Cool theorems proved by undergraduates
Cool theorems proved by undergraduates
Ken Ono
Emory University
Cool theorems proved by undergraduates
Child’s play...
Cool theorems proved by undergraduates
Child’s play...
Thanks to the
,
Cool theorems proved by undergraduates
Child’s play...
Thanks to the
, undergrads and I play by...
Cool theorems proved by undergraduates
Child’s play...
Thanks to the
, undergrads and I play by......shooting off
Cool theorems proved by undergraduates
Child’s play...
Thanks to the
, undergrads and I play by......shooting off
and getting in trouble
,
Cool theorems proved by undergraduates
Child’s play...
Thanks to the
, undergrads and I play by......shooting off
and getting in trouble
....and by proving theorems in number theory.
,
Cool theorems proved by undergraduates
Child’s play...
Our toys include....
Cool theorems proved by undergraduates
Child’s play...
Our toys include....
(Prime numbers)
Cool theorems proved by undergraduates
Child’s play...
Our toys include....
(Prime numbers)
(Partitions)
Cool theorems proved by undergraduates
Child’s play...
Our toys include....
(Prime numbers)
(Partitions)
(Numbers and Number Fields)
Cool theorems proved by undergraduates
Prime Numbers
Some theorems on primes
Cool theorems proved by undergraduates
Prime Numbers
Some theorems on primes
Theorem (Euclid)
There are infinitely many prime numbers.
Cool theorems proved by undergraduates
Prime Numbers
Some theorems on primes
Theorem (Euclid)
There are infinitely many prime numbers.
Theorem (Prime Number Theorem)
If π(x) := #{p ≤ x : prime}, then
π(x) ∼
x
.
ln x
Cool theorems proved by undergraduates
Prime Numbers
Cool theorems proved by undergraduates
Prime Numbers
Theorem (Dirichlet)
If 0 ≤ r < t are integers for which gcd(r , t) = 1, then let
πr ,t (x) := #{p ≤ x : prime and p ≡ r
(mod t)}.
Cool theorems proved by undergraduates
Prime Numbers
Theorem (Dirichlet)
If 0 ≤ r < t are integers for which gcd(r , t) = 1, then let
πr ,t (x) := #{p ≤ x : prime and p ≡ r
Then in terms of Euler’s phi-function, we have
πr ,t (x) ∼
1
x
·
.
φ(t) ln x
(mod t)}.
Cool theorems proved by undergraduates
Prime Numbers
Theorem (Dirichlet)
If 0 ≤ r < t are integers for which gcd(r , t) = 1, then let
πr ,t (x) := #{p ≤ x : prime and p ≡ r
(mod t)}.
Then in terms of Euler’s phi-function, we have
πr ,t (x) ∼
1
x
·
.
φ(t) ln x
Example (Half of the primes are of the following forms:)
3n + 1, 3n + 2
4n + 1, 4n + 3
6n + 1, 6n + 5
Cool theorems proved by undergraduates
Prime Numbers
Arithmetic progressions of primes
Cool theorems proved by undergraduates
Prime Numbers
Arithmetic progressions of primes
Theorem (van der Corput (1939))
There are ∞ly many length 3 arithmetic progressions of primes.
Cool theorems proved by undergraduates
Prime Numbers
Arithmetic progressions of primes
Theorem (van der Corput (1939))
There are ∞ly many length 3 arithmetic progressions of primes.
Example
Here are examples of arithmetic progressions of length 3:
(3, 3 + 2, 3 + 2 · 2) = (3, 5, 7),
(5, 5 + 42, 5 + 2 · 42) = (5, 47, 89),
(43, 43 + 30, 43 + 2 · 30) = (43, 73, 103).
Cool theorems proved by undergraduates
Prime Numbers
How long can these get?
Cool theorems proved by undergraduates
Prime Numbers
How long can these get?
Theorem (Green-Tao (2000s))
For every k there are ∞ly many length k AP of primes.
Cool theorems proved by undergraduates
Prime Numbers
How long can these get?
