Cool theorems proved by undergraduates Ken Ono Emory University - - PowerPoint PPT Presentation

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 (mod t)}. Then in terms of Euler’s phi-function, we have πr,t(x) ∼ 1 φ(t) · x ln x .

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 φ(t) · x 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 := {⌊π⌋, ⌊2π⌋, ⌊3π⌋, . . . }.

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 := {⌊π⌋, ⌊2π⌋, ⌊3π⌋, . . . }. 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 := {⌊π⌋, ⌊2π⌋, ⌊3π⌋, . . . }. 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 := {⌊π⌋, ⌊2π⌋, ⌊3π⌋, . . . }. 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 := {⌊α⌋, ⌊2α⌋, ⌊3α⌋, . . . } α irrational real alg. int., N∗ := {⌊n log log n⌋ : 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 g(z) = z2 − bz − c, α, β = b ± √ b2 + 4c 2 .

Cool theorems proved by undergraduates Recurrence sequences

Lucas Sequences

A Lucas sequence (b, c) has characteristic polynomial and roots g(z) = z2 − bz − c, α, β = b ± √ b2 + 4c 2 . Fact

• 1. un = αn−βn

α−β

vn = αn + βn

• 2. u2k = ukvk
• 3. (b2 + 4c)u2

n = v2 n − 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 ′ be two elliptic curves defined over Q. If E[p] ∼ = E ′[p] for some p > 17, then E and E ′ are isogenous.

Cool theorems proved by undergraduates Recurrence sequences

General Bound

Conjecture (Frey-Mazur) Let E, E ′ be two elliptic curves defined over Q. If E[p] ∼ = E ′[p] for some p > 17, then E and E ′ are isogenous. Theorem (Silliman, Vogt) Assume the Frey Mazur conjecture. Then a pth power un = yp 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

∞

• n=1

(1 − q24n) = q − q25 − q49 + q121 + q169 − q289 · · · .

Cool theorems proved by undergraduates Partitions

Beautiful identities

Euler proved that q

∞

• n=1

(1 − q24n) = q − q25 − q49 + q121 + q169 − q289 · · · . Jacobi proved that q

∞

• n=1

(1 − q8n)3 = q − 3q9 + 5q25 − 7q49 + 9q81 − 11q121 + · · · .

Cool theorems proved by undergraduates Partitions

Beautiful identities

Euler proved that q

∞

• n=1

(1 − q24n) = q − q25 − q49 + q121 + q169 − q289 · · · . Jacobi proved that q

∞

• n=1

(1 − q8n)3 = q − 3q9 + 5q25 − 7q49 + 9q81 − 11q121 + · · · . Gauss proved that q

∞

• n=1

(1 − q16n)2 (1 − q8n) = q+q9+q25+q49+q81+q121+q169+q225+· · ·

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

∞

• n=1

(1 − q2n)(1 + z2q2n−1)(1 + z−2q2n−1) =

• m∈Z

z2mqm2. Modular forms.

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

∞

• n=0

p(n)qn =

∞

• 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 qn = “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) :=

• λ

q|λ| ·

• 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 q10 ·

• h∈H(λ)
• 1 − z

h2

• = q10(1 − z)3

1 − z 4 2 1 − z 9 1 − z 16 2 1 − z 36 1 − z 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) =

• λ

q|λ| ·

• h∈H(λ)
• 1 − 4

h2

• .

Cool theorems proved by undergraduates Nekrasov-Okounkov

A famous identity revisited

Letting z = 4, we consider O4(q) =

• λ

q|λ| ·

• h∈H(λ)
• 1 − 4

h2

• .

We only need λ where every hook h = 2.

Cool theorems proved by undergraduates Nekrasov-Okounkov

A famous identity revisited

Letting z = 4, we consider O4(q) =

• λ

q|λ| ·

• h∈H(λ)
• 1 − 4

h2

• .

We only need λ where every hook h = 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 8|λ| + 1 H(λ) (1 − 4/h2) 1 φ 1 1 9 {1} −3 2 + 1 25 {1, 1, 3} 5 3 + 2 + 1 49 {1, . . . , 5} −7

Cool theorems proved by undergraduates Nekrasov-Okounkov

A famous identity revisited.

λ 8|λ| + 1 H(λ) (1 − 4/h2) 1 φ 1 1 9 {1} −3 2 + 1 25 {1, 1, 3} 5 3 + 2 + 1 49 {1, . . . , 5} −7 And so we have = ⇒ qO4(q8) = q − 3q9 + 5q25 − 7q49 + · · ·

Cool theorems proved by undergraduates Nekrasov-Okounkov

A famous identity revisited.

