Number Theory for SKKU
Final Test
1. (20 points) Find at least two primitive roots
17.
( modulo
)
211
2. (20 points) Compute the Legedre symbol
.
37
3. (20 points) Let p be an odd prime such that p = a2 + b2
for
some integers a and b. Let a be an odd integer. Prove
( )
a
= 1.
p
4. (20 points) Suppose that σ(n) is an odd integer. Show
that n is a square or twice a square. (i.e. n = m2 or
n = 2m2 for some integer m.)
)
∑ µ(d)
∏ (
1
5. (20 points) Prove that
=
d|n d2
p|n 1 − p2 ,
where the product runs through all primes p dividing n.
6. (20 points) Find at least two primitive roots
19.
( modulo
)
198
7. (20 points) Compute the Legedre symbol
.
41
8. (20 points) Let p (
be a)prime such that p ≡ ±1 (mod 4a)
a
where a ≥ 1. Prove
= 1.
p
8. (20 points) Find all n ≥ 1 satisfyng ϕ(n) = 2. (
)
∑ µ2 (d)
∏
1
10. (20 points) Prove that
1
+
=
d|n
p|n
d
p ,
where the product runs through all primes p dividing n.
11. (20 points) Let p be an odd prime
( ) and let a be a
a
primitive root modulo p. Show that
= −1.
p
(
)
2000
12. (20 points) Compute the Legedre symbol
.
97
13. (20 points) Let p be an odd prime such that p = a2 + b2
for
some integers a and b. Let a be an odd integer. Prove
( )
a
= 1.
p
14. (20 points) Let p (̸= 3) be a prime and let S =
{1, 2, · · · , p − 1} be a reduced residue system modulo p.
Let A = {a3 | a ∈ S}. Find the number of elements of A
(a) when p ≡ 1 (mod 3);
(b) when p ≡ 2 (mod
3) )
∏ (
∑ µ(d)
1
15. (20 points) Prove that
p|n 1 − p2 ,
d|n d2 =
where the product runs through all primes p dividing n.
)
∑ µ(d)
∏ (
1
20. (20 points) Prove that
=
1
−
,
2
2
d|n d
p|n
p
where the product runs through all primes p dividing n.
21. (20 points) Let p be an odd prime and let S =
{1, 2, · · · , p − 1} be a reduced residue system modulo p.
Find the number of primitive roots modulo p in (S.
)
2001
22. (20 points) Compute the Legedre symbol
.
47
(hint: 2001 is divisible by 23.)
( )
5
23. (20 points) Find a formula for
where p is a prime
p
with p ̸= 2, 5.
∑
24. (20 points) Prove that d|n 1/d = σ(n)/n, where σ(n)
is the sum of all positive divisors of n.
25. (20 points) Let 2n − 1, n > 2 be a prime. Show that
(a) n is a prime and (b) 2n + 1 is a composite number.
26. (20 points) Let p be an odd prime
( ) and let a be a
a
primitive root modulo p. Show that
= −1.
p
)
( 3 2
2 5 111
.
27. (20 points) Compute the Legedre symbol
53
28. (20 points) Show that d(n) is odd if and only if n is a
perfect square. (d(n) is the number of positive divisors of
n.)
29. (20 points) Let n be a fixed positive integer. Show that
there are only finitely many x such that ϕ(x) = n. (ϕ is an
Euler’s phi function.)
(
)
−3
30. (20 points) Let p be an odd prime. Show that
=
p
1 if and only if p ≡ 1 (mod 3).
31. (10 points each) Let p (̸= 3) be a prime and let S =
{1, 2, · · · , p − 1} be a reduced residue system modulo p. Let
A = {a3 | a ∈ S}. Find the number of elements of A
(a) when p ≡ 1 (mod 3);
(b) when p ≡ 2((mod
) 3)
−7
32. (20 points) Compute the Legedre symbol
, where
p
p ̸= 7 is an odd prime.
33. (20 points) Prove that
)
∑ µ(d) ∏ (
1
=
1 − 100 ,
d100
p
d|n
16. (20 points) Find at least two primitive roots modulo
19.
