubic Fibonacci Identities When Cuboids Carry Weight


  • Mariana Nagy Aurel Vlaicu University of Arad, Romania
  • Simon Robin Cowell Aurel Vlaicu University of Arad, Romania
  • Valeriu Beiu Aurel Vlaicu University of Arad, Romania



Fibonacci number, Fibonacci identity, cubic identity


The aim of this paper is to present a comprehensive survey of cubic Fibonacci identities, trying to uncover as many as possible. From the outset, our rationale for a very careful search on an apparently obscure problem was not only a matter of mathematical curiosity, but also motivated by a quest for 3D Fibonacci spirals.

As we were not able to find any survey on the particular topic of cubic Fibonacci identities we decided to try to fill this void. We started by surveying many Fibonacci identities and recording cubic ones. Obviously, tracing all Fibonacci identities (for identifying a handful) is a daunting task. Checking several hundred we have realized that it is not always clear who the author is. The reason is that in many cases an identity was stated in one article (sometimes without a proof, e.g., as an open problem, or a conjecture) while later being proven in another article, or effectively rediscovered independently by other authors. However, we have done our best to present the identities chronologically. We have supplied our own proof for one identity, having tried, but failed, to find a published proof. For all the other identities, we either proved them on a computer, or else verified by hand their original published proofs. Somehow unexpectedly, our investigations have revealed only a rather small number of cubic Fibonacci identities, representing a tiny fraction of all published Fibonacci identities (most of which are linear or quadratic). Finally, out of these, only a handful of cubic Fibonacci identities are homogeneous.



[2] Adegoke, K. Binomial Fibonacci power sums, Preprints, Jun. 2021, art. 2021050378 [Online]. Available at:

[3] Adegoke, K. Binomial cubic Fibonacci sums, Preprints, Jul. 2021, art. 2021060425 [Online]. Available at:

[4] Azarian, M. K. Fibonacci identities as binomial sums I, Intl. J. Contemp. Math. Sci., vol. 7, no. 38, pp. 1871-1876, 2012.

[5] Azarian, M. K. Fibonacci identities as binomial sums II, Intl. J. Contemp. Math. Sci., vol. 7, no. 42, pp. 2053-2059, 2012.

[6] Beiu, V. Neural addition and Fibonacci numbers, Proc. Intl. Work-Conf. Artif. Natural Neural Nets. (IWANN'99), Alicante, Spain, Jun. 2-4, 1999, pp. 198-207.

[7] Beiu, V., Dragoi, V.-F. and Beiu, R.-M. Why reliability for computing needs rethinking, Proc. IEEE Intl. Conf. Rebooting Computing (ICRC 2020), Atlanta, GA, USA, Dec. 1-3, 2020, pp. 16-25.

[8] Benjamin, A. T., Carnes, T. A. and Cloitre, B. Recounting the sums of cubes of Fibonacci numbers, in Proc. Intl. Conf. Fibonacci Numbers & Their Appls., Braunschweig, Germany, 5-9 Jul. 2004

W. A. Webb (ed.): Applications of Fibonacci Numbers, Winnipeg, Manitoba, Canada: Utilitas Mathematica Publ., Jan. 2009, pp. 45-51 [Online]. Available at:

[9] Benjamin, A. T., Eustis, A. K. and Plott, S. S. The 99th Fibonacci identity, Electr. J. Combinatorics, vol. 15, #R34, pp. 1-13, 2008.

[10] Benjamin, A. T. and Quinn, J. J. Proofs that Really Count - The Art of Combinatorial Proof, Washington, DC, USA: Math. Assoc. Amer., 2003.

[11] Binet, J. "Mémoire sur l'intégration des équations linéaires aux différences finies, d'un ordre quelconque, í  coefficients variables [Memoir on the integration of linear equations with finite differences, of any order, with variable coefficients]," Comptes Rendus des Séances de l'Académie des Sciences, vol. 17, pp. 559-567, 25 Sept. 1843 [Online]. Available at: https://gallica.bnf. fr/ark:/12148/bpt6k2976b/f563

[12] Block, D. Curiosum #330: Fibonacci summations, Scripta Math., vol. 19, no. 2-3, p. 191, 1953.

