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

dalo2m1ezyxlo 2022-06-03 Answered
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 8
over the integers?
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Answers (2)

Ullveruxqte
Answered 2022-06-04 Author has 4 answers
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 ) = 0
I 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.
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June Salas
Answered 2022-06-05 Author has 1 answers
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 55
By 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 = 0
Obviously, the only solution is p = 3, q = 2, r = 1, s = 5, t = 4.
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