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| Order of an Integer (mod m) |
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Definition
The least positive integer x that satisfies
a^x== 1 (mod m)
denoted ordm(a) |
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Definition
When (a,n)=1 and ordn(a) = phi(n).
*reminder: Phi(n)= # of positive integers |
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| A Quadratic Residue (mod m) |
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Definition
| A solution to x^2 == a (mod m) where (a,m)=1. |
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| A Quadratic Non-Residue (mod m) |
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| No solutions exist to x^2 == a (mod m) where (a,m)=1. |
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| A point in the plane whose coordinates are integers. |
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| A "World Champion Approximating Rational Number: that approximates an irrational number. Denoted Cn=Pn/Qn |
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| A Simple Continued Fraction |
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Definition
| A continued fraction expansion whose numerators are all 1's. |
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Definition
| Roots of integral polynomials are algebraic numbers. |
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| The Degree of an Algebraic Number |
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| The smallest degree of polynomial for which the algebraic number is a root. |
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Definition
| A number that is not an algebraic number, meaning it is not a root of an integral polynomial. |
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Definition
| Given a prime p and an integer a, (a,p)=1 then (a/p) = 1 if a is a quadratic residue and (a/p) = -1 if a is a quadratic non-residue. |
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Definition
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Definition
For all x>=8,
ln(2)/4 < pi(x)ln(x)/x <30ln(2) |
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Definition
| If p is an odd prime and a is a positive integer satisfying (a,p)=1, then (a/p)==a^((p-1)/2) (mod p) |
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| The Law of Quadratic Reciprocity |
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Definition
| If p and q are distinct odd primes, (p/q)(q/p)=(-1)^(((p-1)/2)((q-1)/2)) |
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Definition
Let p and q be distinct odd primes, and define M and N as follows:
M = [q/p]+[2q/p]+...+[(p-1)/2*q/p]
N = [p/q]+[2p/q]+...+[(q-1)/2*p/q]
Then M+N = ((p-1)/2)((q-1)/2) |
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Definition
Let p and q be distinct odd primes and reduce the set {q,2q,3q, ...,(p-1)/2q} of integers modulo p.
If the number of resulting least positive residues larger than (p/2) is m, then (q/p)=(-1)^m |
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Definition
| If M = [q/p]+[2q/p]+...+[(p-1)/2*q/p] as in Eisenstein's Lemma, and if m is the least positive residues larger than (p/2), then M and m have the same parity- thatis, they are either both even or odd. |
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| Continued Fraction Algorithm |
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Definition
| Any irrational number X=Xo may be expressed as X=[a0;a1,a2,a3...] where ao=[[Xo]], and if Xn=1/((Xn-1)-[[Xn-1]]), then an=[[Xn]]. |
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Definition
Given any irrational alpha, there exist infinitely many rational numbers p/q, q>0, that satisfy the inequality
|alpha-(p/q)|<1/(sqrt(5)q^2) |
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| Liouville's Theorem, 1844 |
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Definition
| If alpha is irrational and algebraic of degree n, there exists a constant c such that for any rational (p/q), q>0, we must have |alpha-(p/q)|>(c/q^n) |
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| Give the "short proof" of the Law of Quadratic Reciprocity |
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Definition
(p/q)(q/p)=(-1)^m*(-1)^n (by Gauss's Lemma) =(-1)^(m+n) (by rules of exponents)=(-1)^(M+N) (by Parity Lemma)
=(-1)^((p-1)/2)((q-1)/2) (by Eisenstein's Lemma. |
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