[13] Brooke, M. Fibonacci numbers: Their history through 1900, Fib. Quart., vol. 2, no. 2, pp. 149-153, Apr. 1964.

[14] Brother Brousseau, A. A sequence of power formulas, Fib. Quart., vol. 6, no. 1, pp. 81-83, Feb. 1968.

[15] Brother Brousseau, A. Fibonacci and Related Number Theoretic Tables, Santa Clara, CA, USA: Fibonacci Assoc., 1972 [Online]. Available at:

[16] Brother Brousseau, A. Fibonacci numbers and geometry, Fib. Quart., vol. 10, no. 3, pp. 303-318 & 323, Apr. 1972.

[17] Brother Brousseau, A. Ye olde Fibonacci curiosity shoppe, Fib. Quart., vol. 10, no. 4, pp. 441-443, Oct. 1972.

[18] Cassini, J. D. Mathematique [Mathematics], Hist. Acad. Royale Sci., Tom. I (1666-1699), p. 201, 1680 [Online]. Available at:

[19] Castro, T. G., Munteanu, F.-D. and Cavaco-Paulo, A. Electrostatics of tau protein by molecular dynamics, Biomolecules, vol. 9, no. 3, art. 116 (pp. 1-16), Mar. 2019, biom9030116

[20] Catalan, E. C. CLXXXIX. - Sur la série de Lamé (Octobre 1879) [189 - On Lamé's series (October 1879)], Mém. Soc. Roy. Sci. Lií¨ge, ser. 2, vol. 13, pp. 319-321, 1886 [Online]. Available at:

[21] Chandra, P. andWeisstein, E. W. Fibonacci number, from MathWorld - AWolframWeb Resource.

[22] Chen, A. and Chen, H. Identities for the Fibonacci powers, Intl. J. Math. Edu. Sci. & Tech., vol. 39, no. 4, pp. 534-541, 2008.

[23] Chen, H. Excursions in Classical Analysis, Washington, DC, USA: Math. Assoc. Amer., 2010.

[24] Clary, S. and Hemenway, P. D. On sums of cubes of Fibonacci numbers, in Proc. Intl. Conf. Fibonacci Numbers & Their Appls., St. Andrews, Scotland, 20-24 Jul. 1992

G. E. Bergum, A. N. Philippou, and A. F. Horadam (eds.): Applications of Fibonacci Numbers, vol. 5, Dordrecht, Netherlands: Kluwer Acad. Pub., 1993, pp. 123-136.

[25] Cowell, S. R., Nagy, M. and Beiu, V. A proof of a generic Fibonacci identity from Wolfram's MathWorld, Theory Appl. Math. Comp. Sci., vol. 8, no. 1, pp. 60-63, 2018. Available at: http://

[26] de Moivre, A. Miscellanea Analytica de Seriebus et Quadraturis [Various Analyses on Series and Quadratures], London: J. Tonson & J. Watts, 1730 [Online].

[27] Devlin, K. The Man of Numbers: Fibonacci's Arithmetic Revolution, New York, NY, USA:Walker Co., 2011.

[28] Devlin, K. Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World, Princeton, NJ, USA: Princeton Univ. Press, 2017.

[29] Dickson, L. E. History of the Theory of Numbers (Vol. I: Divisibility and Primality), Washington, DC, USA: Carnegie Inst., 1919.

[30] Enright, M. B. and Leitner, D. M. Mass fractal dimension and the compactness of proteins, Phys. Rev. E, vol. 71, no. 1, art. 011912 (pp. 1-9), Jan. 2015.

[31] Fairgrieve, S. and Gould, H. W. Product difference Fibonacci identities of Simson, Gelin-Cesí ro, Tagiuri and generalizations, Fib. Quart., vol. 43, no. 2, pp. 137-141, May 2005.

