Q#167612 reposted. Determine all the possible triples (a,b,c) of positive integers for which ab−c,bc−a and ca−b are powers of 2.


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Question Number 167725 by Rasheed.Sindhi last updated on 23/Mar/22

$You can't use 'macro parameter character #' in math mode$ $D e t e r m i n e a l l t h e p o s s i b l e t r i p l e s$ $(a, b, c) o f p o s i t i v e i n t e g e r s f o r w h i c h$ $a b - c, b c - a a n d c a - b a r e p o w e r s o f$ $2 .$
Commented by Tinku Tara last updated on 24/Mar/22

$a = b = c = 2$
Commented by Tinku Tara last updated on 24/Mar/22

$if a^{2} - 1 is odd then 2^{x} + 2^{y} a need not be$ $odd . It could be 2 (a^{2} - 1) k$
Commented by Tinku Tara last updated on 24/Mar/22

$subtract (1) from (2)$ $a b - c = 2^{α}$ $a c - b = 2^{β}$ $a (b - c) - (b - c) = 2^{a} (2^{b} - 1)$ $(b - c) (a - 1) = 2^{a} (2^{b} - 1)$
Commented by Rasheed.Sindhi last updated on 24/Mar/22

$T {h e r e}^{'} s n o c o n t h e r i g h t s i d e .$ $T h a t m e a n s i t h a s n o e f f e c t o n t h e$ $r e s u l t ?$
	Answered by Rasheed.Sindhi last updated on 24/Mar/22

	$A T r y . . .$ $(e r e p r e s e n t s e v e n a n d o r e p r e s e n t s o d d h e r e)$ $C a s e 0 : N o o n e o f a, b, c i s o d d .$ $L e t a, b, c a r e d e n o t e d a s e_{a}, e_{b}, e_{c}$ $r e s p e c t i v e l y .$ $a b - c = e_{a} e_{b} - e_{c} = 2^{l}$ $e - e = e$ $e = e$ $b c - a = e_{b} e_{c} - e_{a} = 2^{m}$ $e - e = e$ $e = e$ $c a - b = e_{c} e_{a} - e_{b} = 2^{n}$ $e - e = e$ $e = e$ $N o c o n t r a d i c t i o n!$ $C a s e 1 : O n l y o n e o f a, b, c i s o d d .$ $L e t a, b, c a r e d e n o t e d a s o_{a}, e_{b}, e_{c}$ $r e s p e c t i v e l y .$ $a b - c = o_{a} e_{b} - e_{c} = 2^{l}$ $b c - a = e_{b} e_{c} - o_{a} = 2^{m}$ $e - o = e$ $o = e$ $C o n t r d i c t i o n!$ $c a - b = e_{c} o_{a} - e_{b} = 2^{n}$ $C a s e 2 : O n l y t w o o f a, b, c a r e o d d .$ $L e t a, b, c a r e d e n o t e d a s o_{a}, o_{b}, e_{c}$ $r e s p e c t i v e l y .$ $a b - c = o_{a} o_{b} - e_{c} = 2^{l}$ $o - e = e$ $o = e$ $C o n t r d i c t i o n!$ $C a s e 3 : A l l o f a, b, c a r e o d d .$ $L e t a, b, c a r e d e n o t e d a s o_{a}, o_{b}, o_{c}$ $r e s p e c t i v e l y .$ $a b - c = o_{a} o_{b} - o_{c} = 2^{l}$ $o - o = e$ $e = e$ $b c - a = o_{b} o_{c} - o_{a} = 2^{m}$ $o - o = e$ $e = e$ $c a - b = o_{c} o_{a} - o_{b} = 2^{l}$ $o - o = e$ $e = e$ $N o c o n t r a d i c t i o n!$ $∴ E i t h e r a l l t h e t h r e e a r e o d d o r$ $e v e n .$ $\begin{matrix} {\begin{matrix} a, b, c \in E \end{matrix}}_{{\begin{matrix} or \end{matrix}}_{\begin{matrix} a, b, c \in_{}^{} O \end{matrix}}} \end{matrix}$
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