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, r_k=r_k+1q_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_ka+t_kb, 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=m₁...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=pⁿ, further reduction to m=p.

Lecture 6

ord_ma, prime divisors of x²ⁿ+1, calculating ord_ma 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=2²ⁿ+1, Pocklington's test, Proth's theorem on the primality of n=h2^m+1, h < 2^m, 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, 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 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 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=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 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 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=p^my, 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. 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-1² - 2 t_k-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.

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KRM 18th January 2002