Ternary Quadratic Forms And Quadratic Residues

Quadratic Diophantine equations are classical and active topics in number theo-ry.Lagrange's four-square theorem proved in 1770 asserts that each natural number(i.e.,nonnegative integers)is a sum of four squares.In this thesis,we study represen-tations of natural numbers via quadratic polynomials and some permutation problems involving squares over finite fields.For a quadratic polynomial f(x,y,z) with integer coefficients,if each natural num-ber can be written as f(x,y,z)with x,y,z integers,then we call f universal.Using the Siegel-Minkowski formula and spinor exceptional integers in the arithmetic theory of quadratic forms,we prove some conjectures of Zhi-Wei Sun on universal quadratic polynomials,for example,we show that x2+y(y+1)/2+2z(4z+1)is universal.In addition,we study some problems involving representations of arithmetic progressions by diagonal ternary quadratic forms and confirm some conjectures posed by Z.-W.Sun,L.Pehlivan and K.S.Williams,for example,we show that 3x2+5y2+30z2 with x,y,z integers represents all natural numbers congruent to 8 modulo 15.For each integer m? 3,we let Pm(x) denote the generalized m-gonal number(m-2)x2-(m-4)x/2 with x?Z.Given positive integers a,b,c,k and an odd prime number p with p(?)c,we employ the theory of ternary quadratic forms to determine completely when the polynomial ax2+by2+cPpk+2(z)with x,y,z integers represents all but finitely many positive integers.We also use the theory of quadratic forms and related modular forms to study rep-resentation problems on sums of squares with linear restrictions.The 1-3-5 conjecture of Z.-W.Sun states that any n?N N={0,1,2,...} can be written as x2+y2+z2+w2 with w,x,y,z ? N such that x+3y+5z is a square.We show that there is a fi-nite set A of positive integers such that any sufficiently large integer not in the set{16ka:a?A,k?N} can be written as x2+y2+z2+w2 with x,y,z,w ? Z and x+3y+5z ? {4k:k?N}.We also confirm some conjectures of Zhi-Wei Sun on sums of four squares with certain linear restrictions.For example,any positive integer can be written as x2+y2+z2+2w2 with x,y,z,w?Z and x+y+2z+2w=1,and any sufficiently large integer can be written as x2+y2+z2+2w2 with x,y,z,w ? Z and x+2y+3z=1.Let p?1(mod 4) be a prime,and let g?Z is a primitive root modulo p.Let a1<a2<…<a(p-1)2 be all quadratic residues modulo p among 1,…,p-1.Consider the three sequences.A0:12,22,…,((p-1)/2)2,A1:a1,a2,…a(p-1)/2,A2:g2,g4,…,gp-1,where a stands for the residue class a+pZ.For i,j?1,2,3,clearly Ai is a permutation of Aj and we call this permutation ?i,j.We determine the signs of these permutations.Also,we study some permutation problems involving squares in Fp2.