Can the powers of 2 in the denominators of a fraction sum be used to find a contradiction? Let

Caleb Proctor 2022-07-15 Answered
Can the powers of 2 in the denominators of a fraction sum be used to find a contradiction?
Let a 1 , a 2 , b 1 , b 2 , c 1 , c 2 , d 1 , d 2 , e 1 , e 2 be odd integers, with
gcd ( a 1 , a 2 ) = gcd ( b 1 , b 2 ) = gcd ( c 1 , c 2 ) = gcd ( d 1 , d 2 ) = gcd ( e 1 , e 2 ) = 1 ,
and assume n 3 and m 1 are integers such that
( ) a 1 2 3 ( n 1 ) m a 2 + b 1 2 3 ( ( n 1 ) m + 1 ) b 2 + c 1 2 3 ( ( n 1 ) m + 1 ) c 2 + d 1 2 3 n m d 2 + e 1 2 3 n m e 2 = 1.
Is there a way to prove, using only the powers of 2 in the set of denominators, that (⋆) is impossible?
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Answers (1)

Zichetti4b
Answered 2022-07-16 Author has 13 answers
The way it stands now, ( ) is obviously possible. Let n = 3 , m = 2. Then the denominators of five fractions contain (correspondingly, in the same order) 2 12 , 2 15 , 2 15 , 2 18 ,  and  2 18 . Now,
184313 45 2 12 + 1 9 2 15 + 1 1 2 15 + 1 15 2 18 + 1 1 2 18 =
...guess what?
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