Theorem (Green-Tao (2000s))
For every k there are ∞ly many length k AP of primes.
Example
Length 10 example:
199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089.
Cool theorems proved by undergraduates
Prime Numbers
Different problem
Question
Can you find k consecutive primes which end with the digit 1?
Cool theorems proved by undergraduates
Prime Numbers
Different problem
Question
Can you find k consecutive primes which end with the digit 1?
Example
We have the following consecutive primes
181, 191,
241, 251,
..
.
4831, 4861, 4871.
Cool theorems proved by undergraduates
Prime Numbers
A great theorem...
Cool theorems proved by undergraduates
Prime Numbers
A great theorem...
Theorem (Shiu (2000))
Let p1 = 2, p2 = 3, . . . be the primes in order. If gcd(r , t) = 1,
then for every positive integer k there is an n for which
pn ≡ pn+1 ≡ pn+2 ≡ · · · ≡ pn+k ≡ r
(mod t).
Cool theorems proved by undergraduates
Prime Numbers
A cool theorem...
Theorem (Monks, Peluse, Ye)
“Special” sets of primes have arbitrarily long sequences of primes
in any arithmetic progression.
Cool theorems proved by undergraduates
Prime Numbers
A cool theorem...
Theorem (Monks, Peluse, Ye)
“Special” sets of primes have arbitrarily long sequences of primes
in any arithmetic progression.
Corollary
Define the sequence of integers Nπ := {bπc, b2πc, b3πc, . . . }.
Cool theorems proved by undergraduates
Prime Numbers
A cool theorem...
Theorem (Monks, Peluse, Ye)
“Special” sets of primes have arbitrarily long sequences of primes
in any arithmetic progression.
Corollary
Define the sequence of integers Nπ := {bπc, b2πc, b3πc, . . . }.
Let pπ (n) be the nth prime in this sequence, and so
pπ (1) = 3, pπ (2) = 31, pπ (3) = 37, . . . .
Cool theorems proved by undergraduates
Prime Numbers
A cool theorem...
Theorem (Monks, Peluse, Ye)
“Special” sets of primes have arbitrarily long sequences of primes
in any arithmetic progression.
Corollary
Define the sequence of integers Nπ := {bπc, b2πc, b3πc, . . . }.
Let pπ (n) be the nth prime in this sequence, and so
pπ (1) = 3, pπ (2) = 31, pπ (3) = 37, . . . .
Then for every gcd(r , t) = 1 and every k there exists an n for which
Cool theorems proved by undergraduates
Prime Numbers
A cool theorem...
Theorem (Monks, Peluse, Ye)
“Special” sets of primes have arbitrarily long sequences of primes
in any arithmetic progression.
Corollary
Define the sequence of integers Nπ := {bπc, b2πc, b3πc, . . . }.
Let pπ (n) be the nth prime in this sequence, and so
pπ (1) = 3, pπ (2) = 31, pπ (3) = 37, . . . .
Then for every gcd(r , t) = 1 and every k there exists an n for which
pπ (n) ≡ pπ (n + 1) ≡ pπ (n + 2) ≡ pπ (n + k) ≡ r
(mod t).
Cool theorems proved by undergraduates
Prime Numbers
More on strings of prime
Cool theorems proved by undergraduates
Prime Numbers
More on strings of prime
Example
The first 6 consecutive primes ≡ 5 (mod 7) in Nπ is
26402437, 26402507, 26402591, 26402843, 26402899, 26402927.
Cool theorems proved by undergraduates
Prime Numbers
More on strings of prime
Example
The first 6 consecutive primes ≡ 5 (mod 7) in Nπ is
26402437, 26402507, 26402591, 26402843, 26402899, 26402927.
Remark
Special sets includes primes in the sets
Nα := {bαc, b2αc, b3αc, . . . }
α irrational real alg. int.,
N ∗ := {bn log log nc : n = 1, 2, 3, . . . }.
