UMASS AMHERST MATH 471, F. HAJIR
HOMEWORK 5: IDEALS AND CONGRUENCES I
PART UN: Ideals in G
Before we get to the problems, we need a definition.
Definition. A suset I ⊆ G is called an ideal of G if it satisfies the following properties:
1a) 0 ∈ I
1b) If α ∈ I, then −α ∈ I
1c) If α, β ∈ I, then α + β ∈ I.
2) If α ∈ I and λ ∈ G, then αλ ∈ I.
Loosely speaking, when it comes to addition and subtraction, I is closed within itself, but
when it comes to multiplication, I is closed in a very strong way: if any element of G gets
multiplied by an element of I, it gets sucked into I.
1. Prove that if I, J are ideals of G, then I + J and I ∩ J are also ideals of G. Here, I + J
is defined to be I + J = {γ + λ|γ ∈ I, λ ∈ J}.
2. Prove that if γ ∈ G, then (γ) = γG is an ideal of G, called “the principal ideal generated
by γ.”
3. Prove that if I is an ideal of G, then there exists a γ ∈ G such that I = (γ), thus every
ideal of G is principal.
4. Use 2 and 3 to prove that if α, β ∈ G, then there exist γ, λ ∈ G such that (α, β) = (γ)
and (α) ∩ (β) = (λ). We call such γ, λ a gcd and an lcm of α, β, respectively.
5. Prove that if α = π1m1 . . . πrmr and β = 0 π1n1 . . . , πrnr with integers nj ≥ 0 for j =
1, · · · , r, are factorizations of α, β into distinct Gaussian primes, then
r
r
Y
Y
min(mj ,nj )
max(mj ,nj )
πj
,
λ=
πj
γ=
j=1
j=1
are, respectively, gcd and lcm for α, β.
6. Prove that if γ is a gcd for α, β, and λ is an lcm for α, β, then γλ is an associate of αβ.
Thus if you can find a gcd or lcm, then you can find the other immediately.
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PART DEUX: Congruences
1. True or False:
(a) 2 ≡ −5 mod 7
(b) 55 ≡ 73 mod 11
(c) 1111 ≡ 11 mod 11
2. For (a) and (b), compute the smallest non-negative integer in the following residue
classes. For (c), reinterpret (b).
(a) 2283 mod 17
(b) 92006 mod 100
(c) What are the last (i.e. right-most) two digits of 92006 ?
3. Prove or disprove: a ≡ b mod m implies that a2 ≡ b2 mod m2 .
4. Is 32n+5 + 24n+1 divisible by 7 for all n ≥ 1? Prove it.
5. Is the sum of three consecutive cubes always divisible by 9? (i.e. is f (x) = x3 +
(x + 1)3 + (x + 2)3 ≡ 0 mod 9 for all integers x ≥ 1)? To investigate this, make a table for
x = 1, 2, 3, 4, 5, 6, 7, 8, 9.
6. The answer to the previous question is “Yes.” Now let’s show that the calculation we
did is already enough to prove it.
(a) Show that if x ≡ y mod 9, then f (x) ≡ f (y) mod 9.
(b) Show that 5 and 6(a) imply that the sum of three consecutive cubes is always a multiple
of 9.
7. Suppose the integer n has the base ten representation am am−1 · · · a1 a0 , i.e.
n = am 10m + am−1 10m−1 + · · · + 10a1 + a0 .
(a) Show that n ≡ a0 − a1 + a2 − · · · + (−1)m am mod 11;
(b) Show that 11|n if and only if
a0 − a1 + a2 − · · · + (−1)m am ≡ 0 mod 11.
(c) Use (b) to check that 32323271626242003 is divisible by 11;
(d) Use (a) to check the fact the “all-ones-number” 11111 · · · 111 is congruent either to 1
or 0 modulo 11 depending on whether there are oddly many ones or evenly many ones.
8. Prove that if a ≡ b mod m, then gcd(a, m) = gcd(b, m).
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Extra Credit Problems
A. Show that if p is a fixed prime not dividing 10 (i.e. p is neither 2 nor 5), then infinitely
many of the “all-ones-numbers” 1, 11, 111, 1111, . . . are divisible by p.
B. Characterize the odd primes p such that the congruence
x2 ≡ −1
(mod p)
has a solution. Prove your answer.
C. (This is quite hard) Prove the Bertrand Postulate to the effect that for each integer
n ≥ 1, there is a prime in the range n < p ≤ 2n. In other words, prove the validity of the
well-known rap song (with thanks to Patrick Bolland):
Bertrand The Man
Ooh Ah Ah Ooh Ahah Ah
Ooh Ah Ah Ooh Ahah Ah
Bertrand said it
and I’ll say it again
There’s always a prime
Between n and 2n
Ooh Ah Ah Ooh Ahah Ah
Ooh Ah Ah Ooh Ahah Ah
D. For an integer n ≥ 1 and a prime p in the range n < p ≤ 2n, determine
2n
vp (
).
n
E. Prove that for n > 1, the truncated harmonic series
1 1
1
hn := 1 + + + · · · +
2 3
n
is not an integer.
Hint: One method is to consider v2 (hn ). Another is to use Bertrand.
F. With hn as above, define
γ := lim hn − log n.
n→∞
Prove that this limit exists. We call γ the Euler or Euler-Mascheroni constant; it’s a real
number – find an estimate for it that you are happy with. We think that γ 6∈ Q. No one
today knows how to prove that. See what you can find out about it. Can you show that if
γ = a/b with positive integers a, b, then b > 10? How about b > 100? What’s the biggest
such bound for b that you could produce do you think? It is believed that no polynomial
with integer coefficients could have γ as a root. Can we currently prove this even for degree
1 polynomials? Why not?
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