How would one go about solving the system of five equations: p 2 = p + q − 2 r + 2 s + t − 8 q 2 = − p − 2 q − r + 2 s + 2 t − 6 r 2 = 3 p + 2 q + r + 2 s + 2 t − 31 s 2 = 2 p + q + r + 2 s + 2 t − 2 t 2 = p + 2 q + 3 r + 2 s + t − 8over the integers?

Question

How would one go about solving the system of five equations:      p    2    =  p  +  q  −  2  r  +  2  s  +  t  −  8      q    2    =  −  p  −  2  q  −  r  +  2  s  +  2  t  −  6      r    2    =  3  p  +  2  q  +  r  +  2  s  +  2  t  −  31      s    2    =  2  p  +  q  +  r  +  2  s  +  2  t  −  2      t    2    =  p  +  2  q  +  3  r  +  2  s  +  t  −  8over the integers?

Ullveruxqte · Accepted Answer

Not a direct answer but too big for a comment: Rewrite your question as      (                            p          −          1                          −          1                          2                          −          2                          −          1                          8                                      1                          q          +          2                          1                          −          2                          −          2                          6                                      −          3                          −          2                          r          −          1                          −          2                          −          2                          31                                      −          2                          −          1                          −          1                          s          −          2                          −          2                          2                                      −          1                          −          2                          −          3                          −          2                          t          −          1                          8                      )        (                            p                                      q                                      r                                      s                                      t                                      1                      )    =  0I would go for the nullspace of this matrix which is not simplifying much but maybe allow for a cleaner search. For example,   s and   t columns look suspicious.EDIT: A small Matlab routine gave a solution as       (                            p                          q                          r                          s                          t                      )    =      (                            3                          2                          1                          5                          4                      )  EDIT2 : I forgot to write that I have massaged the problem a bit by applying some row manipulations from the left which is the only detail that I wanted to stress but I wrote anything but that.

June Salas · Accepted Answer

If you add all your equations you get      p    2    +      q    2    +      r    2    +      s    2    +      t    2    =  6  p  +  4  q  +  2  r  +  10  s  +  8  t  −  55By completing the square you can rewrite this as  (  p  −  3      )    2    +  (  q  −  2      )    2    +  (  r  −  1      )    2    +  (  s  −  5      )    2    +  (  t  −  4      )    2    =  9  +  4  +  1  +  25  +  16  −  55  =  0Obviously, the only solution is   p  =  3,   q  =  2,   r  =  1,   s  =  5,   t  =  4.

How would one go about solving the system of five equations: p 2 </msup> = p +

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