On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle

The Pascal’s triangle is generalized to “the <i<k</i<-Pascal’s triangle” with any integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<k</mi< <mo<≥</mo< <mn<2</mn< </mrow< </semantics< </math< </inline-formula<. Let <i<p</i< be any prime number. In this article, we prove that for any positive integers <i<n</i< and <i<e</i<, the <i<n</i<-th row in the <inline-formula< <math display="inline"< <semantics< <msup< <mi<p</mi< <mi<e</mi< </msup< </semantics< </math< </inline-formula<-Pascal’s triangle consists of integers which are congruent to 1 modulo <i<p</i< if and only if <i<n</i< is of the form <inline-formula< <math display="inline"< <semantics< <mstyle displaystyle="true" scriptlevel="0"< <mfrac< <mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mi<m</mi< </mrow< </msup< <mo<−</mo< <mn<1</mn< </mrow< <mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo<−</mo< <mn<1</mn< </mrow< </mfrac< </mstyle< </semantics< </math< </inline-formula< with some integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<m</mi< <mo<≥</mo< <mn<1</mn< </mrow< </semantics< </math< </inline-formula<. This is a generalization of a Lucas’ result asserting that the <i<n</i<-th row in the (2-)Pascal’s triangle consists of odd integers if and only if <i<n</i< is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mrow< <mo stretchy="false"<(</mo< <mi<x</mi< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< </msup< <mo<≡</mo< <msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi<x</mi< <mi<p</mi< </msup< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mo<−</mo< <mn<1</mn< </mrow< </msup< </msup< <mrow< <mo stretchy="false"<(</mo< <mi<mod</mi< <mspace width="4pt"<</mspace< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< of binomial expansions which we could prove for any prime number <i<p</i< and any positive integer <i<e</i<. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions. Ausführliche Beschreibung

Gespeichert in:

Autor*in:	Atsushi Yamagami [verfasserIn] Kazuki Taniguchi [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2020

Schlagwörter:	the pe-pascal’s triangle lucas’ result on the pascal’s triangle congruences of binomial expansions

Übergeordnetes Werk:	In: Symmetry - MDPI AG, 2009, 12(2020), 2, p 288
Übergeordnetes Werk:	volume:12 ; year:2020 ; number:2, p 288

Links:	Link aufrufen Link aufrufen Link aufrufen Journal toc

DOI / URN:	10.3390/sym12020288

Katalog-ID:	DOAJ079287948

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520			\|a The Pascal’s triangle is generalized to “the <i<k</i<-Pascal’s triangle” with any integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<k</mi< <mo<≥</mo< <mn<2</mn< </mrow< </semantics< </math< </inline-formula<. Let <i<p</i< be any prime number. In this article, we prove that for any positive integers <i<n</i< and <i<e</i<, the <i<n</i<-th row in the <inline-formula< <math display="inline"< <semantics< <msup< <mi<p</mi< <mi<e</mi< </msup< </semantics< </math< </inline-formula<-Pascal’s triangle consists of integers which are congruent to 1 modulo <i<p</i< if and only if <i<n</i< is of the form <inline-formula< <math display="inline"< <semantics< <mstyle displaystyle="true" scriptlevel="0"< <mfrac< <mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mi<m</mi< </mrow< </msup< <mo<−</mo< <mn<1</mn< </mrow< <mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo<−</mo< <mn<1</mn< </mrow< </mfrac< </mstyle< </semantics< </math< </inline-formula< with some integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<m</mi< <mo<≥</mo< <mn<1</mn< </mrow< </semantics< </math< </inline-formula<. This is a generalization of a Lucas’ result asserting that the <i<n</i<-th row in the (2-)Pascal’s triangle consists of odd integers if and only if <i<n</i< is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mrow< <mo stretchy="false"<(</mo< <mi<x</mi< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< </msup< <mo<≡</mo< <msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi<x</mi< <mi<p</mi< </msup< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mo<−</mo< <mn<1</mn< </mrow< </msup< </msup< <mrow< <mo stretchy="false"<(</mo< <mi<mod</mi< <mspace width="4pt"<</mspace< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< of binomial expansions which we could prove for any prime number <i<p</i< and any positive integer <i<e</i<. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions.
