Number Theory Flashcards

1
Q

Composite

A

An integer >= 2 that is not prime

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2
Q

Highest common factor

A

The largest integer which is a factor of both a and b is the hcf(a,b)

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3
Q

Coprime

A

Two integers a,b are congruent modulo n Id a-b is a multiple of n

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4
Q

Invertible

A
  • An element a in Z/n is invertible modulo n if there exists b in Z/n s.t. ab=1modn.
  • The set of invertible elements in Z/n is written(Z/n)^x.
  • This is a group with operation of multiplication.
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5
Q

Euler-Totient Function

A

The number of invertible elements in Z/n, ie. The order of the group (Z/n)^x

The number of positive integers less than n, coprime to n

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6
Q

Primitive Root (modp)

A

An element a ∈ F^xp is a primitive root modulo p if a generates the multiplicative group F^xp ie. Every element of F^xp is a power of a

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7
Q

Prime Number

A

An integer p>=2 is a prime if the only factors of p are ±1 and ±p

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8
Q

Chinese Remainder Theorem

A

Let n,m be coprime.

Given a ∈ Z/n and b ∈Z/m

There is a unique x ∈ Z/mm s.t. X = a modn and X = b modm

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9
Q

CRT proof

A

Existence: since n and m are coprime, by Euclid’s algorithm there is a h,k s.t. 1=hn + LM
Let x=hnb + kma

We can check X=a(n) X=b(m)
a = hnb + kma (n)
=(1-hn)a = 1 x a = a(n)

Similarly b(m)

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10
Q

Fermat’s Little Theorem

A

A ∈Fp^x and this is a group with p-1 elements
Let n be the order of a in this group IR. The smallest natural number s.t. a^n = 1 modp

Proof:
By Lagrange’s theorem: n divides the order of a which is p-1, so n is a factor of p-1
Therefore p-1 =nm for some m∈Z
Then a^p-1 = a^nm =(a^n)^m = 1^m = 1 modp

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11
Q

Berzout’s Lemma

A

Given a, b ∈Z, and a, b not zero, Ε h,k ∈ Z s.t ha + kb = hcf (a, b)

  • use Euclid’s algorithm to find h and k
  • to find a^-1 : find ha + kn = 1 by euclids then h is a^-1 mod n
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12
Q

There are infinitely many prime numbers

A

Assume that there are only finitely many primes; p1,p2,…..,pn
Let N = p1•p2…•pn +1
Since N >= 2 there is at least one prime which divides N
So pi | N ie. N = 0 (pi)
But clearly N = 1 modpi
Therefore contradiction!
And N doesn’t have a prime factor and must be a prime

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13
Q

Unique factorisation of integers

A

For any positive integer n, there is a factorisation n=p1….pr
This factorisation is unique up to reordering the primes pi….pr

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14
Q

Euler’s Theorem

A

If a is invertible mod(n), ie. a ∈(Z/n)^x then a^@(n) =1 mod(n)

Proof:
Let a ∈(Z/n)^x and let d be the order of a in this group, ie. Smallest d > 0 st. a^d = 1 (n)

By Lagrange’s theorem, d|@(n) ie. @(n) = d x e
a^d = 1 (n) therefore a^@(n) = a^(de) = (a^d)^e = 1^e = 1 (n)

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15
Q

Eratostene’s Sieve

A

Test for n primes

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16
Q

Testing for primitive roots

A

Let p be a prime a ∈Fˣp
To test if α is a quadratic residue mod p:
1. Find the primes q | p-1
2. For each q, calculate a^(p-1)/q mod p
If none of these are 1 modp ⇒ a is a primitive root

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17
Q

Cyclic group

A

A group G is cyclic if ∃ an element g∈G s.t. G = {g^i : i∈ℤ}

18
Q

Generator

A

The element g∈G is called the generator of G

19
Q

Primitive Root

A

A primitive root modulo p is a generator of Fp ( an element of Fo order p-1)

20
Q

Gauss’ Theorem

A

For every prime number p, ∃ a primitive root modulo p. There are φ(p-1) of them.
If P is prime, then Fp is a cyclic group.

