代写Number Theory (MA3Z7) Problem Sheet III帮做R语言

Number Theory (MA3Z7)

Problem Sheet III

1. Solve the linear congruences (i) 2x ≡ 4 (mod 7), (ii) 6x ≡ 3 (mod 21).

Use (i) or otherwise to find all integers x, y such that 2x+ 7y = 53.

2. Prove the converse of Wilson’s Theorem: for n > 1 composite,

[Hint: show that n|(n−1)! unless n = 4 and treat n = 4 separately.]

3. Find all solutions of the simultaneous congruences

4. Find the remainder when 235042 is divided by 37.

5. Let n ∈ N such that n ≡ 3 (mod 4). Show that n cannot be written as the sum of two squares.

Show further that a positive integer ≡ 7 (mod 8) cannot be written as the sum of three squares.

Any guesses for sums of four squares?