Vtyjkyuk

Prove that there exists no integer among the $2^n+1$ numbers

$$pmfrac1k pm frac1k+1pmfrac1k+2...pmfrac1k+n $$

That is,

$$mathbbZ cap leftlbrace fracdelta_0k + fracdelta_1k + 1 + ldots + fracdelta_nk+n : delta_0, ldots, delta_n in lbrace -1, 1 rbrace rightrbrace = emptyset$$

This is a discrete maths homework question with another part preceding it which wants us to prove that "in any block of consecutive positive integers there is a unique integer divisible by
a higher power of $2$ than any of the others". This I could prove (by negation) so you can use this statement to answer the question.

Any suggestions of approaches to the problem are welcome.

Let $m$ be the exponent of this
maximum power of 2,
so $2^m$ divides exactly one denominator.

Consider the lcm of the denominators $d$.
$2^m|d$.
When all the fractions are written in the form
$c/d$, the one whose denominator
is divisible by $2^m$ will have
an odd numerator
and all the others will have
an even numerator.
Therefore their sum will be odd,
so the resulting fraction
will have an odd numerator
and even denominator
and so can not
be an integer.

Factor all the $n+1$ integers $k,k+1,cdots,k+n$ and extract the factor $2$ all together. We have $$i=2^lambda_ip_i,$$ where $i=k,k+1,cdots,k+n.$

Apparently, $lambda_i geq 0$ and $2 nmid p_i.$ Denote $lambda=maxlambda_i.$ Then $lambda>0$ when $ngeq 1.$ Morover, we may prove there exists only one $i$ such that $i=2^lambda p_i$. Otherwise, if $i=2^lambda p_i$ and $j=2^lambda p_j$ but $i neq j$, then there exists another even integer between $p_i $ and $p_j$(since $p_i$ and $p_i$ are two different odd integers), which implies there exists a $l$ between $i$ and $j$ such that $l=2^lambda+1p_l.$ This contradicts the assumption that $lambda$ is the largest.

Now, assume $m=2^lambda p_m.$ Multiply $$pmfrac1kpm frac1k+1pm cdots pm frac1k+n$$

by $X=2^lambda-1p_kp_k+1cdots p_k+n$. Apparently, $M$ could be divided by almost every denominator except for $m$. Thus, we may write $$left(pmfrac1kpm frac1k+1pm cdots pm frac1k+nright)X=Ypmfracp_kp_k+1cdots p_k+n2 p_m,tag1$$
where $Y$ is an integer. Notice that $p_k,p_k+1,cdots,p_k+n$ are all odd integers. Hence, the product of them except for $p_m$ could not be divided by $2$. Therefore, the right-hand side of $(1)$ is not an integer, from which we may complete the proof.