252 5 Number Theory
715.For a positive integernand a real numberx, compute the sum
∑0 ≤i<j≤n⌊
x+i
j⌋
.
716.Prove that for any positive integern,
√
n=⌊√
n+1
√
n+√
n+ 2⌋
.
717.Express
∑n
k= 1 √
kin terms ofnanda=√
n.718.Prove the identity
n(n 2 + 1 )
∑k= 1⌊
− 1 +
√
1 + 8 k
2⌋
=
n(n^2 + 2 )
3,n≥ 1.719.Find all pairs of real numbers(a, b)such thatabn=banfor all positive
integersn.
720.Forpandqcoprime positive integers prove the reciprocity law
⌊
p
q⌋
+
⌊
2 p
q⌋
+···+
⌊
(q− 1 )p
q⌋
=
⌊
q
p⌋
+
⌊
2 q
p⌋
+···+
⌊
(p− 1 )q
p⌋
.
721.Prove that for any real numberxand for any positive integern,
nx≥x
1+
2 x
2+
3 x
3+···+
nx
n.
722.Does there exist a strictly increasing functionf:N→Nsuch thatf( 1 )=2 and
f (f (n))=f (n)+nfor alln?
723.Suppose that the strictly increasing functionsf, g:N→NpartitionNinto two
disjoint sets and satisfy
g(n)=f(f(kn))+ 1 , for alln≥ 1 ,for some fixed positive integerk. Prove thatfandgare unique with this property
and find explicit formulas for them.