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allfields	10.3390/sym12020288 doi (DE-627)DOAJ079287948 (DE-599)DOAJ0db67940865b4ec7a2aa9f55bb01da45 DE-627 ger DE-627 rakwb eng QA1-939 Atsushi Yamagami verfasserin aut On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The Pascal’s triangle is generalized to “the <i<k</i<-Pascal’s triangle” with any integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<k</mi< <mo<≥</mo< <mn<2</mn< </mrow< </semantics< </math< </inline-formula<. Let <i<p</i< be any prime number. In this article, we prove that for any positive integers <i<n</i< and <i<e</i<, the <i<n</i<-th row in the <inline-formula< <math display="inline"< <semantics< <msup< <mi<p</mi< <mi<e</mi< </msup< </semantics< </math< </inline-formula<-Pascal’s triangle consists of integers which are congruent to 1 modulo <i<p</i< if and only if <i<n</i< is of the form <inline-formula< <math display="inline"< <semantics< <mstyle displaystyle="true" scriptlevel="0"< <mfrac< <mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mi<m</mi< </mrow< </msup< <mo<−</mo< <mn<1</mn< </mrow< <mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo<−</mo< <mn<1</mn< </mrow< </mfrac< </mstyle< </semantics< </math< </inline-formula< with some integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<m</mi< <mo<≥</mo< <mn<1</mn< </mrow< </semantics< </math< </inline-formula<. This is a generalization of a Lucas’ result asserting that the <i<n</i<-th row in the (2-)Pascal’s triangle consists of odd integers if and only if <i<n</i< is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mrow< <mo stretchy="false"<(</mo< <mi<x</mi< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< </msup< <mo<≡</mo< <msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi<x</mi< <mi<p</mi< </msup< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mo<−</mo< <mn<1</mn< </mrow< </msup< </msup< <mrow< <mo stretchy="false"<(</mo< <mi<mod</mi< <mspace width="4pt"<</mspace< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< of binomial expansions which we could prove for any prime number <i<p</i< and any positive integer <i<e</i<. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions. the pe-pascal’s triangle lucas’ result on the pascal’s triangle congruences of binomial expansions Mathematics Kazuki Taniguchi verfasserin aut In Symmetry MDPI AG, 2009 12(2020), 2, p 288 (DE-627)610604112 (DE-600)2518382-5 20738994 nnns volume:12 year:2020 number:2, p 288 https://doi.org/10.3390/sym12020288 kostenfrei https://doaj.org/article/0db67940865b4ec7a2aa9f55bb01da45 kostenfrei https://www.mdpi.com/2073-8994/12/2/288 kostenfrei https://doaj.org/toc/2073-8994 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 12 2020 2, p 288
spelling	10.3390/sym12020288 doi (DE-627)DOAJ079287948 (DE-599)DOAJ0db67940865b4ec7a2aa9f55bb01da45 DE-627 ger DE-627 rakwb eng QA1-939 Atsushi Yamagami verfasserin aut On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The Pascal’s triangle is generalized to “the <i<k</i<-Pascal’s triangle” with any integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<k</mi< <mo<≥</mo< <mn<2</mn< </mrow< </semantics< </math< </inline-formula<. Let <i<p</i< be any prime number. In this article, we prove that for any positive integers <i<n</i< and <i<e</i<, the <i<n</i<-th row in the <inline-formula< <math display="inline"< <semantics< <msup< <mi<p</mi< <mi<e</mi< </msup< </semantics< </math< </inline-formula<-Pascal’s triangle consists of integers which are congruent to 1 modulo <i<p</i< if and only if <i<n</i< is of the form <inline-formula< <math display="inline"< <semantics< <mstyle displaystyle="true" scriptlevel="0"< <mfrac< <mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mi<m</mi< </mrow< </msup< <mo<−</mo< <mn<1</mn< </mrow< <mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo<−</mo< <mn<1</mn< </mrow< </mfrac< </mstyle< </semantics< </math< </inline-formula< with some integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<m</mi< <mo<≥</mo< <mn<1</mn< </mrow< </semantics< </math< </inline-formula<. This is a generalization of a Lucas’ result asserting that the <i<n</i<-th row in the (2-)Pascal’s triangle consists of odd integers if and only if <i<n</i< is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mrow< <mo stretchy="false"<(</mo< <mi<x</mi< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< </msup< <mo<≡</mo< <msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi<x</mi< <mi<p</mi< </msup< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mo<−</mo< <mn<1</mn< </mrow< </msup< </msup< <mrow< <mo stretchy="false"<(</mo< <mi<mod</mi< <mspace width="4pt"<</mspace< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< of binomial expansions which we could prove for any prime number <i<p</i< and any positive integer <i<e</i<. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions. the pe-pascal’s triangle lucas’ result on the pascal’s triangle congruences of binomial expansions Mathematics Kazuki Taniguchi verfasserin aut In Symmetry MDPI AG, 2009 12(2020), 2, p 288 (DE-627)610604112 (DE-600)2518382-5 20738994 nnns volume:12 year:2020 number:2, p 288 https://doi.org/10.3390/sym12020288 kostenfrei https://doaj.org/article/0db67940865b4ec7a2aa9f55bb01da45 kostenfrei https://www.mdpi.com/2073-8994/12/2/288 kostenfrei https://doaj.org/toc/2073-8994 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 12 2020 2, p 288
allfields_unstemmed	10.3390/sym12020288 doi (DE-627)DOAJ079287948 (DE-599)DOAJ0db67940865b4ec7a2aa9f55bb01da45 DE-627 ger DE-627 rakwb eng QA1-939 Atsushi Yamagami verfasserin aut On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The Pascal’s triangle is generalized to “the <i<k</i<-Pascal’s triangle” with any integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<k</mi< <mo<≥</mo< <mn<2</mn< </mrow< </semantics< </math< </inline-formula<. Let <i<p</i< be any prime number. In this article, we prove that for any positive integers <i<n</i< and <i<e</i<, the <i<n</i<-th row in the <inline-formula< <math display="inline"< <semantics< <msup< <mi<p</mi< <mi<e</mi< </msup< </semantics< </math< </inline-formula<-Pascal’s triangle consists of integers which are congruent to 1 modulo <i<p</i< if and only if <i<n</i< is of the form <inline-formula< <math display="inline"< <semantics< <mstyle displaystyle="true" scriptlevel="0"< <mfrac< <mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mi<m</mi< </mrow< </msup< <mo<−</mo< <mn<1</mn< </mrow< <mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo<−</mo< <mn<1</mn< </mrow< </mfrac< </mstyle< </semantics< </math< </inline-formula< with some integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<m</mi< <mo<≥</mo< <mn<1</mn< </mrow< </semantics< </math< </inline-formula<. This is a generalization of a Lucas’ result asserting that the <i<n</i<-th row in the (2-)Pascal’s triangle consists of odd integers if and only if <i<n</i< is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mrow< <mo stretchy="false"<(</mo< <mi<x</mi< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< </msup< <mo<≡</mo< <msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi<x</mi< <mi<p</mi< </msup< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mo<−</mo< <mn<1</mn< </mrow< </msup< </msup< <mrow< <mo stretchy="false"<(</mo< <mi<mod</mi< <mspace width="4pt"<</mspace< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< of binomial expansions which we could prove for any prime number <i<p</i< and any positive integer <i<e</i<. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions. the pe-pascal’s triangle lucas’ result on the pascal’s triangle congruences of binomial expansions Mathematics Kazuki Taniguchi verfasserin aut In Symmetry MDPI AG, 2009 12(2020), 2, p 288 (DE-627)610604112 (DE-600)2518382-5 20738994 nnns volume:12 year:2020 number:2, p 288 https://doi.org/10.3390/sym12020288 kostenfrei https://doaj.org/article/0db67940865b4ec7a2aa9f55bb01da45 kostenfrei https://www.mdpi.com/2073-8994/12/2/288 kostenfrei https://doaj.org/toc/2073-8994 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 12 2020 2, p 288