(
)
211
17. (20 points) Compute the Legedre symbol
.
37
18. (20 points) Let p(be )
a prime such that p ≡ ±1 (mod 4a)
a
= 1.
where a ≥ 1. Prove
p
19. (20 points) Suppose that σ(n) is an odd integer. Show
that n is a square or twice a square. (i.e. n = m2 or
n = 2m2 for some integer m.)
p|n
where the product runs through all primes p dividing n.
34. (10 points each)
(
)
556
(a) Compute the Jacobi symbol
.
121
(b) Is the congruence x2 ≡ 556 (mod 121) solvable or not
?
35. (20 points) Suppose that the congruence x2 ≡ −1
(mod p) is solvable, where p is a prime. Then show that
the congruence x2 ≡ −1 (mod ps ) is solvable for any s ≥ 2.
(Hint: Use induction argument.)
36. (15 points) Let 2n − 1 be a prime where n ≥ 1. Show
that n is also a prime.
37. (15 points) Let p be an odd prime. Is the product
of two primitive roots (mod p) also primitive ? Prove the
assertion or give a counterexample.
(
)
−11
38. (20 points) Compute the Legedre symbol
,
p
where p ̸= 11 is an odd prime.
39. (15 points) Let f (n) and g(n) be multiplicative arithmetic functions. Is the product f (n)g(n) also multiplicative
? Prove the assertion or give a counterexample.
40. (15 points) Let 2p(− 1 be )
a Mersenne prime. Compute
2p
the Legendre symbol
2p − 1
41. (20 points) Let p(be )
a prime such that p ≡ ±1 (mod 4a)
a
where a ≥ 1. Prove
= 1.
p
6. (20 points) Let p be an odd prime and a be an integer
with (a, p) = 1. Let S = {±1, ±2, · · · , ± p−1
2 }. For all
1 ≤ i ≤ p−1
,
choose
a
∈
S
satisfying
ia
≡
ai (mod p).
i
2
Prove that ai ̸= ±aj if i ̸= j.
7. (4 points each) Write True or False for the following
statements. There will be −2 points penalty for each of
wrong guesses.
(a) There are infinitely many Carmichael numbers.
(b) Let ab ≡ 1 (mod p) where p is an odd prime. If a
is a quadratic residue (mod p), then b is also a quadratic
residue (mod p).
(c) Let d be a positive integer with ax + by = d. Then
(a, b) = d.
(d) n (> 1) is a prime if and only if ϕ(n) = n − 1.
(e) For a given n, factoring n is easier than determining
whether n is a prime or not.
2007
42. (15 points) Is the congruence x2 ≡ 1592 (mod 169)
solvable or not ?
43. (15 points) Is there a primitive root (mod 16) ? Prove
or disprove.
44. (20 points) Let p be an odd prime and a and b be
integers with p ̸ |ab. If ordp a and ordp b are relatively prime,
then show ordp ab = ordp a · ordp b.
(
)
−5
45. (15 points) Compute the Legedre symbol
, where
p
p ̸= 5 is an odd prime.
46. (15 points) Show that a strong pseudoprime to base b
is a Fermat pseudoprime to base b.
47. (20 points) Let p and q be odd primes
p ≡ −q
( such
) that
( )
a
a
(mod 4a) where a ≥ 1 and p ̸ |a. Prove
=
.
p
q
1. (15 points) Let p be an odd prime and let g be a primitive
root (mod p). Prove that −g is a primitive root (mod p)
if and only if p ≡ 1 (mod 4).
2. (15 points) Let a, b be positive integers with (a, b) = 1.
Then prove that
aϕ(b) + bϕ(a) ≡ 1
(mod ab)
(
)
92
.