Cool theorems proved by undergraduates
Prime Numbers
Cool theorems proved by undergraduates
Recurrence sequences
The Fibonacci Sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 · · ·
Cool theorems proved by undergraduates
Recurrence sequences
The Fibonacci Sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 · · ·
Cool theorems proved by undergraduates
Recurrence sequences
The Fibonacci Sequence
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377 · · ·
Conjecture (Folklore)
These are the only perfect powers!
Cool theorems proved by undergraduates
Recurrence sequences
Perfect Powers in the Fibonacci Sequence
Theorem (Siksek, Bugeaud, Mignotte, Annals of Math 2006)
F0 = 0p , F1 = 1p , F6 = 23 , and F12 = 122 are the only perfect
powers in the Fibonacci sequence.
Cool theorems proved by undergraduates
Recurrence sequences
Perfect Powers in the Fibonacci Sequence
Theorem (Siksek, Bugeaud, Mignotte, Annals of Math 2006)
F0 = 0p , F1 = 1p , F6 = 23 , and F12 = 122 are the only perfect
powers in the Fibonacci sequence.
Proof.
Modularity of elliptic curves :-)
Cool theorems proved by undergraduates
Recurrence sequences
Lucas Sequences
Definition (Lucas Sequence)
A Lucas sequence un is a nondegenerate integral linear binary
recurrence relation defined by
un+2 = bun+1 + cun
with u0 = 0 and u1 = 1.
Cool theorems proved by undergraduates
Recurrence sequences
Lucas Sequences
Definition (Lucas Sequence)
A Lucas sequence un is a nondegenerate integral linear binary
recurrence relation defined by
un+2 = bun+1 + cun
with u0 = 0 and u1 = 1.
Definition (Companion Sequence)
The companion sequence vn is defined by vn+2 = bvn+1 + cvn with
v0 = 2 and v1 = b.
Cool theorems proved by undergraduates
Recurrence sequences
Lucas Sequences
A Lucas sequence (b, c) has characteristic polynomial and roots
√
b ± b 2 + 4c
2
g (z) = z − bz − c,
α, β =
.
2
Cool theorems proved by undergraduates
Recurrence sequences
Lucas Sequences
A Lucas sequence (b, c) has characteristic polynomial and roots
√
b ± b 2 + 4c
2
g (z) = z − bz − c,
α, β =
.
2
Fact
1. un =
αn −β n
α−β
vn = αn + β n
2. u2k = uk vk
3. (b 2 + 4c)un2 = vn2 − 4(−c)n
Cool theorems proved by undergraduates
Recurrence sequences
Specific Examples
Theorem (Silliman, Vogt)
For the following values of b and c:
(b, c) = (3, −2), (5, −6), (7, −12), (17, −72), (9, −20)
the Lucas sequence un has no nontrivial pth powers, except
u2 = 32 in (9, −20).
Cool theorems proved by undergraduates
Recurrence sequences
General Bound
Conjecture (Frey-Mazur)
Let E , E 0 be two elliptic curves defined over Q. If E [p] ∼
= E 0 [p] for
some p > 17, then E and E 0 are isogenous.
Cool theorems proved by undergraduates
Recurrence sequences
General Bound
Conjecture (Frey-Mazur)
Let E , E 0 be two elliptic curves defined over Q. If E [p] ∼
= E 0 [p] for
some p > 17, then E and E 0 are isogenous.
Theorem (Silliman, Vogt)
Assume the Frey Mazur conjecture. Then a pth power un = y p
satisfies
p ≤ Ψ(b, c).
where Ψ(b, c) is an explicit constant.
Cool theorems proved by undergraduates
Recurrence sequences
More Examples
Example
The sequence (3, 1):
0, 1, 3, 10, 33, 109, 360, 1189, 3927, 12970, 42837, 141481, · · ·
We haven’t found any nontrivial perfect powers . . .
Cool theorems proved by undergraduates
Recurrence sequences
More Examples
Example
The sequence (3, 1):
0, 1, 3, 10, 33, 109, 360, 1189, 3927, 12970, 42837, 141481, · · ·
We haven’t found any nontrivial perfect powers . . .