allfieldsGer	10.3390/sym12020288 doi (DE-627)DOAJ079287948 (DE-599)DOAJ0db67940865b4ec7a2aa9f55bb01da45 DE-627 ger DE-627 rakwb eng QA1-939 Atsushi Yamagami verfasserin aut On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The Pascal’s triangle is generalized to “the <i<k</i<-Pascal’s triangle” with any integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<k</mi< <mo<≥</mo< <mn<2</mn< </mrow< </semantics< </math< </inline-formula<. Let <i<p</i< be any prime number. In this article, we prove that for any positive integers <i<n</i< and <i<e</i<, the <i<n</i<-th row in the <inline-formula< <math display="inline"< <semantics< <msup< <mi<p</mi< <mi<e</mi< </msup< </semantics< </math< </inline-formula<-Pascal’s triangle consists of integers which are congruent to 1 modulo <i<p</i< if and only if <i<n</i< is of the form <inline-formula< <math display="inline"< <semantics< <mstyle displaystyle="true" scriptlevel="0"< <mfrac< <mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mi<m</mi< </mrow< </msup< <mo<−</mo< <mn<1</mn< </mrow< <mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo<−</mo< <mn<1</mn< </mrow< </mfrac< </mstyle< </semantics< </math< </inline-formula< with some integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<m</mi< <mo<≥</mo< <mn<1</mn< </mrow< </semantics< </math< </inline-formula<. This is a generalization of a Lucas’ result asserting that the <i<n</i<-th row in the (2-)Pascal’s triangle consists of odd integers if and only if <i<n</i< is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mrow< <mo stretchy="false"<(</mo< <mi<x</mi< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< </msup< <mo<≡</mo< <msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi<x</mi< <mi<p</mi< </msup< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mo<−</mo< <mn<1</mn< </mrow< </msup< </msup< <mrow< <mo stretchy="false"<(</mo< <mi<mod</mi< <mspace width="4pt"<</mspace< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< of binomial expansions which we could prove for any prime number <i<p</i< and any positive integer <i<e</i<. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions. the pe-pascal’s triangle lucas’ result on the pascal’s triangle congruences of binomial expansions Mathematics Kazuki Taniguchi verfasserin aut In Symmetry MDPI AG, 2009 12(2020), 2, p 288 (DE-627)610604112 (DE-600)2518382-5 20738994 nnns volume:12 year:2020 number:2, p 288 https://doi.org/10.3390/sym12020288 kostenfrei https://doaj.org/article/0db67940865b4ec7a2aa9f55bb01da45 kostenfrei https://www.mdpi.com/2073-8994/12/2/288 kostenfrei https://doaj.org/toc/2073-8994 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 12 2020 2, p 288
allfieldsSound	10.3390/sym12020288 doi (DE-627)DOAJ079287948 (DE-599)DOAJ0db67940865b4ec7a2aa9f55bb01da45 DE-627 ger DE-627 rakwb eng QA1-939 Atsushi Yamagami verfasserin aut On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle 2020 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier The Pascal’s triangle is generalized to “the <i<k</i<-Pascal’s triangle” with any integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<k</mi< <mo<≥</mo< <mn<2</mn< </mrow< </semantics< </math< </inline-formula<. Let <i<p</i< be any prime number. In this article, we prove that for any positive integers <i<n</i< and <i<e</i<, the <i<n</i<-th row in the <inline-formula< <math display="inline"< <semantics< <msup< <mi<p</mi< <mi<e</mi< </msup< </semantics< </math< </inline-formula<-Pascal’s triangle consists of integers which are congruent to 1 modulo <i<p</i< if and only if <i<n</i< is of the form <inline-formula< <math display="inline"< <semantics< <mstyle displaystyle="true" scriptlevel="0"< <mfrac< <mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mi<m</mi< </mrow< </msup< <mo<−</mo< <mn<1</mn< </mrow< <mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo<−</mo< <mn<1</mn< </mrow< </mfrac< </mstyle< </semantics< </math< </inline-formula< with some integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<m</mi< <mo<≥</mo< <mn<1</mn< </mrow< </semantics< </math< </inline-formula<. This is a generalization of a Lucas’ result asserting that the <i<n</i<-th row in the (2-)Pascal’s triangle consists of odd integers if and only if <i<n</i< is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mrow< <mo stretchy="false"<(</mo< <mi<x</mi< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< </msup< <mo<≡</mo< <msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi<x</mi< <mi<p</mi< </msup< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mo<−</mo< <mn<1</mn< </mrow< </msup< </msup< <mrow< <mo stretchy="false"<(</mo< <mi<mod</mi< <mspace width="4pt"<</mspace< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< of binomial expansions which we could prove for any prime number <i<p</i< and any positive integer <i<e</i<. