Problem 2

Determine all triples

(a,b,c)

of positive integers such that each of the numbers

ab-c

bc-a

, and

ca-b

is a power of

2

(a power of

2

is an integer of the form

2^n

, where

n

is a nonnegative integer).

Step 1 of 8: Use symmetry and order the variables

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In plain words

Permuting $a,b,c$ only permutes the three expressions, so we may impose $a\le b\le c$ .

(a,b,c)\mapsto(b,a,c)\ \text{fixes}\ \{ab-c,\,bc-a,\,ca-b\}\ \text{as a set}

Detailed analysis

Swapping any two variables permutes the set $\{ab-c,bc-a,ca-b\}$ , so the conditions are symmetric. Assume $a\le b\le c$ and write $ab-c=2^m$ , $ac-b=2^n$ , and $bc-a=2^p$ . The inequalities $ab-c\le ac-b\le bc-a$ follow respectively from $(a+1)(c-b)\ge0$ and $(c+1)(b-a)\ge0$ , so $m\le n\le p$ .