We study the modular resolution method using a new tool called a Polyserie introduced by Wildberger N.J. & Rubine D. to prove an equivalence theorem of the existence and the unicity of the solution of the equation of the form Ht2(x) – Ht(x)+t ≡ 0 (mod tn). These equations admits solutions for a fixed t but when t changes in a range of several values, then the equation becomes parametric and hence the solution should be parametric too. Prof. Wildberger N.J has studied such equations using Catalan numbers sequence using the recurrence formula between their terms Cn+1 = ∑k=0n Ck Cn-k, which allow us to easily prove the main result of the present article. We study Wildberger’s Polynumber Sequences, Binomial Chu-Vandermonde Identity, Truncated Polyseries and finally Modular Resolution as Application.
| Published in | Pure and Applied Mathematics Journal (Volume 14, Issue 6) |
| DOI | 10.11648/j.pamj.20251406.12 |
| Page(s) | 166-175 |
| Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
| Copyright |
Copyright © The Author(s), 2025. Published by Science Publishing Group |
Truncated Polyseries, Binomial Chu-Vandermonde Identity, Polynumber Sequences, Exponential Euler Polyseries, Newton Polyserie Catalan Numbers, Modular Resolution
| [1] | H.Cohen. ACourseinComputational Algebraic Number Theory. Springer, 1993. |
| [2] | F. Q. Gouvea. p-adic Numbers: An Introduction. Springer, 1997. |
| [3] | G. H. Hardy and E. M. Wright. An Introduction to the Theory of Numbers. Oxford University Press, 1979. |
| [4] | R. Lidl and H. Niederreiter. Finite Fields. Cambridge University Press, 1997. |
| [5] | G. Ma, T. S. Aarthy, and O. Ozen. On the quinary homogeneousbi-quadratic equation x4+y4 (x+y)w3 = 14z2 t2. Journal of Fundamental and Applied Sciences, 12(2): 516-524, 2020. |
| [6] | G. Ma, T. S. Aarthy, and O. Ozen. On the system of double equations with three unknowns d+ay+bx+cx2 = z2 y +z = x2. International Journal of Nonlinear Analysis and Applications, 12(1): 575-581, 2021. |
| [7] | G. Ma, V. Sa, and O. Ozen. A Collection of Pellian Equation (Solutions and Properties). Akinik Publications, 1st edition, 2018. |
| [8] | J. Neukirch. Algebraic Number Theory. Springer Science &Business Media, Mar. 2013. |
| [9] | O. Ozen and G. Ma. On the homogeneous cone z 2 + (k + 1)y2 = (k + 1)(k + 3)x2. Pioneer Journal of Mathematics and Mathematical Sciences, 25(1): 9-18, 2019. |
| [10] | V. Sa, G. Ma, T. S. Aarthy, and O. Ozen. On ternary biquadratic diophantine equation 11(x2 y2)+3(xy) = z4. Notes on Number Theory and Discrete Mathematics, 25(3): 65-71, 2019. |
| [11] | V. Sa, G. Ma, and O. Ozen. On non-homogeneous cubic equation with four unknowns xy + 2z2 = 2w3. Purakala: UGC Care Approved Journal, 31(2): 927-933, 2022. |
| [12] | B. Salim, A. M. Ahmed, and O. Ozen. Representation of integers by k-generalized fibonacci sequences and applications in cryptography. Asian-European Journal of Mathematics, 14(9): 21501571-215015711, 2021. |
| [13] | J.-P. Serre. A Course in Arithmetic. Springer, 1973. |
| [14] |
N. J. Wildberger. Math foundations. YouTube playlist on Insights Into Mathematics channel, 2009.
https://youtu.be/HsUxn2l1Wzk?list=PLIljB45xT85DGxj1x_dyaSggbauAgrB6R |
| [15] | N. J. Wildberger. Data structures in mathematics (math foundations 151). YouTube video on Insights Into Mathematics channel, 2015. |
| [16] | N. J. Wildberger. Wild egg maths. YouTube playlist on WildEggMathematicsCourses, 2017. |
