solve for all x ⌊x^2 ⌋ − ⌊x⌋^2 = 100


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Question Number 174960 by alcohol last updated on 14/Aug/22

$s o l v e f o r a l l x$ $⌊ x^{2} ⌋ - ⌊ x ⌋^{2} = 100$
	Answered by mr W last updated on 15/Aug/22

	$s a y x = n + f w i t h 0 ⩽ f < 1$ $⌊ n^{2} + 2 n f + f^{2} ⌋ - n^{2} = 100$ $⌊ 2 n f + f^{2} ⌋ = 100$ $100 = ⌊ 2 n f + f^{2} ⌋ < 2 n + 1$ $\Rightarrow n > 49.5 \Rightarrow n ⩾ 50$ $⌊ x^{2} ⌋ = 100 + n^{2}$ $100 + n^{2} ⩽ x^{2} < 101 + n^{2}$ $\Rightarrow \sqrt{100 + n^{2}} ⩽ x < \sqrt{101 + n^{2}} w i t h n ⩾ 50$
	Commented by alcohol last updated on 15/Aug/22

	$t h a n k y o u$ $p l e a s e c a n y o u e x p l a i n ?$
	Commented by mr W last updated on 15/Aug/22

	$i t h o u g h t i h a v e g i v e n a l l n e c e s s a r y$ $s t e p s .$ $w h e r e d o y o u n e e d m o r e e x p l a n a t i o n ?$
	Commented by alcohol last updated on 15/Aug/22

	$i h a v e n o i d e a a b o u t i t . i f y o u c a n p r o p o s e m e a b o o k$ $t o r e a d i w i l l b e g r a t e f u l$
	Commented by mr W last updated on 15/Aug/22

	$s i n c e y o u a s k e d t h i s q u e s t i o n, i a s s u m e d$ $t h a t y o u k n o w w h a t t h e g r e a t e s t i n t e g e r$ $f u n c t i o n ⌊ x ⌋ i s . i {d o n}^{'} t k n o w a n y b o o k s$ $a b o u t s u c h t h i n g s .$
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