λ 8|λ| + 1 H(λ) (1 − 4/h2) 1 φ 1 1 9 {1} −3 2 + 1 25 {1, 1, 3} 5 3 + 2 + 1 49 {1, . . . , 5} −7 And so we have = ⇒ qO4(q8) = q − 3q9 + 5q25 − 7q49 + · · ·

Jacobi?

= q

∞

• n=1

(1 − q8n)3

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) :=

• λ

q|λ| ·

• h∈H(λ)
• 1 − z

h2

• =

∞

• n=1

(1 − qn)z−1.

Cool theorems proved by undergraduates Nekrasov-Okounkov

Big Identity

Theorem (Nekrasov-Okounkov) If z is complex, then Oz(q) :=

• λ

q|λ| ·

• h∈H(λ)
• 1 − z

h2

• =

∞

• n=1

(1 − qn)z−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) :=

• |λ|=n
• h∈Ha(λ)
• 1 − ab

h2

• .

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) :=

• |λ|=n
• h∈Ha(λ)
• 1 − ab

h2

• .

2 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) :=

• |λ|=n
• h∈Ha(λ)
• 1 − ab

h2

• .

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

• A(a, b; n)qn :=

∞

• n=1

(1 − qan)b (1 − qn) .

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 ∆K :=

• D

if D ≡ 1 (mod 4), 4D

• therwise.

Cool theorems proved by undergraduates Number Fields

Quadratic fields

If D is square-free and K := Q( √ D), then ∆K :=

• D

if D ≡ 1 (mod 4), 4D

• therwise.

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 N2(X) N2(X)/X 102 61 0.6100 . . . 104 6086 0.6086 . . . 106 607925 0.6079 . . .

Cool theorems proved by undergraduates Number Fields

Distribution of quadratic fields

• Notation. Let N2(X) := #{quad fields with |∆| ≤ X}.

X N2(X) N2(X)/X 102 61 0.6100 . . . 104 6086 0.6086 . . . 106 607925 0.6079 . . . Theorem (Easy) We have that lim

X→+∞

N2(X) X = 6 π2 = 0.607927 . . . .

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.

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. Theorem (Easy, Davenport-Heilbronn (70s), Bhargava (2000s)) Linnik’s Conjecture is true for n = 2, 3, 4, 5.

Cool theorems proved by undergraduates 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      X

n+2 4

if 6 ≤ n ≤ 84393, X exp(C√log n) if n > 84393.

Cool theorems proved by undergraduates 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

Cool theorems proved by undergraduates 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.

Cool theorems proved by undergraduates Number Fields

Prime cyclic cases

Theorem (Wright (1989)) If p is an odd prime, then there is a constant c(p) for which Np(Z/pZ; X) = c(p) · X

1 p−1 + O(X 1/p).

Cool theorems proved by undergraduates Number Fields

Prime cyclic cases

Theorem (Wright (1989)) If p is an odd prime, then there is a constant c(p) for which Np(Z/pZ; X) = c(p) · X

1 p−1 + O(X 1/p).

Conjecture (Cohen, Diaz y Diaz, Olivier (2006)) We have that N3(Z/3Z; X) = c(3) · X

1 2 + O(X 1/6).

Cool theorems proved by undergraduates 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 Np(Z/pZ; X) = c(p) · X

1 p−1 + X 1 3(p−1) · Rp(log X) + O(X 1 4(p−1) ),

where Rp(x) is a polynomial of degree ⌊p(p − 2)/3⌋ − 1.

Cool theorems proved by undergraduates Number Fields

A cool theorem...

Theorem (Lee-Oh) Assuming the GRH, if p is an odd prime, then Np(Z/pZ; X) = c(p) · X

1 p−1 + X 1 3(p−1) · Rp(log X) + O(X 1 4(p−1) ),

where Rp(x) is a polynomial of degree ⌊p(p − 2)/3⌋ − 1. Corollary (Lee-Oh) Assuming GRH, the Cohen, Diaz y Diaz, Olivier Conjecture is true.

Cool theorems proved by undergraduates 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

5 10 15 20 25 2 1 1 2 3

x − axis = log10(X) y − axis = log10

• N3(Z/3Z; X) − c(3) · X

1 2

Cool theorems proved by undergraduates 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...