Availability and Installation of Hydrogen
Hydrogen as an energy source is available
because it could be produced from water
Technology of Hydrogen
Economics of Hydrogen
Impact on the environment and humans
Hydrogen impact on animals: aquatic and
terrestr
Math 567 Number Theory I, Solutions 10
p
Throughout e(z ) = e2iz , p is an odd prime and (x) the Legendre symbol modulo p. Also p = n=1 (n)e(n/p),
the Gauss sum formed from . L = L(p) is the least positive quadratic non-residue modulo p and U is the numbe
567 NUMBER THEORY I, FALL 2008, SOLUTIONS 12
1. Show that for arbitrary real or complex numbers c1 , . . . , cq ,
q
cn (n)
q
2
|cn |2
= (q )
n=1
n=1
(n,q )=1
where the sum on the left hand side runs over all Dirichlet characters (mod q ).
q
q
LHS= m=1 n=1
567 Number Theory I, Fall Term 2008, Solutions 13
1. (i) Find all elements A of which commute with S . (ii) Find all elements A of which commute
with T . (iii) Find the smallest n > 0 such that (ST )n = I . (iv) Determine all A in which leave i
xed. (v) D
9. MODULAR FORMS
9.1. Introduction A modular form is an analytic function which satises a certain simple relationship
under the action of Mbius transformations together with some other simple properties, to be dened. The
o
importance of modular forms is t
567 Number Theory I, Fall Term 2008, Solutions 14
1. (i) Prove that if p 1 (mod 3), then
3
p
p 1
2
= 1. By quad. recip.
L
3
p
=
L
(1)
(1) p1 p L = 1.
2
3
(ii) Let M = cfw_n N : p|n = p 1 (mod 3). Prove that if n M, then
x2 + 3 0 (mod 4n) is soluble in x.
Math 567 Number Theory I, Solutions 9
1. (n) denotes the total number of prime factors on n. Show that (n)
2(n) n.
log n
log 2 .
We have
2
2. Show that nx (n) = x + O(log x) for x 2. We have (n) = m|n m = m|n n/m.
n
6
1
x
Thus
(n)/n =
= x mx m2 + O(log
MATH 567 FALL 2008, NUMBER THEORY I, SOLUTIONS 2
1. Let a, b, c Z with a and b not both zero. Prove each of the following. (i) If (a, b) = 1 and
a|bc, then a|c. (ii)
a
b
(a,b) , (a,b)
= 1. (iii) (a, b) = (a + cb, b). (i) There are x and y such that
ax + b
MATH 567 INTRODUCTION TO NUMBER
THEORY I, FALL TERM 2008, SOLUTIONS 11
1. (a) Prove that if x 1, then
(n)
nx n
1 + 1/x. (a) x/n =
nx
(n)
m x
n|m
x
n
= 1. (b) Prove that 1 + 1/x
(n) x/n =
1, so
nx
(n) = 1.
mx n|m
(b) We have x nx (n) =
nx (n) x/n +
nx (n
10 HECKE OPERATORS
10.1. Introduction The motivation here is to nd operators T which act on modular
forms f of given weight 2k in such a way that the Fourier coecients of f and T f satisfy
useful relationships. A crucial rle is played by eigenforms f whic
MATH 567 FALL 2008, NUMBER THEORY II, SOLUTIONS 3
1. Show that if p is a prime number and 1 j p 1, then p divides the binomial
coecient p . p! = j !(p j )! p . Moreover p|LHS but p j !(p j )!.
j
j
2. Show that n|(n 1)! for all composite n > 4. We have n =
Practice 2.1.9: Brain Biology
Name: Jonathan Elisha
Username: jelisha
Date: February 28, 2015
Corpus Callosum
Description
- Thick band of axon fibers that connect and
allow communication between the two
hemispheres; its located in the cerebral cortex
in
MATH 567 FALL 2008, NUMBER THEORY I, PROBLEMS 1
To be submitted by Tuesday, September 2nd
Easier problems
1. Given a|b and c|d, prove that ac|bd.
2. Prove that if n is odd, then n2 1 is divisible by 8.
3. Find the greatest common divisor g of the numbers
MATH 567 FALL 2008, NUMBER THEORY I, PROBLEMS 4
To be submitted by Tuesday, September 23rd
Easier problems
1. Show that if f (x) is a polynomial with integer coecients and if f (a) k
(mod m), then f (a + tm) k (mod m) for every integer t.
