# Infinite odd Engel expansions

2/7 = 1/5 + 1/(5*3) + 1/(5*3*5) + 1/(5*3*5*3) + 1/(5*3*5*3*5) + ... and

3/7 = 1/3 + 1/(3*5) + 1/(3*5*3) + 1/(3*5*3*5) + 1/(3*5*3*5*3) + ...

More generally,

2/(12n+7) = 1/(6n+5) + 1/(6n+5)(4n+3) + 1/(6n+5)(4n+3)(6n+5) + ... and

3/(12n+7) = 1/(4n+3) + 1/(4n+3)(6n+5) + 1/(4n+3)(6n+5)(4n+3) + ...

So the analogue to the odd greedy Egyptian fraction problem for Engel expansion has an immediate negative answer.

The same infinite 2/7 and 3/7 expansions also occur as the CostÃ© prime expansion of these numbers, which is infinite for the same reasons.

3/7 = 1/3 + 1/(3*5) + 1/(3*5*3) + 1/(3*5*3*5) + 1/(3*5*3*5*3) + ...

More generally,

2/(12n+7) = 1/(6n+5) + 1/(6n+5)(4n+3) + 1/(6n+5)(4n+3)(6n+5) + ... and

3/(12n+7) = 1/(4n+3) + 1/(4n+3)(6n+5) + 1/(4n+3)(6n+5)(4n+3) + ...

So the analogue to the odd greedy Egyptian fraction problem for Engel expansion has an immediate negative answer.

The same infinite 2/7 and 3/7 expansions also occur as the CostÃ© prime expansion of these numbers, which is infinite for the same reasons.