By L.-K. Hua

ISBN-10: 3540108181

ISBN-13: 9783540108184

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40 3. Quadratic Residues Therefore Similarly we have so that If we can prove that t(p-1) [kq] t(q-1) k= 1 P 1= 1 L - + L [lP] _P - 1 q - 1 - --2 q or 2 =p-lq-l 2 2 (mod 2), then the theorem will follow. It suffices therefore to prove the following lemma. Lemma. _P- 1q- 1 L [kq] -P + L [IP] -q . 2 2 t(p-1) t(q- 1) k= 1 1= 1 Proof Consider the rectangle with vertices: (O,tq) (0,0), (0, tq), (tp, 0), (tp, tq)

In §1. " We recall that the problem was to solve the simultaneous congruences x == 2 (mod 3), x == 3 (mod 5), x == 2 (mod 7). The meaning of the song here is as follows: Multiply by 70 the remainder of x when divided by 3, multiply by 21 the remainder of x when divided by 5, multiply by 15 (the number of days in half a Chinese (synodic) month) the remainder of x when divided by 7. Add the three results together, and then subtract a suitable multiple of 105 and you shall have the required smallest solution.

Example. The numbers 1,2,4 are quadratic residues and 3,5,6 are quadratic nonresidues mod 7. Definition 2 (Legendre's symbol). Letp be an odd prime, and suppose thatp,tn. We let if n is a quadratic residue mod p, if n is a quadratic non-residue mod p. 1. Let p > 2. There are t

### Introduction to Number Theory by L.-K. Hua

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