331
4. (15 points) Let n ≥ 1 be an odd composite integer and
a be an integer with (a, n) = 1. We say n is an Euler
pseudoprime to base a if
(a)
n−1
(mod n),
≡a 2
n
2006
(a)
where
the
Jacobi
symbol
is a natural extension of the
1. (15 points) Find a primitive root (mod 25).
n( ) ( )( )
( )
a
a
a
a
2. (15 points) Let p be a prime with p ≡ 3 (mod 4). Let g
Legendre symbol defined by
=
···
is a primitive root (mod p). Then show that −g is not a
n
p1
p2
ps
primitive root (mod p).
with n = p1 p2 · · · ps a product of (not necessarily distinct)
3. (15 points) Determine whether the congruence equation primes. Show that an Euler pseudoprime to base a is a
Fermat pseudoprime to base a.
x2 − 4x − 6 ≡ 0 (mod 2011)
5. (20 points) Let p be an odd prime. Prove that there
exists a primitive root (mod p2 ).
has a solution or not. (Hint: 2011 is a prime)
6. (20 points) Let a be a positive integer which is not a
4.
(15 points) Let n > 2 be an integer and
square, i.e., a ̸= n2 for all n = 1, 2, 3, (
· · · .)Show that that
{a1 , a2 , · · · , aϕ(n) } be a reduced residue system (mod n).
a
ϕ(n)
there exists an odd prime p such that
= −1.
∑
p
Prove that
aj ≡ 0 (mod n). (What happen when n = 2 Hint: You may use “Dirichlet’s Theorem on Primes in
j=1
Arithmetic Progressions” which says; Let m, r be positive
?)
integers such that (m, r) = 1. Then there are infinitely
5. (20 points) Let p be an odd prime with p ≡ 2 (mod 3).
many primes among the integers mk + r, k = 1, 2, 3, · · · .
Show that the equation
7. (4 points each) Write True or False for the following
2
2
statements. There will be −2 points penalty for each of
x + 3y = p
wrong guesses.
does not have any integral solution.
(a) Product of two primitive roots is also a primitive root.
3. (15 points) Compute the Legedre symbol
1. (15 points) Is the congruence x2 ≡ 102 (mod 169) solvable or not ?
2. (15 points) Show that, if p is prime and 2p − 1 is composite, then 2p − 1 is a Fermat pseudoprime to the base
2.
3. (15 points) Let p(be )a prime such that p ≡ 1 (mod 4a)
a
= 1.
where a ≥ 1. Prove
p
4. (15 points)
Let n > 1 be an even integer. Show that
√
2008
ϕ(n) ≥ 21 n.
5. (20 points) Let p be an odd prime. Show that there is a
1. (15 points) Find all positive integers n such that 4 ̸ |ϕ(n), primitive root (mod p2 ).
i.e. Find n such that the Euler phi-function ϕ(n) is not 6. (20 points) Let p be a prime and suppose that the condivisible by 4.
gruence equation
Hint: There are infinitely many such integers.
x2 + x + 1 ≡ 0 (mod p)
2. (15 points) Let p > 3 be a prime and let S =
{1, 2, 3, · · · , p − 1}. Find
has a solution. Prove that either p = 3 or p ≡ 1 (mod 3).
∏
7. (4 points each) Write True or False for the following
g (mod p).
statements. There will be −2 points penalty for each of
g∈S,ordp g=p−1
wrong guesses.
(a)
For any
integer n, there exists an odd prime p
That is, find the product of all incongruent primitive roots
( positive
)
n
(mod p).
such that
=1
p
3. (15 points) Is the congruence x2 + x + 3 ≡ 0 (mod 97)
(b) If an odd composite n is a Carmichael number, then n
solvable ? Prove or disprove it.
4. (15 points) Let p be an odd prime and let n ≥ 1 be an is squarefree.
(c) Let p be an odd prime. Then the sum of two quadratic
integer. Show that there is no solution of
residues (mod p) is also a quadratic residue (mod p).
(d) Let n be a positive integer and let a and b be integers
x2 ≡ 1 (mod pn )
such that (ab, n) = 1. Then we have ordn (ab) = ordn a ·
other than x ≡ ±1 (mod pn ).
ordn b.
5. (20 points) Let n > 1 be an odd composite integer and let (e) For given n, factoring n is easier than determining
a be an integer with (a, n) = 1. Show that, if n is a strong whether n is a prime or not.
pseudoprime to base a, then n is also a strong pseudoprime
2010
to base a2 .
