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-

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 \dots

(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}