[32] Ginsburg, J. Curiosum #343: A relationship between cubes of Fibonacci numbers, Scripta Math., vol. 19, no. 2-3, p. 242, Dec. 1953.

[33] Gould, H. W. The functional operator Tf(x) = f(x+a)f(x+b)−f(x)f(x+a+b), Math. Mag., vol. 37, no. 1, pp. 38-46, Jan. 1964.

[34] Hall, R. W. Math for poets and drummers, Math Horizons, vol. 15, no. 3, pp. 10-11 & 24, Feb. 2008.

[35] Halton, J. H. On a general Fibonacci identity, Fib. Quart., vol. 3, no. 1, pp. 31-43, Feb. 1965.

[36] Harris, V. C. On identities involving Fibonacci numbers, Fib. Quart., vol. 3, no. 3, pp. 214-218, Oct. 1965.

[37] Hillman, A. P. (ed), Elementary problems and solutions, Fib. Quart., vol. 8, no. 4, pp. 443-448, Oct. 1970.

[38] Hoggatt Jr., V. E. (ed.), Advanced problems and solutions, Fib. Quart., vol. 1, no. 1, pp. 46-48, Feb. 1963.

[39] Hoggatt Jr., V. E. (ed.), Advanced problems and solutions, Fib. Quart., vol. 1, no. 2, pp. 53-55, Apr. 1963.

[40] Hoggatt Jr., V. E. (ed.), Advanced problems and solutions, Fib. Quart., vol. 1, no. 3, pp. 46-52, Oct. 1963.

[41] Hoggatt Jr., V. E. (ed.), Advanced problems and solutions, Fib. Quart., vol. 1, no. 4, pp. 47-52, Dec. 1963.

[42] Hoggatt, Jr., V. E. and Bergum, G. E. A problem of Fermat and the Fibonacci sequence, Fib. Quart., vol. 15, no. 4, pp. 323-330, Dec. 1977.

[43] Horadam, A. F. Basic properties of a certain generalized sequence of numbers, Fib. Quart., vol. 3, no. 3, pp. 161-176, Oct. 1965.

[44] Johnson, R. C. Fibonacci numbers and matrices, 15 Jun. 2009 [Online]. Available at:

[45] Jumper, J., Evans, R., Pritzel, A., Green, T., Figurnov, M., Ronneberger, O., Tunyasuvunakool, K., Bates, R., Ží­dek, A., Potapenko, A., Bridgland, A., Meyer, C., Kohl, S. A. A., Ballard, A. J., Cowie, A., Romera-Paredes, B., Nikolov, S., Jain, R., Adler, J., Back, T., Petersen, S., Reiman, D., Clancy, E., Zielinski, M., Steinegger, M., Pacholska, M., Berghammer, T., Bodenstein, S., Silver, D., Vinyals, O., Senior, A. W., Kavukcuoglu, K., Kohli, P. and Hassabis, D. Highly accurate protein structure prediction with AlphaFold, Nature, vol. 596, no. 7873, pp. 583-589, 15 Jul. 2021,

[46] Kepler, J. De Nive Sexangula, Frankfurt on Main, Germany: Godfrey Tampach, 1611. See [Translated by C. Hardie, The Six-Cornered Snowflake, Oxford, UK: Clarendon Press, 1966]

[47] Khomovsky, D. I. A method for obtaining Fibonacci identities, Integers, vol. 18, art. A42 (pp. 1-9), May 2018 [Online]. Available at:

[48] Koshy, T. Fibonacci and Lucas Numbers with Applications, New York, NY, USA: J. Wiley & Sons, 2001.

[49] Lang, C. L. and Lang, M. L. Fibonacci numbers and identities, Fib. Quart., vol. 51, no. 4, pp. 330- 338, Nov. 2013.

[50] Lang, W. A215037: Applications of the partial summation formula to some sums over cubes of Fibonacci numbers, Tech. Notes. for A215037, Karlsruhe Inst. Tech. (KIT), Karlsruhe, Germany, 9 Aug. 2012 [Online]. Available at: [See]

[51] Livio, M. The Golden Ratio: The Story of Phi, the World's Most Astonishing Number, New York, NY, USA: Broadway Books, 2002.