21
Q

Quadratic Reciprocity

A

Let p be an odd prime number and a∈Fˣp coprime to p

a is a quadratic residue of x² = a mod p has solutions

22
Q

Legendre Symbol

A

(a/p)

=1 is a is a quadratic residue
=-1 if a is a quadratic non-residue

=0 if p|a

23
Q

Quadratic reciprocity law

A

If p and q are distinct odd primes, then (p/q) = (-1)^((p-1)(q-1)/4)(q/p)

24
Q

First Nebensatz

A

If p=1 mod4 then (-1/p) = 1 and if p =-1 mod p then (-1/p) =(-1)^(p-1)/2

25
Second Nebensatz
If p = ±1 mod 8 then (2/p)=1 | If p= ±3 mod 8 then (2/p)=(-1)^((p^2 -1)/8)
26
Gauss Sum
Let p be an odd prime. The Guass Sun G(p) is: sum from p=1 to 0-1 of (a/p) e^2πi/p
27
Valuation Valuation of x at p
For a non-zero integer n and prime p: Vp(n) = a Where a is the largest power s.t. P^a is a factor of n Vp(0) = ∞ Vp(n/m) = vp(n) - vp(m)
28
Hensel's Lemma
Let p be a prime number and let f∈ℤp[x] be a polynomial | Suppose there is an a₀∈ℤp[x] St. f(a₀) = 0 modp^2c+1 where c = Vp(f'(a₀))
29
P-adic
Congruence modulo p^n where p is a prime number and n is large
30
Proposition
Let p be an odd prime and let a be an integer coprime to p If x²≡a (p) has solutions mod p then it has solutions mod p^n for all n
31
Proposition
Le a be an odd number. If the congruence x₂≡a (8) has solutions mod 8, then it has solutions mod(2^n) for all n>0 True iff. a≡1mod8
32
Teichmuller lift
The teichmuller lift of a to ,ℤ/pn is defined to be T(a) = a^(p^n-1) ∈ℤ/pn
33
If T(x) = a modp^n then T(x) ≡ a^p modp^n+1
To calculate T(x) mod p^n | First calculate T(x) modp^n-1 and then raise it to the power p
34
Every element of (ℤ/p^n)^x can be written uniquely:
T(x) exp(py) Where x∈Fp and y ∈ℤ/p^n-1 Py = log(aT⁻¹(x)) a=T(x)exp(py)
35
ℤp
Ring of p-adic integers The set of all p-adic integers
36
P-Adic integer
Let p be any prime; | p-adic integer means a p-adically convergent series ∑ai where Vp(ai) →∞
37
P-adic integer
A p-adic integer is a p-adically convergent series - 2 p-Adic integers are equal if they are congruent mod p^n for every n -the set of p-adic integers ℤp is a ring since sums and products of p-adically convergent series are also p-adically convergent series
38
Square-free
An integer d is caller square-free if the only factor of d which is a square is 1
39
Unit / invertible elements in a ring
A unit is an element in a ring with an inverse | Ie. U in a ring R, is a unit of there exists an element U⁻¹ in R St. UU⁻¹ =1
40
Irreducible
An element P in R is irreducible if P is not a unit and for all factorisations; P =AB and either A is a unit or B is a unit
41
Norm-Euclidean Quadratic Rings
A quadratic ring ℤ[α] is norm Euclidean if: For every A, B ∈ℤ[α] with B ≠ 0 ∃ Q, R ∈ℤ[α st: 1. A = QB + R 2. |N(R)|
42
Irreducible element
An element P ∈ℤ[α] is called irreducible if: 1. P is not a unit 2. If P =AB with A, B ∈ℤ[α] then either A or B is a unit 2. Says that the only factors of P are units and unit multiples of P