Show that 2x^7 −4x^4 +4x^2 =6x^6 +3 Has no solution in N


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Question Number 89906 by Ar Brandon last updated on 19/Apr/20

$S h o w t h a t$ $2 x^{7} - 4 x^{4} + 4 x^{2} = 6 x^{6} + 3$ $H a s n o s o l u t i o n i n N$
	Answered by MJS last updated on 20/Apr/20

	$transform to$ $x^{7} - 3 x^{6} - 2 x^{4} + 2 x^{2} - \frac{3}{2} = 0$ $p = x^{7} - 3 x^{6} - 2 x^{4} + 2 x^{2} \land q = - \frac{3}{2}$ $x \in N \Rightarrow p \in N but q \notin N$ $or$ $x, m, n \in N$ $2 x^{7} - 4 x^{4} + 4 x^{2} = 2 (x^{7} - 2 x^{4} + 2 x^{2}) = 2 m$ $6 x^{6} + 3 = 3 (2 x^{6} + 1) = 3 (2 n + 1)$ $2 m = 3 (2 n + 1)$ $\frac{2}{3} m = 2 n + 1$ $let m = 3 k; k \in N$ $2 k = 2 n + 1$ $even = uneven$ $impossible$
	Commented by Ar Brandon last updated on 20/Apr/20

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