Question

Show that the following equations have no integer solutions: (4) 25 + 12x2 + 24x + 1 = 0. (1) x4 + x + 1 = 0; (2) 12x + 5 = y2; (3) x? +6x +1 = 0;

Answer 1

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1)
2° ++1=0

Modulo we have
2 = r mod 2
so equation becomes
2.1 +1 = 0 (mod 2) =1=0 (mod 2)
which is impossible so this equation has no integer solution

2)
12.c +5=y

Modulo 3 we get
12.c + 5 = y2 (mod 3) = 2 = y2 (mod 3)
which is impossible as squares modulo 3 are 0 or 1. So this equation has no integer solution

3)
r' + 6.6 +1=0

' = r mod 7
and so equation becomes
I + 6.6 +1 = 0 (mod 7) = 72 +1=0 mod 7) =1=0 (mod 7)
so this is impossible