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions. the pe-pascal’s triangle lucas’ result on the pascal’s triangle congruences of binomial expansions Mathematics Kazuki Taniguchi verfasserin aut In Symmetry MDPI AG, 2009 12(2020), 2, p 288 (DE-627)610604112 (DE-600)2518382-5 20738994 nnns volume:12 year:2020 number:2, p 288 https://doi.org/10.3390/sym12020288 kostenfrei https://doaj.org/article/0db67940865b4ec7a2aa9f55bb01da45 kostenfrei https://www.mdpi.com/2073-8994/12/2/288 kostenfrei https://doaj.org/toc/2073-8994 Journal toc kostenfrei GBV_USEFLAG_A SYSFLAG_A GBV_DOAJ GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_95 GBV_ILN_105 GBV_ILN_110 GBV_ILN_151 GBV_ILN_161 GBV_ILN_170 GBV_ILN_206 GBV_ILN_213 GBV_ILN_230 GBV_ILN_285 GBV_ILN_293 GBV_ILN_602 GBV_ILN_2005 GBV_ILN_2009 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2055 GBV_ILN_2111 GBV_ILN_4012 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4249 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4367 GBV_ILN_4700 AR 12 2020 2, p 288
language	English
source	In Symmetry 12(2020), 2, p 288 volume:12 year:2020 number:2, p 288
sourceStr	In Symmetry 12(2020), 2, p 288 volume:12 year:2020 number:2, p 288
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	the pe-pascal’s triangle lucas’ result on the pascal’s triangle congruences of binomial expansions Mathematics
isfreeaccess_bool	true
container_title	Symmetry
authorswithroles_txt_mv	Atsushi Yamagami @@aut@@ Kazuki Taniguchi @@aut@@
publishDateDaySort_date	2020-01-01T00:00:00Z
hierarchy_top_id	610604112
id	DOAJ079287948
language_de	englisch
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Let <i<p</i< be any prime number. In this article, we prove that for any positive integers <i<n</i< and <i<e</i<, the <i<n</i<-th row in the <inline-formula< <math display="inline"< <semantics< <msup< <mi<p</mi< <mi<e</mi< </msup< </semantics< </math< </inline-formula<-Pascal’s triangle consists of integers which are congruent to 1 modulo <i<p</i< if and only if <i<n</i< is of the form <inline-formula< <math display="inline"< <semantics< <mstyle displaystyle="true" scriptlevel="0"< <mfrac< <mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mi<m</mi< </mrow< </msup< <mo<−</mo< <mn<1</mn< </mrow< <mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo<−</mo< <mn<1</mn< </mrow< </mfrac< </mstyle< </semantics< </math< </inline-formula< with some integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<m</mi< <mo<≥</mo< <mn<1</mn< </mrow< </semantics< </math< </inline-formula<. This is a generalization of a Lucas’ result asserting that the <i<n</i<-th row in the (2-)Pascal’s triangle consists of odd integers if and only if <i<n</i< is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mrow< <mo stretchy="false"<(</mo< <mi<x</mi< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< </msup< <mo<≡</mo< <msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi<x</mi< <mi<p</mi< </msup< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mo<−</mo< <mn<1</mn< </mrow< </msup< </msup< <mrow< <mo stretchy="false"<(</mo< <mi<mod</mi< <mspace width="4pt"<</mspace< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< of binomial expansions which we could prove for any prime number <i<p</i< and any positive integer <i<e</i<. 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callnumber-first	Q - Science
author	Atsushi Yamagami
spellingShingle	Atsushi Yamagami misc QA1-939 misc the pe-pascal’s triangle misc lucas’ result on the pascal’s triangle misc congruences of binomial expansions misc Mathematics On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle
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topic_title	QA1-939 On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle the pe-pascal’s triangle lucas’ result on the pascal’s triangle congruences of binomial expansions
topic	misc QA1-939 misc the pe-pascal’s triangle misc lucas’ result on the pascal’s triangle misc congruences of binomial expansions misc Mathematics
topic_unstemmed	misc QA1-939 misc the pe-pascal’s triangle misc lucas’ result on the pascal’s triangle misc congruences of binomial expansions misc Mathematics
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title	On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle
