MIDTERM 1 FOR NUMBER THEORY
Date: Thursday, Nov 23, 2000 Instructor: Shu-Yen Pan
No credit will be given for an answer without reasoning.
1.
(i) [5%] Prove that n2− 1 is divisible by 8 if n is odd.
(i) [5%] Prove that n3− n is divisible by 6 for every integer n.
2. [10%] Suppose that m is odd. Show that
−(m − 1)
2 ,−(m − 3)
2 , . . . ,m − 3 2 ,m − 1
2 is a complete residue system modulo m.
3. [10%] Solve the congruence equation x2+ x + 7 ≡ 0 (mod 27).
4. [10%] Let p be a prime and let gcd(a, p) = gcd(b, p) = 1. Suppose that x2 ≡ a (mod p) and x2≡ b (mod p) are not solvable. Show that x2≡ ab (mod p) is solvable.
5.
(i) [5%] Evaluate the Legendre symbol¡−35
101
¢.
(ii) [5%] Find the number of integers in the set S = {1, 2, . . . , 2100} that are divisible by neither 3 nor 5.
6. [10%] If n is an even integer, show thatP
d|nµ(d)φ(d) = 0 where µ(d) is the M¨obius function and φ(d) is the number of elements in a reduced residue system modulo d.
7. [10%] Suppose that f and g are multiplicative arithmetic functions. Show that f ∗ g is also multi- plicative.
8. [10%] Prove that if x2+ y2= z2, then one of x, y is a multiple of 3 and one of x, y, z is a multiple of 5.
9. [10%] Two quadratic forms f (x, y) and g(X, Y ) are said to be equivalent if there is an integral transformation with determinant ±1 that carries f (x, y) to g(X, Y ). Prove that the following quadratic forms
ax2+ bxy + cy2, ax2− bxy + cy2, cx2+ bxy + ay2 are equivalent.
10. [10%] Let p be a prime and k be an integer. Prove that
1k+ 2k+ · · · + (p − 1)k≡
( 0 (mod p), if k is not divisible by p − 1;
−1 (mod p), if k is divisible by p − 1.
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