2022 Term 2
Term 2 --- Week 1
8 : Elizabeth: Question 6 from NRICH (same as last Term)
Question 3 from worksheet 2 ... link to appear)
Discussion of the sets of numbers NN, QQ, RR, CC.
9 : Zoltan : The Horrible Game
(1) y = (x-10)^2+1.
Lead to exploration of parabolas and symmetry.
(2) y = next prime after x.
Lead to the "prime generating polynomial" n^2+n+41.
10: Ralph : Exploring Permutations
Image notation, cycle notation and conversion between 2 types.
Enumerating all possible permutations on a set with 3 elements.
Composition of permutations. Exercises.
Filling out the full composition table for S_3.
Properties of a group explored with this example.
11&12: Peter: Mathematical structures (groups, rings & fields),
Example, when is Z/nZ a field? What about Q[x]/<f(x)>?
Term 2 --- Week 2
8 : Tamiru : Discussed modular arithmetic
9 : Philip : Number bases, introduction and word problems.
10: Zoltan : Intermediate AMC 2002 Q14-30
11&12: Peter: Continued groups rings and fields.
Generalised Euclidean algorithm
Fermat's Little Theorem
Term 2 --- Week 3
8 : Tamiru : Solved problems related to modular arithmetic
9 : Zoltan : Intermediate AMC 2002 Q14-30
10: Zoltan : Intermediate AMC 2002 Q14-30
11&12: Peter: Generalised Euclidean algorithm. FLT
Chinese remainder theorem ... Z/NZ = Z/pZ x Z/qZ
|(Z/NZ)^*| = phi(N) = (p-1)(q-1)
RSA
Term 2 --- Week 4
8 : Elizabeth: Graph theory introduction
9 : Zoltan : Continued Intermediate AMC 2002 Q14-30
Horrible game -- y=x(x+1) too easy
-- y=2^x+(-1)^x very hard, but
spawned 5 different solutions.
Mentioned why (-1)^2 = 1.
Discontinuous function y = 0 for x in Q, 1 for x not in Q.
10: Ralph : Axioms of the integers (secretly revealing a group).
Some lemmas required for an ultimate proof of the
Theorem : (-1)x(-1) = +1.
11&12: Peter: Factoring --- L_N(a,b), Fibonacci, Fermat, Kraitchik,
Pollard rho, NFS (mentioned Shor).
NFS examples : 1999 512 bits, 2010 768 bits.
Term 2 --- Week 5
8 : David : The game of life. Examples by hand as well as the
iphone app.
9 : Tamiru : Modular arithmetic, if a=b mod n & c=d mod n proved
(1) a+c=b+d mod n
(2) a*c=b*d mod n
10: Peter : Groups, Rings, Fields, axioms and examples.
Z, Q, Z[x], Q[x], ... , R[x].
11&12: Ralph: Exploring Permutations ---
Image notation, cycle notation and conversion between 2 types.
Enumerating all possible permutations on a set with 3 elements.
Composition of permutations. Exercises.
Filling out the full composition table for S_3.
Properties of a group explored with this example.
Term 2 --- Week 6
8&9 : Tamiru : Modular arithmetic.
What are the last two digits of the integer 17^198?
Compute the remainder when
2018^(2019^2020) + 2019^(2020^2021) + 2020^(2020^2020) +
+ 2021^(2020^2019) + 2022^(2021^2020)
is divided by 2020.
10: Peter : Rings Z[x], Z/NZ for N prime or composite.
In R, ab=0 --> there does not exist a^(-1) or b^(-1).
Modular arithmetic and generalised euclidean algorithm.
Lemma In Z/NZ there exists a^(-1) iff gcd(a,N) = 1.
11&12: Ralph : Discussed the symmetries of an equilateral triangle.
Showed how to tell if two symmetries are the same or different.
Discussed cyclic groups, dihedral groups and related them
to symmetries of regular n-gons.
Compared D_3 to C_3 and C_6 and D_3.
Mentioned the simple group classification theorem and monster.
Term 2 --- Week 7
8&9 : Tamiru : Group Theory
10: Peter : In Z/NZ there exists a^(-1) iff gcd(a,N)=1.
Extended Euclidean Algorithm example xa+yb = (a,b)
Thm : Z/NZ is a field iff N is prime.
|(Z/pZ)*| = p-1.
Chinese Remainder Theorem : If N=pq then Z/NZ =~ Z/pZ x Z/qZ.
If N=pq then |(Z/NZ)*| = (p-1)(q-1).
If a in R* (finite) then a^|R*| = 1.
RSA encryption - decryption algorithm.
11&12: Ralph : Discussed 1-1, onto, homomorphisms and isomorphisms.
Worked on the example of <Z/4Z,+> =~ <(Z/5Z)*,x>.
Monster group.
Relationship to modular j-Invariant.