FINAL FOR NUMBER THEORY
Date: Thursday, January 18, 2001 Instructor: Shu-Yen Pan
No credit will be given for an answer without reasoning.
1. [10%] Prove that√ 2 +√
3 is a root of x4− 10x2+ 1 = 0, and hence establish that it is irrational.
2. [10%] Suppose that p is an odd rational prime. Show that µ3
p
¶
=
( 1, if p ≡ 1 (mod 3);
−1, if p ≡ 2 (mod 3).
where¡a
p
¢denotes the Legendre symbol.
3. [10%] Recall that the function µ is defined by
µ(n) =
1, if n = 1;
0, if a2|n for some rational integer a > 1;
(−1)r, if n = p1p2· · · pr where p1, . . . , pr are distinct rational primes.
Find a positive rational integer n such that µ(n) + µ(n + 1) + µ(n + 2) = 3.
4. [10%] Show that the primitive solutions of x2+ y2 = z2 with y even are x = r2− s2, y = 2rs, z = r2+ s2, where r and s are arbitrary integers of opposite parity with r > s > 0 and gcd(r, s) = 1.
5. [10%] Let {ai} and {bi} be increasing sequences of real numbers. We say that ai is asymptotic to bi, write ai ∼ bi, if and only if limi→∞ai/bi= 1. Prove that ai ∼ bi implies ln ai ∼ ln bi, but the converse is not true.
6. [10%] Let a0, a1, a2, . . . be a sequence of rational integers, all positive except perhaps a0. Recall that we define the sequence of rational integers {kn} by k−2= 1, k−1 = 0 and ki= aiki−1+ ki−2for i ≥ 0.
Show that kn/kn−1= han, an−1, . . . , a2, a1i for n ≥ 1.
7. [10%] Let k and r be positive rational integers with k > 1 and r > 1. Prove that there is a rational prime whose digital representation to base r has exactly k digits.
8. [10%] Show that 3 is a prime in Q(√
−1), but not a prime in Q(√ 6).
9. [10%] The rational prime 13 can be factored in two ways in Q(√
−3), 13 = 7 +√
−3
2 · 7 +√
−3
2 =¡
1 + 2√
−3¢¡
1 − 2√
−3¢ . Explain why this does not conflict with the fact that Q(√
−3) has the unique factorization property.
10. [10%] Let p(n) denote the number of partitions of n. Compute p(9).
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