- Lecture 1
- Revision of divisibility, congruences, congruence or residue classes, Euclid's algorithm: r
_{0}=a, r_{1}=b, r_{k}=r_{k+1}q_{k+1}+r_{k+2}, 0 < r_{k+2}<r_{k+1}and gcd(a,b), the sequences s_{k},t_{k}, where r_{k}=s_{k}a+t_{k}b, 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 m
_{i}, where m=m_{1}...m_{t}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=p
^{n}, further reduction to m=p. - Lecture 6
- ord
_{m}a, prime divisors of x^{2n}+1, calculating ord_{m}a efficiently, primitive root (mod m), primitive roots (mod p) exist. - Lecture 7
- A primitive root/primality test, Pepin's test for primality of F
_{n}=2^{2n}+1, Pocklington's test, Proth's theorem on the primality of n=h2^{m}+1, h < 2^{m}, primitive roots (mod p^{n}) and (mod 2p^{n}), primitive roots exist (mod m) if and only if m=2, 4, p^{n}and 2p^{n}, binomial congruences, solubility of x^{n}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 x
^{2}-1 (mod p) if p=4n+1. - Lecture 9
- Multiplicativity of the Legendre symbol, n
_{p}(the least quadratic nonresidue mod p), n_{p}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 2
^{2n}+1, converse of Pepin's theorem, evaluating the Legendre symbol, bad way:- factoring, good way:- Jacobi symbol . - Lecture 12
- Reduced residues (mod 2
^{n}), Tonelli's √d (mod p) algorithm, calculating the integer part of a^{1/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=x
^{2}+y^{2}, uniqueness of such a representation, extension to p=x^{2}+ny^{2}, n=2,3,5. - Lecture 14
- Pseudoprimes and strong pseudoprimes, Lucas-Lehmer test for primality of 2
^{p}-1. - Lecture 15
- Continuation of Lucas-Lehmer, finite simple continued fractions, 2 x 2 matrix notation.
- Lecture 16
- Partial quotients, convergents p
_{n}/q_{n}, 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 d
^{1/2}, d rational, periodic simple continued fractions represent quadratic irrationals and conversely. - Lecture 19
- Pell's equations x
^{2}-dy^{2}=1, x^{2}-dy^{2}=±1 and associated multiplicative groups U and V, η and η_{0}, (the least elements > 1 of U, V respectively), U={±η^{ n}| n an integer}, V={±η_{0}^{ n}| n an integer}, η_{0}^{2}=η if x^{2}-dy^{2}= -1 has a solution. - Lecture 20
- Finding η
_{0}from the simple continued fraction for √d, the equation x^{2}-dy^{2}=±4, the associated multiplicative group W, and least element η_{1}> 1, x^{2}-py^{2}=-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=p
^{m}y, 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 p
^{n}u, u a unit, p-adic value |x|_{p}, limits, canonical series expansion a_{0}+a_{1}p + ··· of a p-adic integer, Cauchy sequences, ie. a_{n+1}- a_{n}tends to zero, Cauchy sequences are bounded, completeness of the p-adic numbers. - Lecture 23
- Pointed out that Serret's algorithm gives r
_{k-1}^{2}- 2 t_{k-1}^{2}=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.

KRM 18th January 2002