Theorem (Silliman, Vogt)
Assume the Frey-Mazur Conjecture. There are no nontrivial
powers in the sequences (b, c) = (3, 1), (5, 1), and (7, 1).
Cool theorems proved by undergraduates
Recurrence sequences
Cool theorems proved by undergraduates
Partitions
Beautiful identities
Cool theorems proved by undergraduates
Partitions
Beautiful identities
Euler proved that
q
∞
Y
(1 − q 24n ) = q − q 25 − q 49 + q 121 + q 169 − q 289 · · · .
n=1
Cool theorems proved by undergraduates
Partitions
Beautiful identities
Euler proved that
q
∞
Y
(1 − q 24n ) = q − q 25 − q 49 + q 121 + q 169 − q 289 · · · .
n=1
Jacobi proved that
q
∞
Y
(1 − q 8n )3 = q − 3q 9 + 5q 25 − 7q 49 + 9q 81 − 11q 121 + · · · .
n=1
Cool theorems proved by undergraduates
Partitions
Beautiful identities
Euler proved that
q
∞
Y
(1 − q 24n ) = q − q 25 − q 49 + q 121 + q 169 − q 289 · · · .
n=1
Jacobi proved that
q
∞
Y
(1 − q 8n )3 = q − 3q 9 + 5q 25 − 7q 49 + 9q 81 − 11q 121 + · · · .
n=1
Gauss proved that
q
∞
Y
(1 − q 16n )2
= q+q 9 +q 25 +q 49 +q 81 +q 121 +q 169 +q 225 +· · ·
(1 − q 8n )
n=1
Cool theorems proved by undergraduates
Partitions
Such rare identities have been discovered by:
Cool theorems proved by undergraduates
Partitions
Such rare identities have been discovered by:
Crazy combinatorial manipulation of power series.
Higher identities such as Jacobi’s
∞
Y
(1 − q 2n )(1 + z 2 q 2n−1 )(1 + z −2 q 2n−1 ) =
n=1
Modular forms.
X
m∈Z
2
z 2m q m .
Cool theorems proved by undergraduates
Partitions
Nekrasov-Okounkov Theory
Cool theorems proved by undergraduates
Partitions
Nekrasov-Okounkov Theory
Deeper structure for such identities.
Cool theorems proved by undergraduates
Partitions
Nekrasov-Okounkov Theory
Deeper structure for such identities.
One doesn’t have to multiply out and combine terms.
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Partitions
Definition
A nonincreasing sequence of positive integers summing to n is a
partition of n.
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Partitions
Definition
A nonincreasing sequence of positive integers summing to n is a
partition of n.
p(n) := # {partitions of n} .
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Partitions
Definition
A nonincreasing sequence of positive integers summing to n is a
partition of n.
p(n) := # {partitions of n} .
Example
{Partitions of 4} = {4 , 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1}
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Partitions
Definition
A nonincreasing sequence of positive integers summing to n is a
partition of n.
p(n) := # {partitions of n} .
Example
{Partitions of 4} = {4 , 3 + 1, 2 + 2, 2 + 1 + 1, 1 + 1 + 1 + 1}
=⇒ p(4) = 5.
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Generating function for p(n)
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Generating function for p(n)
Lemma
We have that
∞
X
n=0
n
p(n)q =
∞
Y
n=1
1
.
1 − qn
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Wishful thinking
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Wishful thinking
Question
Is there a “combinatorial theory” of infinite products where
coefficient of q n = “formula in partitions of n”?
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Hooklengths of partitions
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Hooklengths of partitions
Definition
Hooks are the maximal
∗ ∗ ∗···∗
∗
∗
..
.
∗
in the Ferrers board of λ.
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Hooklengths of partitions
Definition
Hooks are the maximal
∗ ∗ ∗···∗
∗
∗
..
.
∗
in the Ferrers board of λ.
Hooklengths are their lengths, and
H(λ) = { multiset of hooklengths of λ}
Cool theorems proved by undergraduates
Nekrasov-Okounkov
An example
Example (λ = 5 + 3 + 2)
7 6 4 2 1
4 3 1
2 1
Cool theorems proved by undergraduates
Nekrasov-Okounkov
An example
Example (λ = 5 + 3 + 2)
7 6 4 2 1
4 3 1
2 1
An so H(λ) = {1, 1, 1, 2, 2, 3, 4, 4, 6, 7}.