2. Prove that an
MATH 567 FALL 2008, NUMBER THEORY II, PROBLEMS 3
To be submitted by Tuesday, September 16th
Easier problems
1. Show that if p is a prime number and 1 j p 1, then p divides the binomial
coecient p .
j
2. Show that n|(n 1)! for all composite n > 4.
3. Exhib
MATH 567 NUMBER THEORY I, PROBLEMS 7
To be submitted by Tuesday, October 14th
Throughout this problem sheet, p denotes an odd prime number.
Easier problems
1. Let g be a primitive root modulo p. Prove that the quadratic residues are
precisely the residue
Math 567 Fall 2008, Number Theory I, Solutions 1
1. Given a|b and c|d, prove that ac|bd. By denition there are u, v Z such that
b = au, d = cv . Thus bd = aucv = (ac)(uv ).
2. Prove that if n is odd, then n2 1 is divisible by 8. Since n is odd, n = 4k + r
MATH 567 NUMBER THEORY I, SOLUTIONS 7
Throughout this problem sheet, p denotes an odd prime number.
1. Let g be a primitive root modulo p. Prove that the quadratic residues are precisely the residue
classes g 2k with 0 k < 1 (p 1). Show that the sum of th
MATH 567 FALL 2008, NUMBER THEORY I, SOLUTIONS 4
1. Show that if f (x) is a polynomial with integer coecients and if f (a) k (mod m),
d
then f (a + tm) k (mod m) for every integer t. Write f (x) = j =0 aj xj with aj Z. We
have (a + tm)j =
(mod m).
j
j
h=0
Math 567 Fall 2008 Number Theory I, Solutions 5
1. Prove that if (a, m) = (a 1, m) = 1, then 1 + a + a2 + + a(m)1 0 (mod m), and
deduce that every prime other than 2 or 5 divides innitely many of the integers 1, 11, 111,
1111, . . . . (1 + a + a2 + + a(m)
Math 567 Fall 2008 Number Theory I, Solutions 6
2
1. Show that
m|n
d(m)
=
m|n
d(m)3 . d and d3 are multiplicative, hence so are the sums on either side. Moreover it
k
j =0 (j
is easily proved by induction on k that
+ 1) = 1 (k + 1)(k + 2) and
2
k
j =0 (j
MATH 567 NUMBER THEORY I, PROBLEMS 9
To be submitted by Tuesday 28th October
Easier problems
1. (n) denotes the total number of prime factors on n. Show that (n)
2. Show that
nx
3. Let D(x) =
d(n)
nx n
that
=
( n)
n
=
2
6x
log n
log 2 .
+ O(log x) for x
Math 567 Number Theory I, Solutions 9
1. (n) denotes the total number of prime factors on n. Show that (n)
2(n) n.
log n
log 2 .
We have
2
2. Show that nx (n) = x + O(log x) for x 2. We have (n) = m|n m = m|n n/m.
n
6
1
x
Thus
(n)/n =
= x mx m2 + O(log
MATH 567 NUMBER THEORY I, SOLUTIONS 7
Throughout this problem sheet, p denotes an odd prime number.
1. Let g be a primitive root modulo p. Prove that the quadratic residues are precisely the residue
classes g 2k with 0 k < 1 (p 1). Show that the sum of th
Math 567 Fall 2008 Number Theory I, Solutions 6
2
1. Show that
m|n
d(m)
=
m|n
d(m)3 . d and d3 are multiplicative, hence so are the sums on either side. Moreover it
k
j =0 (j
is easily proved by induction on k that
+ 1) = 1 (k + 1)(k + 2) and
2
k
j =0 (j
Math 567 Fall 2008 Number Theory I, Solutions 5
1. Prove that if (a, m) = (a 1, m) = 1, then 1 + a + a2 + + a(m)1 0 (mod m), and
deduce that every prime other than 2 or 5 divides innitely many of the integers 1, 11, 111,
1111, . . . . (1 + a + a2 + + a(m)
MATH 567 FALL 2008, NUMBER THEORY I, SOLUTIONS 4
1. Show that if f (x) is a polynomial with integer coecients and if f (a) k (mod m),
d
then f (a + tm) k (mod m) for every integer t. Write f (x) = j =0 aj xj with aj Z. We
have (a + tm)j =
(mod m).
j
j
h=0
MATH 567 FALL 2008, NUMBER THEORY II, SOLUTIONS 3
1. Show that if p is a prime number and 1 j p 1, then p divides the binomial
coecient p . p! = j !(p j )! p . Moreover p|LHS but p j !(p j )!.
j
j
2. Show that n|(n 1)! for all composite n > 4. We have n =