MATH 4400 - SAMPLE EXAM II PROBLEMS
1) State the quadratic reciprocity law. Then calculate
193
1307
.
2) Let n be a positive integer. Let p be a prime divisor of n2 + 3 i.e. n2 ≡ −3 (mod p)
different from 2 or 3. Use
2 n
−3
=
p
p
and the quadratic reciprocity to conclude that p ≡ 1 (mod 3).
3) Let n be a product of any set S of primes, including 2 but not 3. For example, n = 2 · 5.
Show that any prime divisor of n2 + 3 is not in the set S.
4) Use the descent procedure procedure to find a solution of the equation x2 +y 2 = 29 starting
with 122 + 12 = 5 · 29.
5) Use the mathematical induction to prove that the sum of the first n odd integers is equal
to n2 .
6) A number Pn is called pentagonal, if Pn pebbles can be arranged in the shape as pictured,
with n pebbles along each edge. Write down a simple formula for Pn , n-th pentagonal number,
following the two steps: First find two integers a and b such that Pn+1 − Pn = an + b, and
then use the formula 1 + 2 + . . . + (n − 1) = n(n − 1)/2.
i
i i
y i
P2 = 5
i
i i
i y
i
y y i
y y i
P3 = 12
i
i i
i y
i
i
i
y y
y y
y i
y y y i
y y y i
P4 = 22
5) In this problem we will find some square numbers which are also pentagonal. The equation
m2 = Pn , after substituting n = (x+1)/6, and m = y/2 becomes√the Pell equation x2 −6y 2 =
1, which has (5, 2) as a basic solution. Of course, the powers (5+2 6)k will give all solutions to
the Pell equation, but not all of these will give rise to square-pentagonal numbers. Calculate
several low degree powers to see what is going on, and determine the first non-trivial squarepentagonal number.
1