• Toronto MTH 609 Number Theory
• UBC MATH 312 Introduction to Number Theory
• UCSD Math 105 (Basic Number Theory)
• UCLA Mathematics Math 296B

$$F(x_1, \ldots, x_n)=0$$

$$F(x_1, \ldots, x_n) \equiv 0 (\bmod p)$$

$$f(x) \equiv f_1(x) \ldots f_r(x)(\bmod p)、$$

$$f\left(x_1 \ldots x_n\right) \equiv 0(\bmod p)、$$

## 下面是一些经典的CONGRUENCES WITH PRIME MODULUS题目

Problem 1. Show that an integer of the form $8 n+7$ cannot be represented as the sum of three squares.
Problem 2. Using the properties of the Legendre symbol, show that the congruence

$$\left(x^2-13\right)\left(x^2-17\right)\left(x^2-221\right) \equiv 0(\bmod m)$$

is solvable for all $m$. It is clear that the equation $\left(x^2-13\right)\left(x^2-17\right)\left(x^2-221\right)=0$ has no integral solutions.

Problem 3. Show that the equation $a_1 x_1+\cdots+a_n x_n=b$, where $a_1, \ldots, a_n, b$ are integers, is solvable in integers if and only if the corresponding congruence is solvable for all values of the modulus $m$.
Problem 4. Prove the analogous assertion for systems of linear equations.

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