| [17] |
N. J. Wildberger. Box arithmetic. YouTube video on Insights Into Mathematics channel, 2021.
https://youtu.be/4xoF2SRp194?list=PLIljB45xT85B0aMG-G9oqj-NPIuBMnq8z |
| [18] |
N. J. Wildberger. Solving polynomial equations. YouTube video on Wild Egg Maths channel, 2021.
https://youtu.be/XHC1YLh67Z0?list=PLzdiPTrEWyz7PpsRFHuGb3EhwZtEOdRjV |
| [19] |
N. J. Wildberger. Algebraic calculus two. YouTube playlist on Wild Egg Maths channel, 2022.
https://youtu.be/HXdUfOTwDlc?list=PLzdiPTrEWyz7cyg5JdKBpzB4fTQT7f-Xl |
| [20] | N. J. Wildberger and D. Rubine. A hyper-catalan series solution to polynomial equations, and the geode. The American Mathematical Monthly, 132(5): 383-402, May 2025. |
APA Style
Brahimi, M. (2025). Modular Resolution by Polyseries. Pure and Applied Mathematics Journal, 14(6), 166-175. https://doi.org/10.11648/j.pamj.20251406.12
ACS Style
Brahimi, M. Modular Resolution by Polyseries. Pure Appl. Math. J. 2025, 14(6), 166-175. doi: 10.11648/j.pamj.20251406.12
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Download@article{10.11648/j.pamj.20251406.12,
author = {Mahdi-Tahar Brahimi},
title = {Modular Resolution by Polyseries
},
journal = {Pure and Applied Mathematics Journal},
volume = {14},
number = {6},
pages = {166-175},
doi = {10.11648/j.pamj.20251406.12},
url = {https://doi.org/10.11648/j.pamj.20251406.12},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20251406.12},
abstract = {We study the modular resolution method using a new tool called a Polyserie introduced by Wildberger N.J. & Rubine D. to prove an equivalence theorem of the existence and the unicity of the solution of the equation of the form Ht2(x) – Ht(x)+t ≡ 0 (mod tn). These equations admits solutions for a fixed t but when t changes in a range of several values, then the equation becomes parametric and hence the solution should be parametric too. Prof. Wildberger N.J has studied such equations using Catalan numbers sequence using the recurrence formula between their terms Cn+1 = ∑k=0n Ck Cn-k, which allow us to easily prove the main result of the present article. We study Wildberger’s Polynumber Sequences, Binomial Chu-Vandermonde Identity, Truncated Polyseries and finally Modular Resolution as Application.
},
year = {2025}
}
TY - JOUR T1 - Modular Resolution by Polyseries AU - Mahdi-Tahar Brahimi Y1 - 2025/11/10 PY - 2025 N1 - https://doi.org/10.11648/j.pamj.20251406.12 DO - 10.11648/j.pamj.20251406.12 T2 - Pure and Applied Mathematics Journal JF - Pure and Applied Mathematics Journal JO - Pure and Applied Mathematics Journal SP - 166 EP - 175 PB - Science Publishing Group SN - 2326-9812 UR - https://doi.org/10.11648/j.pamj.20251406.12 AB - We study the modular resolution method using a new tool called a Polyserie introduced by Wildberger N.J. & Rubine D. to prove an equivalence theorem of the existence and the unicity of the solution of the equation of the form Ht2(x) – Ht(x)+t ≡ 0 (mod tn). These equations admits solutions for a fixed t but when t changes in a range of several values, then the equation becomes parametric and hence the solution should be parametric too. Prof. Wildberger N.J has studied such equations using Catalan numbers sequence using the recurrence formula between their terms Cn+1 = ∑k=0n Ck Cn-k, which allow us to easily prove the main result of the present article. We study Wildberger’s Polynumber Sequences, Binomial Chu-Vandermonde Identity, Truncated Polyseries and finally Modular Resolution as Application. VL - 14 IS - 6 ER -
Department of Mathematics, University of Mohamed Boudiaf, Msila, Algeria
