Tìm cặp số nguyên tố $(p,q)$ sao cho $p^2+q^3$ và $q^2+p^3$ chính phương
#1
Posted 06-11-2011 - 22:17
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Discovery is a child’s privilege. I mean the small child, the child who is not afraid to be wrong, to look silly, to not be serious, and to act differently from everyone else. He is also not afraid that the things he is interested in are in bad taste or turn out to be different from his expectations, from what they should be, or rather he is not afraid of what they actually are. He ignores the silent and flawless consensus that is part of the air we breathe – the consensus of all the people who are, or are reputed to be, reasonable.
Grothendieck, Récoltes et Semailles (“Crops and Seeds”).
#2
Posted 06-11-2011 - 23:01
$p^2+q^3=k^2;\;\;(k\in\mathbb{N}^*)$
$q^3=(k+p)(k-p)$
Vì $gcd(k+p;k-p) \in \{1,p,2p\}$ nên ta có: $\begin{cases}q^2=k+p \\ q=k-p\end{cases}\;\; (1)$
hoặc $q^2=p^2\;\;(2)$
(1) Suy ra: $q(q-1)=2p\Rightarrow p=q=3$ vì trường hợp $q=2\Rightarrow p=1$ (loại)
(2) Thay vào ta có $p^2+p^3=k^2\Rightarrow p^2(p+1)=k^2\Rightarrow p+1=m^2;\;\;(k\in\mathbb{N}^*)$
Suy ra $p=(m-1)(m+1)\Rightarrow m-1=1\Rightarrow p=3$
Thử lại $3^2+3^3=9+27=36=6^2$
Vậy $(p,q)=(3,3)$ là cặp duy nhất cần tìm!
Edited by UEVOLI, 06-11-2011 - 23:18.
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