PROVE that the numbers of types 4k+2 & 4k+3 are NOT perfect □s.


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Question Number 161528 by Rasheed.Sindhi last updated on 19/Dec/21

$PROVE that the numbers of types$ $4 k + 2 & 4 k + 3 are NOT perfect ◻ s .$
	Answered by mr W last updated on 19/Dec/21

	$(1) t y p e 4 k + 3$ $a s s u m e i t c a n b e a p e r f e c t s q u a r e .$ $t h a t m e a n s t h e r e e x i s t s s u c h a n o d d$ $n u m b e r 2 n + 1, s u c h t h a t$ $4 k + 3 = {(2 n + 1)}^{2}$ $\Rightarrow 4 k + 3 = 4 n^{2} + 4 n + 1$ $\Rightarrow \frac{1}{2} = n^{2} + n - k = i n t e g e r$ $t h i s i s a c o n t r a d i c t i o n, t h e r e f o r e$ $4 k + 3 c a n n o t b e a p e r f e c t s q u a r e .$ $(2) t y p e 4 k + 2$ $s i m i l a r l y$
	Commented by Rasheed.Sindhi last updated on 19/Dec/21

	$N i c e s i r! T h a n X!$
	Commented by Rasheed.Sindhi last updated on 19/Dec/21

	$F o r t y p e 4 k + 2$ $4 k + 2 = 2 (2 k + 1)$ $∵ 2 o c c u r s o n c e o n l y$ $B u t i n p e r f e c t s q u a r e a n y p r i m e$ $f a c t o r o c c u r s i n e v e n n u m b e r t i m e s$ $∴ 4 k + 2 i s n o t a p e r f e c t s q u a r e .$
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