6. (20 points) Let p be a prime with p ≡ 3 (mod 7). Show 1. (15 points) Is the congruence x2 ≡ 1001 (mod 2003)
that the equation
solvable or not ?
x2 + 7y 2 = p
(Hint: 2003 is a prime and 1001 is a composite.)
2. (15 points) Let n > 1 be an odd composite integer and
does not have any integral solution.
7. (4 points each) Write True or False for the following let a be an integer with (a, n) = 1. Prove that, if n is
statements. There will be −2 points penalty for each of a strong pseudo-prime to base a, then n is also a strong
pseudo-prime to base ā, where ā is an arithmetic inverse of
wrong guesses.
(a) If n is a Fermat pseudoprime to base a, then it is a a (mod n).
3. (15 points) Find all positive integers n such that ϕ(n) =
strong pseudoprime to base a.
4.
(b) Suppose that one round of Miller-Rabin Primality Test
says that n is a (probabilistic) prime. Then n is always a 4. (15 points) Is there a primitive root (mod 35) ? Prove
or disprove it.
( )
prime.
3
(c) Let m > 1. Then m is a prime if and only if (m − 1)! ≡ 5. (15 points) Let p > 3 be a prime. Prove that
=1
p
−1 (mod m).
( )
if and only if p ≡ ±1 (mod 12).
8
(d) Let p be an odd prime. Then one has
= 1 if and 6. (15 points) Let p be an odd prime and r be an integer
p
with (r, p) = 1. Show that r is a primitive root (mod p)
only if p ≡ ±1 (mod 8).
p−1
(e) Let ab ≡ 1 (mod p) where p is an odd prime. If a is if and only if r q ̸≡ 1 (mod p) for all prime divisors q of
a primitive root (mod p), then b is also a primitive root p − 1.
7. (3 points each) Write True or False for the following
(mod p).
statements. There will be −2 points penalty for each of
2009
wrong guesses.
(b) Let n > 1 be an odd integer. Suppose that one round of
Miller-Rabin Primality Test says that n is composite. Then
n is always a composite integer.
(c) For any m ≥ 2, there exists a primitive root (mod m).
(d) For any n ≥ 1, there are only finitely many k satisfying
ϕ(k) = n.
(e) An odd integer a ≥ 1 is called a Carmichael number if
an−1 ≡ 1 (mod n) for all n ≥ 1 with (a, n) = 1.
(a) Let n = p1 p2 · · · ps be a product of(odd
) primes (not
necessarily distinct). A Jacobi symbol( na) ( is )defined
( as
)
(a)
a
a
a
a product of Legendre symbols n = p1
·
·
·
p2
ps .
(2)
n2 −1
Then one has n = (−1) 8 .
(b) There exist infinitely many n > 1 such that n is a strong
pseudo-prime to base b for all b with (b, n) = 1.
(c) For all integers n > 2, ϕ(n) is even.
(d) Let n > 1 be an odd integer. Suppose that one round of
Miller-Rabin Primality Test says that n is composite. Then
n is always a composite integer.
2011
1. (15 points) Show that if p is an odd prime, then
12 32 · · · (p − 4)2 (p − 2)2 ≡ (−1)
p+1
2
(mod p)
2. (15 points) Let n > 1 and σ(n) be the sum of all positive
divisors of n. Prove that σ(n) is a multiplicative arithmetic
function.
3. (15 points) Let p be an odd prime and let a be a quadratic
non-residue (mod p) with ordp a = d. Then show that
d−1 ( k )
∑
a
k=0
p
=0
4. (20 points) Suppose that n is a strong pseudoprime
to base a, but not a strong pseudoprime to base b, where
(a, n) = (b, n) = 1. Is n a strong pseudoprime to base ab ?
Prove or disprove the assertion.
5. (10 points each)
(1) Find a primitive root (mod 25).
(2) Express all the primitive roots (mod 25) using your
answer of (1). State how many primitive roots (mod 25)
exist.
6. (20 points) Let p be an odd prime such that p = a2 + b2
for
some integers a and b. Let a be an odd integer. Prove
( )
a
= 1.
p