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title_full	On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle
author_sort	Atsushi Yamagami
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author_browse	Atsushi Yamagami Kazuki Taniguchi
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title_sort	on a generalization of a lucas’ result and an application to the 4-pascal’s triangle
callnumber	QA1-939
title_auth	On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle
abstract	The Pascal’s triangle is generalized to “the <i<k</i<-Pascal’s triangle” with any integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<k</mi< <mo<≥</mo< <mn<2</mn< </mrow< </semantics< </math< </inline-formula<. Let <i<p</i< be any prime number. In this article, we prove that for any positive integers <i<n</i< and <i<e</i<, the <i<n</i<-th row in the <inline-formula< <math display="inline"< <semantics< <msup< <mi<p</mi< <mi<e</mi< </msup< </semantics< </math< </inline-formula<-Pascal’s triangle consists of integers which are congruent to 1 modulo <i<p</i< if and only if <i<n</i< is of the form <inline-formula< <math display="inline"< <semantics< <mstyle displaystyle="true" scriptlevel="0"< <mfrac< <mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mi<m</mi< </mrow< </msup< <mo<−</mo< <mn<1</mn< </mrow< <mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo<−</mo< <mn<1</mn< </mrow< </mfrac< </mstyle< </semantics< </math< </inline-formula< with some integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<m</mi< <mo<≥</mo< <mn<1</mn< </mrow< </semantics< </math< </inline-formula<. This is a generalization of a Lucas’ result asserting that the <i<n</i<-th row in the (2-)Pascal’s triangle consists of odd integers if and only if <i<n</i< is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mrow< <mo stretchy="false"<(</mo< <mi<x</mi< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< </msup< <mo<≡</mo< <msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi<x</mi< <mi<p</mi< </msup< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mo<−</mo< <mn<1</mn< </mrow< </msup< </msup< <mrow< <mo stretchy="false"<(</mo< <mi<mod</mi< <mspace width="4pt"<</mspace< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< of binomial expansions which we could prove for any prime number <i<p</i< and any positive integer <i<e</i<. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions.
abstractGer	The Pascal’s triangle is generalized to “the <i<k</i<-Pascal’s triangle” with any integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<k</mi< <mo<≥</mo< <mn<2</mn< </mrow< </semantics< </math< </inline-formula<. Let <i<p</i< be any prime number. In this article, we prove that for any positive integers <i<n</i< and <i<e</i<, the <i<n</i<-th row in the <inline-formula< <math display="inline"< <semantics< <msup< <mi<p</mi< <mi<e</mi< </msup< </semantics< </math< </inline-formula<-Pascal’s triangle consists of integers which are congruent to 1 modulo <i<p</i< if and only if <i<n</i< is of the form <inline-formula< <math display="inline"< <semantics< <mstyle displaystyle="true" scriptlevel="0"< <mfrac< <mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mi<m</mi< </mrow< </msup< <mo<−</mo< <mn<1</mn< </mrow< <mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo<−</mo< <mn<1</mn< </mrow< </mfrac< </mstyle< </semantics< </math< </inline-formula< with some integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<m</mi< <mo<≥</mo< <mn<1</mn< </mrow< </semantics< </math< </inline-formula<. This is a generalization of a Lucas’ result asserting that the <i<n</i<-th row in the (2-)Pascal’s triangle consists of odd integers if and only if <i<n</i< is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mrow< <mo stretchy="false"<(</mo< <mi<x</mi< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< </msup< <mo<≡</mo< <msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi<x</mi< <mi<p</mi< </msup< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mo<−</mo< <mn<1</mn< </mrow< </msup< </msup< <mrow< <mo stretchy="false"<(</mo< <mi<mod</mi< <mspace width="4pt"<</mspace< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< of binomial expansions which we could prove for any prime number <i<p</i< and any positive integer <i<e</i<. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions.
