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REFERENCES

[1]Gerck, E. “Algorithms for Quantum Computation: Derivatives of Discontinuous Functions.” Mathematics 2023, 11, 68. https://doi.org/10.3390/math1101006.

[2]Vandersypen, L. M. K. et al. Experimental realization of Shor’s quantum factoring algorithm using nuclear magnetic resonance. Nature 414, 883–887 (2001).

[3]Martín-López, E. et al. Experimental realization of Shor’s quantum factoring algorithm using qubit recycling. Nat. Photon.

6, 773–776 (2012).

[4]Bocharov, A., Roetteler, M., & Svore, K. M. Factoring with qutrits: Shor’s algorithm on ternary and metaplectic quantum architectures. Phys. Rev. A 96, 012306

(2017).

[5]Smolin, J. A., Smith, G. & Vargo, A. Oversimplifying quantum factoring. Nature 499, 163–165 (2013).

[6]Dattani, Nikesh S., & Bryans, Nathaniel. Quantum factorization of 56153 with only 4 qubits. arXiv preprint arXiv:1411.6758v3 (2014).

[7]Yan, S. Y. Quantum Algorithms for Integer Factorization. In:

Quantum Computational Number Theory. Springer, Cham

59–119 (2015).

[8]Peng, X. et al. Quantum Adiabatic Algorithm for Factorization and Its Experimental Implementation. Phys. Rev. Lett. 101, 220405 (2008).

31

[9]Burges, C. J. C. Factoring as Optimization. Microsoft Research MSR-TR-200 (2002).

[10]Wang, B., Yang, X. & Zhang, D. Research on Quantum Annealing Integer Factorization Based on Different

Columns. Front. Phys. 10:914578 (2022).

[11]Dridi, R. & Alghassi, H. Prime Factorization using Quantum Annealing and Computational Algebraic Geometry.

Sci. Rep. 7, 43048 (2017).

[12]Xu, N. et al. Quantum Factorization of 143 on a Dipolar-

Coupling Nuclear Magnetic Resonance System. Phys.

Rev. Lett. 108, 130501 (2012).

[13]Li, Zhaokai et al. High-fidelity adiabatic quantum computation using the intrinsic Hamiltonian of a spin system: Application to the experimental factorization of 291311. arXiv preprint arXiv:1706.08061 (2017).

[14]Xu, K. et al. Experimental Adiabatic Quantum Factorization under Ambient Conditions Based on a Solid-State

Single Spin System. Phys. Rev. Lett. 118, 130504 (2017).

[15]Dhaulakhandi, R., Bikashi, K. B., Seo, F. J. Factorization of large tetra and penta prime numbers on IBM quantum processor. Published in arxiv.2304.04999. 2023.

[16]Gerck, E. “Pathologically breaking RSA by quantum computation”. Published in https://www.researchgate.net/publication/374155658/, 2023.

Accessed October 12, 2023.

[17]Gerck, E. “SECURE EMAIL TECHNOLOGIES X.509 / PKI,

PGP, IBE, AND ZSENTRY/ZMAIL: A Usability & Security

32

Comparison”. Published in book: “Corporate Email Management”, Chapter: 12, Publisher: ICFAI University Press. 2008. Also at “https://www.researchgate.net/publication/286458410/”. Accessed October 12, 2023.

[18]Gerck, E, and Cruz, C.H.B.. “Eigenvalue translation method for mode calculations”. Published in Applied Optics 18(9):1341. 1979. Also at “https://www.researchgate.net/publication/294090734/”. Accessed October 12, 2023.

[19]Gerck,E. “Prony method revised” 1979. Published in Applied Optics 18(18):3075. Accessed October 12, 2023.

[20]Gerck, E. and Gerck, E. V. “New Physics with the

Euler-Lagrange Equation: Going Beyond Newton”. Published in https://www.researchgate.net/publication/333677283/, 2019. Accessed October 12, 2023.

[21]Oppenheim, A.V. , Schafer, R. W., Yuen, C. K. “Digital Signal Processing”. 1978. Published in IEEE Transactions on

Systems, Man, and Cybernetics.

[22]Gerck, E. “Tri-State+ communication symmetry using the algebraic approach”. Published in https://www.researchgate.net/publication/350950693/ , 2019.

Accessed October 12, 2023.

[23]Konstantogiannis, S.

https://www.researchgate.net/profile/Spiros-Konstantogiannis, accessed June 8, 2023.

33

[24]Cesàro, E. (1894). "Sur une formule empirique de M. Pervouchine". Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences. 119: 848–849.

[25]Dusart, P. “The k^{th} prime is greater than k(\ln k + \ln\ln k-1) for k ≥ 2”. 1999, Mathematics of Computation 68(225): 411-416. Also at DOI: 10.1090/S0025-5718-99-01037-6. Accessed June 8, 2023.

[26]Lamé, G. 1884 "Note sur la limite du nombre des divisions dans la recherche du plus grand commun diviseur entre deux nombres entiers". Comptes rendus des séances de l'Académie des Sciences. 19: 867–870. 1884.

[27]Gerck, E., Gallas, J. A. C., and d'Oliveira, A.B. “Solution of the Schrödinger equation for bound states in closed form”.

1982. Published in Phys. Rev. A 26, 662. Also at https://doi.org/10.1103/PhysRevA.26.662. Accessed October 12, 2023.

[28]Gerck, E., and d'Oliveira, A.B.”Continued fraction calculation of the eigenvalues of tridiagonal matrices arising from the Schroedinger equation”. Published by Elsevier in

Journal of Computational and Applied Mathematics

Volume 6, Issue 1, 1980, Pages 81-82. Also available at https://www.sciencedirect.com/science/article/pii/0771050X809

00200. See as “Matrix-Variational Method: An Efficient

Approach to Bound State Eigenproblems”, Report number:

EAV-12/78. Affiliation: Laboratorio de Estudos Avancados, IAE,

CTA, S. J. Campos, SP, Brazil; available online at https://www.researchgate.net/publication/286625459/. Accessed October 12, 2023. In unpublished work, in 1977, cited as [3] in Report number: EAV-12/78, Gerck notes his observations on the “exponential difference”, that the method is based on a new piecewise representation of the second-derivative operator on

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a set of functions that obey the boundary conditions, which was disclosed in Report number: EAV-12/78.

[29] Gerck, E. and Gerck, A. Quickest Calculus, https://www.researchgate.net/publication/365130122/, accessed

June 19, 2013, also available printed at Barnes & Noble, in https://www.barnesandnoble.com/w/quickest-calculus-ann-voge l-gerck/1142516805/, accessed October 12, 2023.

[30]RSA Laboratories, “RSA Challenge Numbers”. Published in https://web.archive.org/web/20130921041734/http://www.emc.com/e mc-plus/rsa-labs/historical/the-rsa-challenge-numbers.htm. Accessed October 12, 2023.

[31]Gerck, E. “Non-Boolean Logic in Medicine”. Published in https://www.researchgate.net/publication/369560023/, 2019. Accessed October 12, 2023.

[32]Epps, J., Ying, H. & Huttley, G.A. “Statistical methods for detecting periodic fragments in DNA sequence data”. Published in Biol Direct 6, 21 (2011). https://doi.org/10.1186/1745-6150-6-21. Accessed October 12, 2023.

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