[52] Lucas, F. í‰. A. Note sur le triangle arithmétique de Pascal et sur la série de Lamé, [Note on the arithmetic triangle of Pascal and on the series of Lamé] Nouv. Corresp. Math., vol. 2, pp. 70-75, 1876.

[53] Lucas, F. í‰. A. Recherches sur plusieurs ouvrages de Léonard de Pise et sur diverses questions d'arithmétique supérieure [Research on several publications of Leonardo of Pisa and on various questions of advanced arithmetic], Bullettino di bibliografia e di storia delle scienze matematiche e fisiche, vol. 10, S. 129-193, S. 239-293, Mar., Apr., & May 1877 [Online]. Available at:

[54] Mandelbrot, B. B. The Fractal Geometry of Nature, New York, NY, USA: W. H. Freeman and Co., 1982.

[55] Melham, R. S. Sums of certain products of Fibonacci and Lucas numbers, Fib. Quart., vol. 37, no. 3, pp. 248-251, Aug. 1999.

[56] Melham, R. S. A Fibonacci identity in the spirit of Simson and Gelin-Cesí ro, Fib. Quart., vol. 41, no. 2, pp. 142-143, May 2003.

[57] Melham, R. S. Some conjectures concerning sums of odd powers of Fibonacci and Lucas numbers, Fib. Quart., vol. 46/47, no. 4, pp. 312-315, Nov. 2008/2009.

[58] Melham, R. S. On product difference Fibonacci identities, Integers, vol. 11, art. A10 (pp. 1-8), Jan. 2011, [Online]. Available at:

[59] Mitchison, G. J. Phyllotaxis and the Fibonacci series, Science, vol. 196, no. 4287, pp. 270-275, 15 Apr. 1977.

[60] Munteanu, F.-D., Cavaco-Paulo, A. and Beiu, V. Studies of solvated ions in confined spaces, Intl. Conf. New Trends Sensing-Monitoring-Telediag. Life Sci. (NTSMT-LS), Bucharest, Romania, Sept. 2017 (abstract in J. Med. Life, vol. 10, no. 2, pp. 1-33, 2017).

[61] Nagy, M., Cowell, S. R. and Beiu, V. Are 3D Fibonacci spirals for real ? - From science to arts and back to sciences, in Proc. IEEE Intl. Conf. Comp. Comm. & Ctrl. (ICCCC), Oradea, Romania, 8-12 May 2018, pp. 91-96.

[62] Nagy, M., Cowell, S. R. and Beiu, V. 3D Fibonacci Spirals, under review, 2021.

[63] d'Ocagne, P. M. Sur une suite récurrente (Séance du 18 novembre 1885) [On a recurrent series (Meeting of 18 November 1885)], Bul. Soc. Math. France, vol. 14, pp. 20-41, 1886.

[64] OEIS Foundation Inc., Fibonacci numbers, The On-Line Encyclopedia of Integer Sequences,

[65] OEIS Foundation Inc., Lucas numbers, The On-Line Encyclopedia of Integer Sequences,

[66] OEIS Foundation Inc., Padovan numbers, The On-Line Encyclopedia of Integer Sequences,

[67] OEIS Foundation Inc., Pell numbers, The On-Line Encyclopedia of Integer Sequences,

[68] OEIS Foundation Inc., Third power of Fibonacci numbers, The On-Line Encyclopedia of Integer Sequences,

[69] Ollerton, R. L. Fibonacci cubes, Intl. J. Math. Edu. Sci. & Tech., vol. 37, no. 6, pp. 754-756, 2006.

[70] Padovan, R. Dom Hans van der Laan: Modern Primitive, Amsterdam, Netherlands: Architectura & Natura Press, Dec. 1994.

[71] Pin˙ gala, D. Chandah. s¯astra [Art of Prosody], c. 200 or 450 BC. [Pinglacarya, Chandahsastram, New Delhi, India: Parimal Pub., 2006.]

