Number Theory

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Ring
A set R of complex numbers is a ring if
1. x+y,x-y,xy are in R for all x,y in R.
2. 0,1 in R
Factor in a ring
Let a,b be elements of a ring R. We say a is a factor of b in R if there exists a z in R such that b=az.
Unit
An element u of a ring R is a unit if u is a factor of 1 in R. i.e. if 1/u is in R.
Units of Z[i]
1,-1,i,-i
Associate
Let x be an element of a ring R. The associates of x are the elements of R of the form ux where u is a unit in R.
Irreducible
A non-zero, non-unit element x of R is irreducible if the only factors of x in R are the units of R and the associates of x in R.
The norm of a
The norm of a N(a)=aa where a is the complex conjugate of a.
N:Z[i]->Z
Euclidean Ring
A ring R of complex numbers is a Euclidean Ring (ER) if
1. |a|^2 is in R for every a in R.
2. Given a fraction of the form
z=a/d where a,d in R and d not equal to 0, there exists a q in R such that |z-q|^2<1.
Highest common factor
Let a,b be in R. A highest common factor (HCF) of a and b is a H in R such that
1. H|a and H|b
2. If there exists a x such that x|a and x|b then x|H
Prime
A non zero non unit element p in a ring R is prime if for all a,b in R, p|ab implies that p|a or p|b
Unique Factorization Domain
A ring R is a Unique Factorization Domain (UFD) if for every non zero non unit a in R,
1. There exist irreducibles p1,p2,....,pn in R such that a=p1p2...pn, and
2. If there are irreducibles q1,q2,...,qn in R such that a=q1q2...ql then l=n and there exists a one-to-one correspondence between the ps and the qs so that corresponding numbers are associate.
Order of a modulo p
Let p be a prime and a an integer with a not congruent to 0 modulo p. The order of a modulo p is the smallest natural number d such that
a^d is congruent to 1 modulo p.
Primitive Root
Let p be a prime. A primitive root modulo p is an integer z with order p-1 modulo p.
f(X) is congruent to g(X) modulo n
Let f(X) and g(X) be polynomials with integer coefficients and let n be a natural number. Then we say that f(X) is congruent to g(X) modulo n if the coefficients of each power of X are congruent modulo n.
Degree modulo n of f(X)
Let n be a natural number and f(X) be a polynomial with integer coefficients. Then the degree modulo n of f(X) is the largest d such that the coefficient of X^d is not congruent to 0 modulo n.
Lagrange's Theorem
Let p be a prime and f(X) be a polynomial with integer coefficients and d the degree modulo p with d >= 0. Then the equation
f(X) congruent to 0 modulo p
has at most d distinct solutions modulo p.
Quadratic residue modulo p
Let a be an integer and p a prime where a is NOT congruent to 0 modulo p. Then a is a quadratic residue modulo p if there exists an integer x such that
x^2 is congruent to a modulo p.
If there doesn't exist such an x then a is a quadratic non-residue.
Legendre symbol (a/p)
(a/p)=
1 if a is a quadratic residue
-1 if a is a quadratic nonresidue
0 if a is congruent to 0 modulo p
Euler's Criterion
Let a be an integer and p be an ODD prime. Then
(a/p) is congruent to a^[1/2(p-1)] modulo p.
Least Residue modulo p of a
Let a be an integer and let p be an odd prime. The least residue modulo p of a is the unique integer b such that
1. -1/2(p-1) =< b =< 1/2(p-1) and
2. b is congruent to a modulo p
Gauss' Lemma
Let p be an odd prime and a be an integer with a not congruent to 0 mod p. Let u be the number of numbers
1a, 2a, ..., 1/2(p-1)a
that have a negative least residue. Then
(a/p)=(-1)^u
(2/p)
(2/p)=
1 if p is congruent to +-1 modulo 8
-1 if p is congruent to +-3 modulo 8
Gauss' Law of Quadratic Reciprocity
Let p,q be distinct odd primes and integers. Then
(p/q)=(-1)^[1/2(p-1) * 1/2(q-1)].
i.e. (p/q) = (q/p) unless p is congruent to q is congruent to 3 modulo 4, in which case
(p/q)=-(q/p).
List of irreducible Gaussian integers
1. 1+i
2. prime numbers p with p congruent to 3 modulo 4
3. the irred. factors of the prime numbers p where p congruent to 1 modulo 4.
4. all the associates of the above
Wallis' Equation
x^4 + 9 = y^2
Coprime in a UFD
Two elements of a UFD R are coprime if their only common factors are units.
Theorem: The only integer solutions to
y^3=x^2+2 are (x,y)=(+-5,3)
Steps of Proof:
Step 1. Prove y odd (consider congruences modulo 4)
Step 2. x+sqrt{-2} and x-sqrt{-2} coprime (let a be a common factor- factor of difference)
Step 3. Show x+sqrt{-2} a cube power
(x+sqrt{-2} = un^3)
Step 4. (x,y)=(+-5,3)
ord_p(a)
Let a be a non zero element of R, a UFD, and p be an irreducible element in R. Then ord_p(a) is defined as the number of associates of p that appear in the factorization of a into irreducibles.
ord_p(a) where a is a unit
0
Modular arithmetic in a ring definition
Let a,b be in R, a ring, and n be a non zero element of R. Then we say a is congruent to b modulo n if n is a factor of (a-b) in R.
Complete set of residues in a ring
A complete set of residues modulo n is a subset S of a ring R such that every element of R is congruent modulo n to exactly one element of S.
Fermat's little theorem in Gaussian integers
Let p be a prime number and a be in Z[i]. Then
a^p congruent to
->a mod p if p congruent to +-1 mod 4
->a* mod p if p congruent to +-3 mod 4
Fermat's little theorem in R=Z[sqrt3]
Let p be a prime number and a be in Z[sqrt3].
Then a^p congruent to
->a mod p if p congruent to +-1 mod 12
->a* mod p if p congruent to +- 5 mod 12
nth Mersenne number
Mn= 2^n -1
Theorem (Lucas Test)
Define a sequence ri by r0=4 and ri+1=(ri)^2-2.
Let n>=3 be a prime.
Mn is prime iff r(n-2) is congruent to 0 mod Mn.
aka Mn is a factor of r(n-2)
Order in a ring
Let R be a ring, a be a non zero element of R and n be in R. Then the order of a modulo n in R is the smallest natural number d such that
a^d is congruent to 1 modulo n.
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