Zagier Problems
Three problems were posed by Don Zagier after each of his five lectures at the Colloquium. You can see the problems below and click on the link for the answer to each one.
If you want a more readable copy of the problems or solutions, they are available as:
Problems : AMS Tex file or dvi file or postscript file
Solutions: AMS Tex file or dvi file or postscript file
First Day: Number Theory
Second Day: Algebra
Third Day: Polynomials
Fourth Day: Geometry
Fifth Day: Sequences and recursions
Supplementary problem
During the conference, a further problem was posed by one of the participants, namely, to prove that in any covering of a square checkerboard (of necessarily even edge length) by 2 × 1 dominos, there would always be a horizontal or vertical line through the checkerboard not passing through any domino.
During the ensuing discussion, a generalisation was conjectured by another participant (viz., me), namely, that the same is true for any rectangular checkerboard.
Since I know that I was not the only person who tried to prove these assertions, it may be worth communicating that both are in fact wrong, counterexamples of smallest size for the generalisation and for the original assertion being the 5 × 6 and 8 × 8 checkerboards:
If you have any comments, you can mail me.
Don Zagier 1996
If you want a more readable copy of the problems or solutions, they are available as:
Problems : AMS Tex file or dvi file or postscript file
Solutions: AMS Tex file or dvi file or postscript file
First Day: Number Theory
- Somebody incorrectly remembered Fermat's little theorem as saying that the congruence holds for all if is prime. Describe the set of integers for which this property is in fact true.
Solution
- Prove or disprove: for every odd number there is a prime of the form .
Solution
- Show that if (for ) is integral, then it is a perfect square.
Solution
Second Day: Algebra
- Show that there is no ring (= commutative, with 1) with exactly five units.
Solution
- Show that if a group has only finitely many elements of finite order, then these form a subgroup.
Solution
- Let and denote the direct product and direct sum of countably many copies of , respectively, i.e. is the set of all sequences of integers with the abelian group structure given by componentwise addition, and is the subgroup consisting of all sequences with for sufficiently large. Show that Hom is isomorphic to .
Solution
Third Day: Polynomials
- Associate to a prime the polynomial whose coefficients are the decimal digits of the prime (e.g. for the prime 9403). Show that this polynomial is always irreducible.
Solution
- Suppose that in has only real roots. Show that the polynomial , has the same property.
Solution
- Let be an odd prime. Show that the polynomial is twice a square in .
Solution
Fourth Day: Geometry
- Let and 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 and are also rational. Give a counterexample to this statement when and are not assumed to be adjacent.
Solution
- A rectangle 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 has the same property.
Solution
- Mark an angle on a pie-plate, and pick another angle .
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 . Show that, whatever the values of and , this operation has finite order (i.e., after a finite number of iterations every piece of the pie is in its original position).
Solution
Fifth Day: Sequences and recursions
- Define a sequence by the recursion
with initial values (1, 1, 1, 1, 1). Prove the statement that all of the are integral.
Solution
- Define a sequence by the formula
Show that the statement "none of the are integral" is false, but that the first counterexample is approximately 102019025 .
Solution
- Define a sequence by the recursion
.Show that the statement "all of the are integral" is false, but that the first counterexample is approximately 10178485291567 .
Solution
Supplementary problem
During the conference, a further problem was posed by one of the participants, namely, to prove that in any covering of a square checkerboard (of necessarily even edge length) by 2 × 1 dominos, there would always be a horizontal or vertical line through the checkerboard not passing through any domino.
During the ensuing discussion, a generalisation was conjectured by another participant (viz., me), namely, that the same is true for any rectangular checkerboard.
Since I know that I was not the only person who tried to prove these assertions, it may be worth communicating that both are in fact wrong, counterexamples of smallest size for the generalisation and for the original assertion being the 5 × 6 and 8 × 8 checkerboards:
1 2 3 3 1 1 1 1 2 1 1 2 2 3 1 2 1 2 2 3 2 3 2 3 3 4 1 3 3 3 1 3 1 3 and 2 3 1 1 2 4 1 2 , respectively. 4 2 2 3 1 2 1 4 4 3 2 3 3 2 4 1 1 4 4 2 1 3 1 3 1 1 4 1 2 3 1 2 4 3 2 2 2 4 4 2 4 3 4 1 1 1 3 3 1 1 4 1
If you have any comments, you can mail me.
Don Zagier 1996