September 4, 2007

**Number Theory Problems (for next week)**

1. Show that if m and n are positive integers then

S(m,n) = 1/m + 1/(m+1) + 1/(m+2) + … + 1/(m+n)

is NOT an integer.

2. Let a and b be relatively prime positive integers.

(a) Suppose that n > ab – a – b. Show

that there are non-negative integers x, y such that

ax + by = n.

(b) Suppose that n = ab – a – b. Are there non-negative integers x, y such that

ax + by = n ?

3. Consider the following two-person game. A number of stones are lying on a table. The two players move alternately. The player whose turn it is takes away x stones, where x is any positive square integer. The player that takes the last stone wins.

Prove that there are infinitely many initial situations for which the second player has a winning strategy.

4. Show that there are infinitely many positive integers n such that the largest prime divisor

of n^{4}
+ 1 is greater than 2n.

(end)