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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’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
Article ID: 2624
Vol 2, Issue 2, 2024
DOI: https://doi.org/10.54517/mss.v2i2.2624
Vol 2, Issue 2, 2024
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
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Cairo University, Egypt
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