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 n4 + 1 is greater than 2n.