If p, q are integers so that q>p>1, show that 1/p+ 1/(p+1)+ 1/(p+2)+ ... +1/q cannot be an integer.
The final inequality is erroneous - the correct end to the proof is given in the next comment.
Let m be the least common multiple of p, (p+1),..., q. Rewrite the given sum as a/m, where
a = (m/p) + (m/(p+1)) + ... + (m/q)
It suffices to show that a is odd, since m is even.
Let k be the highest integer such that 2^k divides m. We will
show that exactly one of the numbers (m/p), (m/(p+1)),..., (m/q) is
odd.
It is easy to see that at least one of the
numbers must be odd, since one of the numbers p, (p+1),..., q is
divisible by 2^k. Suppose that more than one of the numbers is
odd. Then, we have
p <= x*2^k < y*2^k <= q
for some odd x, y. But then we have a number that is divisible by 2^(k+1), strictly between p and q:
p <= x*2^k <
((x+y)/2)*2^{k+1) < y*2^k <= q,
contradicting the definition of k. So a is odd as claimed.
Edited on October 6, 2004, 4:36 pm
Solution
