### Lecture Summary, MP313 Number Theory III, Semester 2, 1999

Lecture 1
Revision of divisibility, congruences, congruence or residue classes, Euclid's algorithm: r0=a, r1=b, rk=rk+1qk+1+rk+2, 0 < rk+2 <rk+1 and gcd(a,b), the sequences sk,tk, where rk=ska+tkb, inverse of b (mod m), reduced residue class.
Lecture 2
Euler's phi function, complete set (mod m), the complete set ak+b, gcd(a,m)=1, reduced set (mod m), the reduced set kb, gcd(b,m)=1, solving a linear congruence ax b (mod m), multiplicativity of Euler's function, formula for Euler's function.
Lecture 3
Euler-Fermat theorem, summing Euler's function over the divisors of n, two proofs - one involving partitioning, the other via the multiplicative function obtained by summing a given one over the divisors of n, d(n), sigma(n) and perfect numbers.
Lecture 4
Characterisation of even perfect numbers, the Möbius function, summing the Möbius function over the divisors of n (two proofs), the Möbius inversion formula, application to formula for Euler's function, Chinese remainder theorem, 1-1 correspondence between residue classes mod m and the cartesian product of the residue classes mod mi, where m=m1...mt is a product of pairwise relatively prime moduli, the restriction of this to the group of reduced residue classes (mod m),
Lecture 5
Solving polynomial congruences (mod m) - using the Chinese remainder theorem to reduce the problem to m=pn, further reduction to m=p.
Lecture 6
ordma, prime divisors of x2n+1, calculating ordma efficiently, primitive root (mod m), primitive roots (mod p) exist.
Lecture 7
A primitive root/primality test, Pepin's test for primality of Fn=22n+1, Pocklington's test, Proth's theorem on the primality of n=h2m+1, h < 2m, primitive roots (mod pn) and (mod 2pn), primitive roots exist (mod m) if and only if m=2, 4, pn and 2pn, binomial congruences, solubility of xn a (mod p).
Lecture 8
The Legendre symbol , Euler's criterion, trapdoor functions, Shanks' baby steps-giant steps algorithm, quadratic residues and non-residues (mod p), evaluation of , solving x2 -1 (mod p) if p=4n+1.
Lecture 9
Multiplicativity of the Legendre symbol, np (the least quadratic nonresidue mod p), np is prime, , Blum numbers pq, quadratic residues (mod pq), principal square roots (mod pq), random bit strings and bitwise encryption.
Lecture 10
Gauss' lemma, evaluation of , quadratic reciprocity.
Lecture 11
Applications of quadratic reciprocity: Evaluation of , primitive roots mod special primes, prime factors of 22n+1, converse of Pepin's theorem, evaluating the Legendre symbol, bad way:- factoring, good way:- Jacobi symbol .
Lecture 12
Reduced residues (mod 2n), Tonelli's √d (mod p) algorithm, calculating the integer part of a1/m, testing for perfect powers.
Lecture 13
Euclid's algorithm revisited, Thue's theorem on small solutions of ax y (mod b), Serret's algorithm for p=x2+y2, uniqueness of such a representation, extension to p=x2+ny2, n=2,3,5.
Lecture 14
Pseudoprimes and strong pseudoprimes, Lucas-Lehmer test for primality of 2p-1.
Lecture 15
Continuation of Lucas-Lehmer, finite simple continued fractions, 2 x 2 matrix notation.
Lecture 16
Partial quotients, convergents pn/qn, using Euclid's algorithm to find the simple continued fraction of a/b, uniqueness of the partial quotients, nested sequence property of the convergents, infinite simple continued fractions, nth complete quotient.
Lecture 17
Irrationality of an infinite simple continued fraction, finding the simple continued fraction of an irrational, [1,1,...] and [1,2,1,2,...],quadratic irrationals, reduced quadratic irrationals, an algorithm for finding the simple continued fraction of a quadratic irrational.
Lecture 18
Justification of the previous algorithm, purely periodic simple continued fractions represent reduced quadratic irrationals and conversely, simple continued fraction of d1/2, d rational, periodic simple continued fractions represent quadratic irrationals and conversely.
Lecture 19
Pell's equations x2-dy2=1, x2-dy2=±1 and associated multiplicative groups U and V, η and η0, (the least elements > 1 of U, V respectively), U={±η n | n an integer}, V={±η0 n | n an integer}, η02=η if x2-dy2= -1 has a solution.
Lecture 20
Finding η0 from the simple continued fraction for √d, the equation x2-dy2=±4, the associated multiplicative group W, and least element η1 > 1, x2-py2=-1 is soluble if p=4n+1 is prime, definition of p-adic integers in terms of coherent sequences, addition and multiplication of p-adic integers.
Lecture 21
contains all rationals of the form a/b with p not dividing b, p-adic units, px=0 implies x=0, nx=0 implies n=0 or x=0 if n is an integer, every nonzero p-adic integer has the form x=pmy, m 0, y a unit, is an integral domain, , the field of p-adic numbers.
Lecture 22
Every nonzero p-adic number has the form pnu, u a unit, p-adic value |x|p, limits, canonical series expansion a0+a1p + ··· of a p-adic integer, Cauchy sequences, ie. an+1 - an tends to zero, Cauchy sequences are bounded, completeness of the p-adic numbers.
Lecture 23
Pointed out that Serret's algorithm gives rk-12 - 2 tk-12=p if p=8n ± 1, completed proof of completeness, Hensel's lemma, versions 1,2,3. Application to square roots in .
Lecture 24
Example of 2-adic expansion arising from the 3x+1 mapping, compactness of , canonical p-adic expansion of a rational number and its periodic nature, the binomial series, example of (1+7/9)1/2 = -4/3 in , p=7.
Lecture 25
Strassman's theorem, further study.