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.

 

 

 

 

 

 

(end)