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is called lucky if it has at least two distinct prime divisors and can be written in the form:![\[
n = p_1^{\alpha_1} + \cdots + p_k^{\alpha_k}
\]](http://latex.artofproblemsolving.com/7/4/4/744a5ccaeb9476ebd7d999c395762cb6e99a7a71.png)
where 
are distinct prime numbers that divide . (Note: it is possible that has other prime divisors not among .) Prove that for every prime number 
, there exists a lucky number such that 
.
, suppose that 
is expressed by the sum of 
consecutive odd numbers. Express the minimum term in terms of and 
.
(for 
) whose sum up to 
terms computes to:
(i).
in terms of 
, then we have:
.
Prove that
,such that 
P
,
