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2024

Vol 2, No 2 (2024)

Vol 2, No 1 (2024)

2023

Vol 1, No 1 (2023)

Featured Articles

Why the big bang never happened

If general relativity is correct, then the origin of the universe is a simple mathematical problem. The Friedmann equation in cosmology is a well-structured ordinary diﬀerential equation, and the global properties of its solutions can be qualitatively analyzed by the phase-trajectory method. In this paper we show that the total energy density of matter in the universe is positive, and the total pressure near the Big Bang is negative. By analyzing the global properties of the solutions to the Friedmann equation according to these two conditions of state functions, we ﬁnd that the Big Bang is impossible, and the space must be a closed 3-dimensional sphere, the cosmological constant is likely to be zero, and the evolution of the universe should be cyclic. The analysis and the proof are simple and straight forward, therefore these conclusions should be reliable.

Fermat surfaces and hypercubes

Fermat’s last theorem appears not as a unique property of natural numbers but as the bottom line of extended possible issues involving larger dimensions and powers when observed from a natural vector space viewpoint. The fabric of this general Fermat’s theorem structure consists of a well-defined set of vectors associated with dimensional vector spaces and the Minkowski norms one can define there. Here, a special vector set is studied and named a Fermat surface. Besides, a connection between Fermat surfaces and hypercubes is unveiled.

On matrix strong Diophantine 27-Tuples and matrix elliptic curves

Joachim Moussounda Mouanda, Kouakou Kouassi Vincent

Article ID: 2624
Vol 2, Issue 2, 2024

DOI: https://doi.org/10.54517/mss.v2i2.2624

Received: 14 March 2024; Accepted: 26 June 2024; Available online: 7 July 2024;
Issue release: 15 November 2024

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Abstract

We introduce an algorithm which allows us to prove that there exists an infinite number of matrix strong Diophantine -tuples. We show that Diophantine quadruples generate matrix elliptic (or hyperelliptic) curves which have each matrix points.

Keywords

Matrices of integers; Diophantine m-tuples; elliptic curves

References

1. Bashmakova IG. Diophantus of Alexandria, Arithmetics and the Book of Polygonal Numbers. Moscow: Nauka; 1974.

2. Euler L. Theorematum quorundam arithmeticorum demonstrationes. Novi Commentarii academiae scientiarum Petropolitanae. 1738; 10: 125-146.

3. Baker A, Davenport H. The equations 3x^2-2=y^2 and 8x^2-7=z^2. The Quarterly Journal of Mathematics. 1969; 20(1): 129-137. doi: 10.1093/qmath/20.1.129

4. Hoggatt VE, Bergum GE. A problem of Fermat and the Fibonacci sequence, Fibonacci Quart. Available online: https://www.mathstat.dal.ca/FQ/Scanned/15-4/hoggatt1.pdf (accessed on 7 March 2024).

5. Joseph A, Jr Hoggatt VE, Straus EG. On Euler’s solution of a problem of Diophantus. Fibonacci Quarterly. 1979; 17(4): 333339.

6. Veluppillai M. A Collection of Manuscripts Related to the Fibonacci sequence, The Fibonacci Association. Santa Clara; 1980. pp. 71-75.

7. Kedlaya K. Solving constrained Pell equations. Mathematics of Computation. 1998; 67(222): 833-842. doi: 10.1090/s0025-5718-98-00918-1

8. Dujella A. The problem of the extension of a parametric family of Diophantine triples. Publicationes Mathematicae Debrecen. 1997; 51(3-4): 311-322. doi: 10.5486/pmd.1997.1886

9. Dujella A, Petho A. A Generalization of a Theorem of Baker and Davenport. The Quarterly Journal of Mathematics. 1998; 49(3): 291-306. doi: 10.1093/qmathj/49.3.291

10. Dujella A. A proof of the Hoggatt-Bergum conjecture. Proceedings of the American Mathematical Society. 1999; 127(7): 1999-2005. doi: 10.1090/s0002-9939-99-04875-3

11. Fujita Y. The extensibility of Diophantine pairs{k−1,k+1}. Journal of Number Theory. 2008; 128(2): 322-353. doi: 10.1016/j.jnt.2007.03.013

12. Dujella A. There are only finitely many Diophantine quintuples. Journal für die reine und angewandte Mathematik (Crelles Journal). 2004; 2004(566). doi: 10.1515/crll.2004.003

13. He B, Togbé A, Ziegler V. There is no Diophantine Quintuple.Transactions of the American Mathematical Society. arXiv. 2019; arXiv:1610.04020.

14. Dujella A, Petričcević V. Strong Diophantine Triples. Experimental Mathematics. 2008; 17(1): 83-89. doi: 10.1080/10586458.2008.10129020

15. Siegel CL. The integer solutions of the equation y^2=ax^n+bx^(n-1)+⋯+k. Journal of the London Mathematical Society. 1926.

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Copyright (c) 2024 Joachim Moussounda Mouanda, Kouakou Kouassi Vincent

License URL: https://creativecommons.org/licenses/by/4.0/

Editor-in-Chief

Prof. Youssri Hassan Youssri

Cairo University, Egypt

Processing Speed (2023)

- 68 days: received to online

News & Announcements

2024-09-20

Highly Read Article Recommendation

Since the journal launch in 2023, we have been proud to publish a plethora of insightful findings in the realms of mathematics and systems science....

2024-07-10

Meeting our Editor-in-Chief and Associate Editor

It is with great pride that we introduce Prof. Youssri Hassan Youssri and Prof. Ali Akgül, who are the Editor-in-Chief and Associate Editor of our esteemed journal.

2024-01-05

Articles in the First Issue of Volume 1 - Now Available Online

We are delighted to announce that the articles in the first issue of Volume 1 of Mathematics and Systems Science is now live and accessible online.

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