PROMYS Guessing Game 2007
(Algebraic Number Theory and primes of the form p = x2
+ m*y2)
Rules:
Guess the number 1 <= n <= 100 by using the following
hints (in order). If you correctly answer the questions then you will
have a unique answer for n at the end. All work must be done by hand, so no
calculators/computers are allowed! Clues are generally in increasing
order of difficulty. Have Fun! =)
Clues:
- All of the prime divisors of n are inert in the maximal totally
real subfield of K := Q(\sqrt{-6},\sqrt{-30}),
and
at least one of the prime divisors of n ramifies in K.
- The degree d of the maximal unramified abelian extension of
Q(\sqrt{-39}) divides n.
- The number n is not constructible.
- Let m be the ratio described in
Clue 2 (i.e. m = n/d). Then the ninth Mersenne prime p :=
261
- 1 is not of
the form x2 + m*y2,
where x and y are integers.
Check yourself:
- Number
of possibilities,
List
of Possibilities, Explanation
- Number
of possibilities,
List
of Possibilities, Explanation (in progress)
- Number
of possibilities,
List
of Possibilities, Explanation (coming soon)
- Number
of possibilities,
List
of Possibilities, Explanation (coming soon)
For a complete computational solution in the SAGE Computer Algebra
system (this solution is cheating), click here. (coming soon)