Tìm n thuộc N để (n^2-3)^2 +16 là 1 số chính phương

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1. $$\eqalign{
   & {\left( {{n^2} – 3} \right)^2} + 16 = {k^2}\,\,\left( {k \in Z} \right) \cr
   & \Leftrightarrow {k^2} – {\left( {{n^2} – 3} \right)^2} = 16 \cr
   & \Leftrightarrow \left( {k – {n^2} + 3} \right)\left( {k + {n^2} – 3} \right) = 16 \cr
   & Do\,\,k,\,\,n \in Z \Rightarrow k – {n^2} + 3 \in U\left( {16} \right) = \left\{ { \pm 1; \pm 2; \pm 4; \pm 8; \pm 16} \right\} \cr
   & TH1:\,\,\left\{ \matrix{
   k – {n^2} + 3 = 1 \hfill \cr
   k + {n^2} – 3 = 16 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
   k = 8 \hfill \cr
   {n^2} = 10\,\,\left( {Loai} \right) \hfill \cr} \right. \cr
   & TH2:\,\,\left\{ \matrix{
   k – {n^2} + 3 = – 1 \hfill \cr
   k + {n^2} – 3 = – 16 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
   k = – {{17} \over 2} \hfill \cr
   {n^2} = – {9 \over 2}\, \hfill \cr} \right.\,\,\left( {Loai} \right) \cr
   & TH3:\,\,\left\{ \matrix{
   k – {n^2} + 3 = 2 \hfill \cr
   k + {n^2} – 3 = 8 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
   k = 5 \hfill \cr
   {n^2} = 6\, \hfill \cr} \right.\,\,\left( {Loai} \right) \cr
   & TH4:\,\,\left\{ \matrix{
   k – {n^2} + 3 = – 2 \hfill \cr
   k + {n^2} – 3 = – 8 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
   k = – 5 \hfill \cr
   {n^2} = 0\, \Leftrightarrow n = 0 \hfill \cr} \right.\,\,\left( {tm} \right)\, \cr
   & TH5:\,\,\left\{ \matrix{
   k – {n^2} + 3 = 4 \hfill \cr
   k + {n^2} – 3 = – 4 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
   k = 0 \hfill \cr
   {n^2} = – 1\, \hfill \cr} \right.\,\,\left( {Loai} \right)\, \cr
   & TH6:\,\,\left\{ \matrix{
   k – {n^2} + 3 = – 4 \hfill \cr
   k + {n^2} – 3 = 4 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
   k = 0 \hfill \cr
   {n^2} = 7 \hfill \cr} \right.\,\,\left( {Loai} \right)\, \cr
   & TH7:\,\,\left\{ \matrix{
   k – {n^2} + 3 = 8 \hfill \cr
   k + {n^2} – 3 = 2 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
   k = 5 \hfill \cr
   {n^2} = 0 \Leftrightarrow n = 0\, \hfill \cr} \right.\,\,\left( {tm} \right) \cr
   & TH8:\,\,\left\{ \matrix{
   k – {n^2} + 3 = – 8 \hfill \cr
   k + {n^2} – 3 = – 2 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
   k = – 5 \hfill \cr
   {n^2} = 6\, \hfill \cr} \right.\,\,\left( {Loai} \right) \cr
   & TH9:\,\,\left\{ \matrix{
   k – {n^2} + 3 = 16 \hfill \cr
   k + {n^2} – 3 = 1 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
   k = {{17} \over 2} \hfill \cr
   {n^2} = – {9 \over 2}\, \hfill \cr} \right.\,\,\left( {Loai} \right) \cr
   & TH10:\,\,\left\{ \matrix{
   k – {n^2} + 3 = – 16 \hfill \cr
   k + {n^2} – 3 = – 1 \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
   k = – {{17} \over 2} \hfill \cr
   {n^2} = {{21} \over 2}\, \hfill \cr} \right.\,\,\left( {Loai} \right) \cr
   & Vay\,\,n = 0 \cr} $$

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