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 
divides 
.
consecutive positive integers is divisible by 
such that 
divides 
.![\[A=\frac{(m+3)^{n}+1}{3m}\]](http://latex.artofproblemsolving.com/f/5/8/f58ed8713c887510943d543c89f83640a546401f.png)
is an integer. Prove that 
is odd.
be divisible by 
.
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 
.
.
Y by UpvoteFarm, Adventure10, Mango247
Let 
be a sequence of positive integers such that ![\[0 < a_{n+1}-a_{n}\le 2001 \;\; \text{for all}\;\; n \in \mathbb{N}.\]](//latex.artofproblemsolving.com/7/7/8/77870f7f51a33f04415ef37f9cc0a315a9e08dbf.png)
Show that there are infinitely many pairs 
of positive integers such that 
and 
.
be a sequence of positive integers such that ![\[0 < a_{n+1}-a_{n}\le 2001 \;\; \text{for all}\;\; n \in \mathbb{N}.\]](http://latex.artofproblemsolving.com/7/7/8/77870f7f51a33f04415ef37f9cc0a315a9e08dbf.png)
Show that there are infinitely many pairs 
of positive integers such that 
and 
.
where 
and 
