A Conjecture Congenetic with Fermat’s Last Theorem

## Keywords

Conjecture
Positive integer
Diophantine equation
Fermat’s last theorem
Indeterminate equation

## How to Cite

Duan, J.-S., & Wang, J.-L. . (2022). A Conjecture Congenetic with Fermat’s Last Theorem. Journal of Advances in Applied & Computational Mathematics, 9, 129–134. https://doi.org/10.15377/2409-5761.2022.09.9  ## Abstract

We propose the conjecture that for any positive integers r and n with n > 2, there do not exist 2r + 1 consecutive positive integers in natural order such that the sum of n-th powers of the first r + 1 integers equals the sum of n-th powers of the subsequent r integers, i.e., there are no positive integers r, m and n, where r < m and n > 2, satisfying (m r)n + (m r + 1)n + … + mn = (m + 1)n + (m + 2)n + … + (m + r)n. We prove that the conjecture is true for the cases n = 3 and n = 4. We also verified by using Mathematica that the conjecture is true for the cases 3 < n < 10 and m < 5000.

## References

Ridders CJF. Fermat’s last theorem. Internat. J. Math. Ed. Sci. Tech., 1981; 12(4): 407-409. https://doi.org/10.1080/0020739810120405

Cox DA. Introduction to Fermat’s last theorem. Amer. Math. Monthly, 1994; 101(1): 3-14.https://doi.org/10.1080/00029890.1994.11996897

Krantz SG, Parks HR. Fermat’s last theorem, In: A Mathematical Odyssey, 2014; Springer, Boston, MA. https://doi.org/10.1007/978-1-4614-8939-9_13

Ribenboim P. 13 Lectures on Fermat’s Last Theorem, 1979; Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9342-9

Rosen M. Remarks on the history of Fermat’s last theorem 1844 to 1984, In: Modular Forms and Fermat’s Last Theorem, 1997; Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1974-3

Wiles AJ. Modular elliptic curves and Fermat’s last theorem. Ann. Math., 1995; 141: 443-551. https://annals.math.princeton.edu/1995/141-3/p01

Kleiner I. From Fermat to Wiles: Fermat’s last theorem becomes a theorem. Elem. Math., 2000; 55: 19-37. https://link.springer.com/article/10.1007/PL00000079

Yu HB. Problems and Solutions in Mathematical Olympiad, 2022; World Scientific and East China Normal University, Shanghai. https://doi.org/10.1142/12089

Keng HL. Introduction to Number Theory, 1982; Springer, Berlin. https://doi.org/10.1007/978-3-642-68130-1

Kaufman R. Limits on legs of Pythagorean triples and Fermat’s last theorem. College Math. J., 2020; 51(1): 53-56. https://www.tandfonline.com/doi/abs/10.1080/07468342.2020.1674620?tab =permissions&scroll=top

Weisstein EW. Pythagorean triple, From MathWorld–A Wolfram Web Resource. http://mathworld.wolfram.com/PythagoreanTriple.html

Stillwell J. Mathematics and Its History, Third Edition, 2010; Springer, New York. https://link.springer.com/book/10.1007/978-1-4419-6053-5

Imhausen A. Mathematics in Ancient Egypt, A Contextual History, 2016; Princeton University Press, Princeton.

Wolfram Mathematica. https://www.wolfram.com/mathematica/?source=nav

Torrence B, Torrence EA. The Student’s Introduction to Mathematica and the Wolfram Language, Third Edition, 2019; Cambridge University Press, Cambridge. https://www.wolfram.com/books/profile.cgi?id=9737 