PROBLEMS VI
DUE FRIDAY 28 APRIL, 2017
P
−(n!)2
. Apply Liouville’s Theorem to show that θ is
1. (i) Let θ = ∞
n=0 10
transcendental.
(ii) Let (pn ) be the sequence of prime numbers, so that p1 = 2, p2 = 3, and so
on. Define the primorial function pn # for each prime number pn by putting
pn # = pn pn−1 . . . 3 · 2. Prove that the real number
∞
X
Θ=
2−pn #
n=1
is transcendental.
2. (i) Show that n2 + (n + 1)2 is a perfect square for infinitely many n.
(ii) Construct an example of such n with n > 100.
3. Note that the equation x2 − 5y 2 = 1 has a solution (x, y) = (9, 4).
(i) Find a second solution of the equation x2 − 5y 2 = 1 with x and y both
positive;
(ii) Show that x2 − 5y 2 = 1 has infinitely many integral solutions;
(iii) Find a solution of the equation u2 − 5v 2 = 5, and show that there are
infinitely many integral solutions of this equation.
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