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Nekrasov-Okounkov q-series
Definition
For z ∈ C, let
Oz (q) :=
X
λ
q |λ| ·
Y h∈H(λ)
1−
z .
h2
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Example (λ = 5 + 3 + 2)
7 6 4 2 1
4 3 1
2 1
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Example (λ = 5 + 3 + 2)
7 6 4 2 1
4 3 1
2 1
The λ-contribution to Oz (q) is
Y z q 10 ·
1− 2
h
h∈H(λ)
z 2 z
z 2 z z 1−
1−
1−
1−
= q 10 (1 − z)3 1 −
4
9
16
36
49
Cool theorems proved by undergraduates
Nekrasov-Okounkov
A famous identity revisited
Cool theorems proved by undergraduates
Nekrasov-Okounkov
A famous identity revisited
Letting z = 4, we consider
O4 (q) =
X
λ
q
|λ|
·
Y h∈H(λ)
4
1− 2
h
.
Cool theorems proved by undergraduates
Nekrasov-Okounkov
A famous identity revisited
Letting z = 4, we consider
O4 (q) =
X
λ
q
|λ|
·
Y h∈H(λ)
We only need λ where every hook h 6= 2.
4
1− 2
h
.
Cool theorems proved by undergraduates
Nekrasov-Okounkov
A famous identity revisited
Letting z = 4, we consider
O4 (q) =
X
λ
q
|λ|
·
Y h∈H(λ)
4
1− 2
h
.
We only need λ where every hook h 6= 2.
=⇒ {Triangular partitions 1 + 2 + · · · + k}
Cool theorems proved by undergraduates
Nekrasov-Okounkov
A famous identity revisited.
Cool theorems proved by undergraduates
Nekrasov-Okounkov
A famous identity revisited.
λ
0
1
2+1
3+2+1
8|λ| + 1
1
9
25
49
H(λ)
φ
{1}
{1, 1, 3}
{1, . . . , 5}
Q
(1 − 4/h2 )
1
−3
5
−7
Cool theorems proved by undergraduates
Nekrasov-Okounkov
A famous identity revisited.
λ
0
1
2+1
3+2+1
8|λ| + 1
1
9
25
49
H(λ)
φ
{1}
{1, 1, 3}
{1, . . . , 5}
Q
(1 − 4/h2 )
1
−3
5
−7
And so we have
=⇒ qO4 (q 8 ) = q − 3q 9 + 5q 25 − 7q 49 + · · ·
Cool theorems proved by undergraduates
Nekrasov-Okounkov
A famous identity revisited.
λ
0
1
2+1
3+2+1
H(λ)
φ
{1}
{1, 1, 3}
{1, . . . , 5}
8|λ| + 1
1
9
25
49
Q
(1 − 4/h2 )
1
−3
5
−7
And so we have
=⇒ qO4 (q 8 ) = q − 3q 9 + 5q 25 − 7q 49 + · · ·
Jacobi?
=
q
∞
Y
(1 − q 8n )3
n=1
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Big Identity
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Big Identity
Theorem (Nekrasov-Okounkov)
If z is complex, then
Oz (q) :=
X
λ
q |λ| ·
Y h∈H(λ)
1−
∞
z Y
=
(1 − q n )z−1 .
h2
n=1
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Big Identity
Theorem (Nekrasov-Okounkov)
If z is complex, then
Oz (q) :=
X
λ
q |λ| ·
Y h∈H(λ)
1−
∞
z Y
=
(1 − q n )z−1 .
h2
n=1
Remark
Letting z = 0 gives the generating function for p(n).
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Euler, Gauss, and Jacobi-type identities
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Euler, Gauss, and Jacobi-type identities
Question
1 When do the sums below vanish?
A(a, b; n) :=
X
Y
|λ|=n h∈Ha (λ)
ab
1− 2
h
.