[72] Pisano, L. / Fibonacci, Liber Abaci [Book of Calculations], Biblioteca Nazionale Centrale di Firenze, Conventi Sopressi C.1.2616, 1202 (revised manuscript, 1228).

[73] Pond, J. C. Generalized Fibonacci summations, Fib. Quart., vol. 6, no. 2, pp. 97-108, Apr. 1968.

[74] Sadegh, S., Higgins, J. L., Mannion, P. C., Tamkun, M. M. and Krapf, D. Plasma membrane is compartmentalized by a self-similar cortical actin meshwork, Phys. Rev. X, vol. 7, no. 1, art. 011031 (pp. 1-10), Mar. 2017.

[75] Shah, J. A history of Pi ˙ ngala's combinatorics, Gan. ita Bh¯arat¯ı: Bulletin Indian Soc. History Maths., vol. 35, no. 1-2, Jun.-Dec. 2013 [Online]. Available at: https://web.northeastern. edu/shah/publications.html

[76] Sigler, L. E. Fibonacci's Liber Abaci - A Translation into Modern English of Leonardo Pisano's Book of Calculation, New York, NY, USA: Springer, 2002.

[77] Simson, R. LVI. An explication of an obscure passage in Albert Girard's commentary upon Simon Stevin's work, Phil. Trans. Royal Soc. London, vol. 48, pp. 368-377, Jan. 1753.

[78] Singh, A. N. On the use of series in Hindu mathematics, Osiris, vol. 1, pp. 606-628, Jan. 1936.

[79] Singh, P. The so-called Fibonacci numbers in ancient and medieval India, Historia Math., vol. 12, no. 3, pp. 229-244, Aug. 1985.

[80] Stevin, S. Les Oeuvres Mathematiques [Mathematical Publications] (with Albert Girard's commentary), Leyden, Netherlands: Bonaventure & Abraham Elsevier, 1634 [Online]. Available at:

[81] Stewart, I. Mathematical recreations: Tales of a neglected number, Sci. Amer., vol. 274, no. 6, pp. 102-103, Jun. 1996.

[82] Subba Rao, K. Some properties of Fibonacci numbers, Amer. Math. Month., vol. 60, no. 10, pp. 680-684, Dec. 1953.

[83] Sun, B. Y., Xie, M. H. Y. and Yang, A. L. B. Melham's conjecture on odd power sums of Fibonacci numbers, Quaestiones Math., vol. 39, no. 7, pp. 945-957, 2016.

[84] Tagiuri, A. Di alcune successioni ricorrenti a termini interi e positive [On some recurrent sequences with positive integer terms], Periodico di Mat., vol. 16, no. 3, pp. 1-12, 1901.

[85] Velankar H. D. (ed.), Vr.ttaj¯atisamuccaya [Collection of Mora- and Syllable-Counting Meters] of Virah¯a ˙ nka: Kavi Virah¯a˙nka Kr.ta Sat.¯ıka Vr.ttaj¯atisamuccaya, Jodhpur, India: Rajasthan Oriental Res. Inst., 1962.

[86] Vinson, J. E. Modulo m properties of the Fibonacci sequence, MA thesis, Oregon State Univ., Jun. 1961 [Online]. Available at: thesis_or_dissertations/zc77ss77r

[87] Vorob'ev, N. N. Chisla Fibonachchi [Russian], Moscow-Leningrad, Russia: Gostekhteoretizdat, 1951. [English translation: Fibonacci numbers, New York, NY, USA: Pergamon Press, 1961.]

[88] Wilson, E. M. The scales from the slopes of mountain Meru and other recurrent sequences - The scales of mountain Meru personal notes, 1992 [Online]. Available at: wilsonmeru.html

[89] Yazlik, Y., Yilmaz, N. and Taskara, N. On the sums of powers of k-Fibonacci and k-Lucas sequences, Selí§uk J. Appl. Math., pp. 47-50, 2012.

[90] Zeitlin, D. On identities for Fibonacci numbers, Amer. Math. Month., vol. 70, no. 9, pp. 987-991, 1963.

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