AoPS Academy
consecutive positive integers is divisible by 
for which there exists an integer 
such that 
divides 
.
and be natural numbers such that ![\[A=\frac{(m+3)^{n}+1}{3m}\]](http://latex.artofproblemsolving.com/f/5/8/f58ed8713c887510943d543c89f83640a546401f.png)
is an integer. Prove that 
is odd.
and be natural numbers and let 
be divisible by 
. Show that 
is divisible by .
and 
be natural numbers having the property ![\[n_{2}\; \vert \; 2^{n_{1}}-1, n_{3}\; \vert \; 2^{n_{2}}-1, \cdots, n_{k}\; \vert \; 2^{n_{k-1}}-1, n_{1}\; \vert \; 2^{n_{k}}-1.\]](http://latex.artofproblemsolving.com/2/0/7/20707ef12d0e08bec8067225a9e9128bbc1d2d45.png)
Show that 
.
, only finitely much, and number them 
. Now 
with 
rational. So, we can express 
as a (nonnegative) linear combination of rationals and irrationals with higher indices (over 
).
be the smallest index for which this is impossible, then 
with 
and necessarily 
irrational (else 
irrational but must have higher index). But then 
, which is a nonnegative linear combination of 
. Now, if the coefficient in that combination of 
is not 
, then we can solve it for as a (nonnegative) linear combination of 
, contradiction. So appears in coefficient and, by nonnegativity, the rest appears in coefficient 
, so 
, contradiction.
and thus every 
is a rational multiple of 
. But then 
with 
rational and 
irratiional. But then 
, and thus 
would be irrational, contradiction. So there can be no irrational number in .
as a vectorial space over 
. Let 
be the subspace generated by 
. If 
, take 
in such that 
is a basis for .
can be written as 
, with 
.
whose coefficient of 
is maximal. It is the average o two other numbers. Clearly, both of them have the same coefficient as . as large as possible.
be a rational number with 
and 
.
.
![\[\sum_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}\]](http://latex.artofproblemsolving.com/d/4/5/d4511807f1e04ac2fa8aae36d65f9e0f5ba11c6f.png)
is not divisible by 
for any integer 
.
is composite, then 
.
.![$ [0,1]$](http://latex.artofproblemsolving.com/b/3/8/b38b2d1b674bd60d23ac11b288abb1f8a57017cc.png)
with the following property: for all 
,
with 
such that 
.