abstract_unstemmed	The Pascal’s triangle is generalized to “the <i<k</i<-Pascal’s triangle” with any integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<k</mi< <mo<≥</mo< <mn<2</mn< </mrow< </semantics< </math< </inline-formula<. Let <i<p</i< be any prime number. In this article, we prove that for any positive integers <i<n</i< and <i<e</i<, the <i<n</i<-th row in the <inline-formula< <math display="inline"< <semantics< <msup< <mi<p</mi< <mi<e</mi< </msup< </semantics< </math< </inline-formula<-Pascal’s triangle consists of integers which are congruent to 1 modulo <i<p</i< if and only if <i<n</i< is of the form <inline-formula< <math display="inline"< <semantics< <mstyle displaystyle="true" scriptlevel="0"< <mfrac< <mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mi<m</mi< </mrow< </msup< <mo<−</mo< <mn<1</mn< </mrow< <mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo<−</mo< <mn<1</mn< </mrow< </mfrac< </mstyle< </semantics< </math< </inline-formula< with some integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<m</mi< <mo<≥</mo< <mn<1</mn< </mrow< </semantics< </math< </inline-formula<. This is a generalization of a Lucas’ result asserting that the <i<n</i<-th row in the (2-)Pascal’s triangle consists of odd integers if and only if <i<n</i< is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mrow< <mo stretchy="false"<(</mo< <mi<x</mi< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< </msup< <mo<≡</mo< <msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi<x</mi< <mi<p</mi< </msup< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mo<−</mo< <mn<1</mn< </mrow< </msup< </msup< <mrow< <mo stretchy="false"<(</mo< <mi<mod</mi< <mspace width="4pt"<</mspace< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< of binomial expansions which we could prove for any prime number <i<p</i< and any positive integer <i<e</i<. We think that this article is fit for the Special Issue “Number Theory and Symmetry,” since we prove a symmetric property on the 4-Pascal’s triangle by means of a number-theoretical property of binomial expansions.
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Let <i<p</i< be any prime number. In this article, we prove that for any positive integers <i<n</i< and <i<e</i<, the <i<n</i<-th row in the <inline-formula< <math display="inline"< <semantics< <msup< <mi<p</mi< <mi<e</mi< </msup< </semantics< </math< </inline-formula<-Pascal’s triangle consists of integers which are congruent to 1 modulo <i<p</i< if and only if <i<n</i< is of the form <inline-formula< <math display="inline"< <semantics< <mstyle displaystyle="true" scriptlevel="0"< <mfrac< <mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mi<m</mi< </mrow< </msup< <mo<−</mo< <mn<1</mn< </mrow< <mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo<−</mo< <mn<1</mn< </mrow< </mfrac< </mstyle< </semantics< </math< </inline-formula< with some integer <inline-formula< <math display="inline"< <semantics< <mrow< <mi<m</mi< <mo<≥</mo< <mn<1</mn< </mrow< </semantics< </math< </inline-formula<. This is a generalization of a Lucas’ result asserting that the <i<n</i<-th row in the (2-)Pascal’s triangle consists of odd integers if and only if <i<n</i< is a Mersenne number. As an application, we then see that there exists no row in the 4-Pascal’s triangle consisting of integers which are congruent to 1 modulo 4 except the first row. In this application, we use the congruence <inline-formula< <math display="inline"< <semantics< <mrow< <msup< <mrow< <mo stretchy="false"<(</mo< <mi<x</mi< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mi<e</mi< </msup< </msup< <mo<≡</mo< <msup< <mrow< <mo stretchy="false"<(</mo< <msup< <mi<x</mi< <mi<p</mi< </msup< <mo<+</mo< <mn<1</mn< <mo stretchy="false"<)</mo< </mrow< <msup< <mi<p</mi< <mrow< <mi<e</mi< <mo<−</mo< <mn<1</mn< </mrow< </msup< </msup< <mrow< <mo stretchy="false"<(</mo< <mi<mod</mi< <mspace width="4pt"<</mspace< <msup< <mi<p</mi< <mi<e</mi< </msup< <mo stretchy="false"<)</mo< </mrow< </mrow< </semantics< </math< </inline-formula< of binomial expansions which we could prove for any prime number <i<p</i< and any positive integer <i<e</i<. 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On a Generalization of a Lucas’ Result and an Application to the 4-Pascal’s Triangle

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