The art of mathematics begins when you understand the theory of numbers...

1. Fermat's little theorem might have said that the congruence a^n+1= a (mod n)

holds for all a if n
is prime. Can you describe the set of integers n for which this property
is in fact true?

2. Can you prove or disprove: for every odd number k there is a prime of
the form 2^n k+1

3. Can you show that if (a^2 + b^2)/(ab+1) (for a,b in N) is integral, then
it is a perfect square.

Show that you know al little about Algebra ...

1. Why is it that there is no ring (= commutative, with 1) with
exactly five units?

2. If a group has only finitely many elements of finite
order, how do these form a subgroup?

3. Given: Let P and S denote the direct product and direct sum of
countably many copies of Z, respectively,

i.e. P is the set of
all sequences of integers (a1, a2, a3, ... ) with the abelian
group structure given by

componentwise addition, and S is the
subgroup consisting of all sequences with a subi = 0 for i
sufficiently large.

Describe how Hom subZ(P, Z) is isomorphic to S.

That you know how to use Polynomials ...

1. Associate to a prime the polynomial whose coefficients are the
decimal digits of the prime

(e.g. 9 x^3 + 4 x^2 + 3 for the prime
9403). Show that this polynomial is always irreducible.

2. Suppose that b subn x^n + b subn-1 x^n-1 + ... + b sub1 x + b sub0 in R[x] has
only real roots.

Show that the polynomial b subn/n! x^n + b subn-1/(n-1)!
x^n-1 + ... + b sub1/1! x + b sub0, has the same property.

3. Let p = 2m - 1 be an odd prime.

Show that the polynomial (1-x)^m + 1 + x^m is twice a square in F subp[x].

and basic structural Geometry ...

1. Let A and B be two adjacent vertices of an equilateral
polygon.

If the angles at all other vertices are known to be
rational (when measured in degrees),

show that the angles at A
and B are also rational.

Give a counterexample to this statement
when A and B are not assumed to be adjacent.

2. A rectangle R is the union of finitely many smaller rectangles
(non-overlapping except on their boundaries),
each one of which
has at least one rational side. Show that R has the same
property.

3. Mark an angle a (0 < a < 2 ) on a pie-plate, and pick another
angle b (0 < b < 2 ).

Define an operation on the pie as follows: cut out the slice of
pie over the marked angle,

lift it up, turn it over, replace it,
and rotate the whole pie on the plate by the angle b.

Can you show how,
whatever the values of a and b, this operation has finite order

(i.e., after a finite number of iterations every piece of the pie
is in its original position).

Finally, defining some sequences and recursions ...

1. Given: sequence t1, t2, t3, ... by the recursion

tn+5 = (tn+4 tn+1 + tn+3 tn+2)/tn

with initial values (1,1,1,1,1). Prove the statement that all of
the tn are integral

2. Define a sequence u1, u2, u3, ... by the formula

Show that the statement "none of the un are integral" is false,
but that the first counterexample is approximately 102019025.

3. Define a sequence v1, v2, v3, ... by the recursion

vn =(2+v12 + ... + vn-12)/n.

Show that the statement "all of the vn are integral" is false,
but that the first counterexample is approximately
10178485291567.