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Euler, Gauss, and Jacobi-type identities
Question
1 When do the sums below vanish?
A(a, b; n) :=
X
Y
|λ|=n h∈Ha (λ)
2
ab
1− 2
h
.
Are there more identities of Euler, Gauss and Jacobi-type?
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Euler, Gauss, and Jacobi-type identities
Question
1 When do the sums below vanish?
A(a, b; n) :=
X
Y
|λ|=n h∈Ha (λ)
ab
1− 2
h
.
2
Are there more identities of Euler, Gauss and Jacobi-type?
3
If so, find them all.
Cool theorems proved by undergraduates
Nekrasov-Okounkov
A cool theorem
Cool theorems proved by undergraduates
Nekrasov-Okounkov
A cool theorem
Theorem (Clader, Kemper, Wage)
The list of all pairs (a, b) for which “almost all” the A(a, b; n)
vanish are
(1, 2), (1, 3), (1, 4), (1, 5), (1, 7), (1, 9), (1, 11), (1, 15), (1, 27),
(2, 2), (2, 3), (2, 5), (2, 7), (3, 3), (3, 5), (3, 9), (4, 5), (4, 7),
(7, 9), (7, 15).
Cool theorems proved by undergraduates
Nekrasov-Okounkov
A cool theorem
Theorem (Clader, Kemper, Wage)
The list of all pairs (a, b) for which “almost all” the A(a, b; n)
vanish are
(1, 2), (1, 3), (1, 4), (1, 5), (1, 7), (1, 9), (1, 11), (1, 15), (1, 27),
(2, 2), (2, 3), (2, 5), (2, 7), (3, 3), (3, 5), (3, 9), (4, 5), (4, 7),
(7, 9), (7, 15).
Remark
These pairs are the Euler, Gauss and Jacobi identities for
X
A(a, b; n)q n :=
∞
Y
(1 − q an )b
.
(1 − q n )
n=1
Cool theorems proved by undergraduates
Nekrasov-Okounkov
Cool theorems proved by undergraduates
Number Fields
Number fields
Wrong Definition
This is a number field.
Cool theorems proved by undergraduates
Number Fields
Number fields
Wrong Definition
This is a number field.
.
Cool theorems proved by undergraduates
Number Fields
Number fields
Wrong Definition
This is a number field.
.
Definition
A finite dimensional field extension of Q is called a number field.
Cool theorems proved by undergraduates
Number Fields
An invariant
Remark
The discriminant ∆K ∈ Z \ {0} of a number field K does:
Cool theorems proved by undergraduates
Number Fields
An invariant
Remark
The discriminant ∆K ∈ Z \ {0} of a number field K does:
“Measures” the volume of algebraic integers.
Cool theorems proved by undergraduates
Number Fields
An invariant
Remark
The discriminant ∆K ∈ Z \ {0} of a number field K does:
“Measures” the volume of algebraic integers.
Controls some properties of primes (ramification).
Cool theorems proved by undergraduates
Number Fields
Quadratic fields
√
If D is square-free and K := Q( D), then
(
D
if D ≡ 1 (mod 4),
∆K :=
4D
otherwise.
Cool theorems proved by undergraduates
Number Fields
Quadratic fields
√
If D is square-free and K := Q( D), then
(
D
if D ≡ 1 (mod 4),
∆K :=
4D
otherwise.
Example
For example, we have
Q(i)
=⇒
∆ = −4
√
Q( 2)
=⇒
∆ = 8.
Cool theorems proved by undergraduates
Number Fields
Distribution of quadratic fields
Notation. Let N2 (X ) := #{quad fields with |∆| ≤ X }.
Cool theorems proved by undergraduates
Number Fields
Distribution of quadratic fields
Notation. Let N2 (X ) := #{quad fields with |∆| ≤ X }.
X
102
104
106
N2 (X )
61
6086
607925
N2 (X )/X
0.6100 . . .
0.6086 . . .
0.6079 . . .
