Diophantine equation 25

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$${Published}$$  $${Online}$$  $${First}$$  $${(14/2/2024)}$$
$${Latest}$$  $${additions}$$  $${(14/2/2024)}$$
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$${Diophantine}$$ $${equation}$$ $${25}$$
$${(25.1)}$$  $${~~~\sum\limits_{i=1}^3\ a_i^5}$$ $${=}$$ $${\sum\limits_{j=1}^3\ b_j^5}$$
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$${case(25.1)}$$  $${~~~\sum\limits_{i=1}^3\ a_i^5}$$ $${=}$$ $${\sum\limits_{j=1}^3\ b_j^5}$$

詳細は後日で…
$${\small ab(a+b)(a^2+ab+b^2)=cd(c+d)(c^2+cd+d^2)}$$
を満たす下記の解

$${a=p^{16}−3p^{11}q^5−5p^6q^{10}−pq^{15}\\b=6p^{11}q^5+2pq^{15}\\c=p^{15}q+5p^{10}q^6+3p^5q^{11}−q ^{16}\\d=p^{15}q−5p^{10}q^6+3p^5q^{11}+q^{16}}$$
とした時、

$${x_1 =a~~,~~x_2 =b~~,~~x_3 =−a−b\\y_1 =c~~,~~y_2 =d~~,~~y_3 =−c−d}$$
が解となる。
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$${REFERENCES}$$
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$${\small【AJAI~CHOUDHRY~AND\\~~~~~~JAROSLAW~WROBLEWSKI】}$$
$${\footnotesize「A~QUINTIC~DIOPHANTINE~EQUATION\\WITH~APPLICATIONS~TO~\\TWO~DIOPHANTINE~SYSTEMS\\CONCERNING~FIFTH~POWERS」}$$
$${\scriptsize ROCKY~MOUNTAIN\\JOURNAL~OF~MATHEMATICS}$$
$${\small Volume 43, Number 6, 2013,~p1893-1899}$$

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【目次001】・【目次002】・【目次003】
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