Unique polynomial solution of $m/n=1/x+1/y+1/z$ for $n \equiv b {\rm mod}\, a$ if $(a,m)=1$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911823348891648 |
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| author | Schuh, Bernd R. |
| author_facet | Schuh, Bernd R. |
| contents | Necessary and sufficient conditions for the existence of an integer solution of the diophantine equation $m/n=1/x(λ)+1/y(λ)+1/z(λ)$ with $n=b+aλ$ are explicitly given for a,b coprime and a not a multiple of m . The solution has the form $x(λ)=kn(λ)$, $y(λ)=n(λ)(s+rλ)$, $z(λ)=(kl/r)(s+rλ)$ where parameters $k,l,s,r\in \mathbb{Z}$ obey certain conditions depending on $a,b$. The conditions imply restrictions for some choices of $a,b$ which differ from the ones known in the case $m=4$. E.g., the modulus must be of the form $l(mk-1)$. One can also deduce that primes of the form $1+4K$ are excluded as modulus. Also if $a=p\ne m$ is prime and $b=a+1$, i.e., $n\equiv 1{\rm mod}\, p$, polynomial solutions are shown to be impossible. All results are valid for integers $m \ge 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01307 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unique polynomial solution of $m/n=1/x+1/y+1/z$ for $n \equiv b {\rm mod}\, a$ if $(a,m)=1$ Schuh, Bernd R. General Mathematics Necessary and sufficient conditions for the existence of an integer solution of the diophantine equation $m/n=1/x(λ)+1/y(λ)+1/z(λ)$ with $n=b+aλ$ are explicitly given for a,b coprime and a not a multiple of m . The solution has the form $x(λ)=kn(λ)$, $y(λ)=n(λ)(s+rλ)$, $z(λ)=(kl/r)(s+rλ)$ where parameters $k,l,s,r\in \mathbb{Z}$ obey certain conditions depending on $a,b$. The conditions imply restrictions for some choices of $a,b$ which differ from the ones known in the case $m=4$. E.g., the modulus must be of the form $l(mk-1)$. One can also deduce that primes of the form $1+4K$ are excluded as modulus. Also if $a=p\ne m$ is prime and $b=a+1$, i.e., $n\equiv 1{\rm mod}\, p$, polynomial solutions are shown to be impossible. All results are valid for integers $m \ge 4$. |
| title | Unique polynomial solution of $m/n=1/x+1/y+1/z$ for $n \equiv b {\rm mod}\, a$ if $(a,m)=1$ |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2404.01307 |