Cool theorems proved by undergraduates
Number Fields
Distribution of quadratic fields
Notation. Let N2 (X ) := #{quad fields with |∆| ≤ X }.
X
102
104
106
N2 (X )
61
6086
607925
N2 (X )/X
0.6100 . . .
0.6086 . . .
0.6079 . . .
Theorem (Easy)
We have that
N2 (X )
6
= 2 = 0.607927 . . . .
X →+∞
X
π
lim
Cool theorems proved by undergraduates
Number Fields
General Number Fields
Conjecture (Linnik)
If we let Nn (X ) := #{degree n fields with |∆| ≤ X }, then
Nn (X ) ∼ cn · X .
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Number Fields
General Number Fields
Conjecture (Linnik)
If we let Nn (X ) := #{degree n fields with |∆| ≤ X }, then
Nn (X ) ∼ cn · X .
Theorem (Easy, Davenport-Heilbronn (70s), Bhargava (2000s))
Linnik’s Conjecture is true for n = 2, 3, 4, 5.
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Number Fields
General results
Cool theorems proved by undergraduates
Number Fields
General results
Theorem (Schmidt (1995), Ellenberg-Venkatesh (2006))
We have that
Nn (X ) n
 n+2

X 4
if 6 ≤ n ≤ 84393,

 exp(C √log n)
X
if n > 84393.
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Number Fields
Related Functions
Notation. If G is a finite group, then let
Nn (G ; X ) :=
number of deg n fields with |∆| ≤ X
whose Galois closure has Galois group G
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Number Fields
Related Functions
Notation. If G is a finite group, then let
Nn (G ; X ) :=
number of deg n fields with |∆| ≤ X
whose Galois closure has Galois group G
Remark
Estimating Nn (G ; X ) is easy only for n = 2 and G = Z/2Z.
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Number Fields
Prime cyclic cases
Theorem (Wright (1989))
If p is an odd prime, then there is a constant c(p) for which
1
Np (Z/pZ; X ) = c(p) · X p−1 + O(X 1/p ).
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Number Fields
Prime cyclic cases
Theorem (Wright (1989))
If p is an odd prime, then there is a constant c(p) for which
1
Np (Z/pZ; X ) = c(p) · X p−1 + O(X 1/p ).
Conjecture (Cohen, Diaz y Diaz, Olivier (2006))
We have that
1
N3 (Z/3Z; X ) = c(3) · X 2 + O(X 1/6 ).
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Number Fields
A cool theorem...
Cool theorems proved by undergraduates
Number Fields
A cool theorem...
Theorem (Lee-Oh)
Assuming the GRH, if p is an odd prime, then
1
1
1
Np (Z/pZ; X ) = c(p) · X p−1 + X 3(p−1) · Rp (log X ) + O(X 4(p−1) ),
where Rp (x) is a polynomial of degree bp(p − 2)/3c − 1.
Cool theorems proved by undergraduates
Number Fields
A cool theorem...
Theorem (Lee-Oh)
Assuming the GRH, if p is an odd prime, then
1
1
1
Np (Z/pZ; X ) = c(p) · X p−1 + X 3(p−1) · Rp (log X ) + O(X 4(p−1) ),
where Rp (x) is a polynomial of degree bp(p − 2)/3c − 1.
Corollary (Lee-Oh)
Assuming GRH, the Cohen, Diaz y Diaz, Olivier Conjecture is true.
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Number Fields
Linear regression when p = 3: Slope ∼
1
6
Cool theorems proved by undergraduates
Number Fields
Linear regression when p = 3: Slope ∼
1
6
3
2
1
5
10
15
20
25
-1
-2
x − axis = log10 (X )
1
y − axis = log10 N3 (Z/3Z; X ) − c(3) · X 2
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In conclusion...
Our summer toys include....
Cool theorems proved by undergraduates
In conclusion...
Our summer toys include....
(Prime numbers)
(Partitions)
(Numbers and Number Fields)
Cool theorems proved by undergraduates
In conclusion...
So, we basically....
Cool theorems proved by undergraduates
In conclusion...
So, we basically....
....prove theorems, and...explode stuff...