課程名稱︰初等數論
課程性質︰選修
課程教師︰朱樺
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰99/01/12
考試時限（分鐘）：180
是否需發放獎勵金：
（如未明確表示，則不予發放）
試題 :
(1) (10%) Solve the following system of equations.
3x^2 - x ≡ -4 (mod 7)
{
4x - 24 ≡ 0 (mod 30).
(2) (10%) Find all integral solutions of the equation 35x + 55y + 77z = 3.
(3) (10%) Factor 1155-660i into a product of Gaussian prime numbers.
(4) (10%) The sequence {a_n} satisfies the recurrence relation:
a_1 = 1, a_2 = 2, a_3 = 3, a_n = 2a_n-1 + 3a_n-2 - 6a_n-3 (for n>3)
Find a formula for the nth term a_n.
(5) (10%) Let F_n be the nth Fibonacci number, F_0 = 0. Find the generatng
function for the sequence {F_n/n!}.
(6) (10%) Let p be a prime. Find the sum [1^n + 2^n +...+ (p-1)^n] (mod p).
(7) (10%) Describe all positive integral solutions of the equation x^2-6y^2=19
(8) (10%) Find all prime numbers p such that -14 is a QR modulo p.
(9) (10%) Find all positive integers n such that the Euler funtion ψ(n) = 24.
(10) (10%) Consider the elliptic curve E: y^2 = x^3 + 8.
(a) Find 7 integral points on E.
(b) Find the p- defects a_p of E for p = 2,3,5,7.
(11) (10%) Suppose that n is a positive integer that can be expressed as sum
of two relative prime number squares. Show that every positive
divisor of n must also have this property.
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