On Two Conjectures Concerning Class Numbers Of Binary Quadratic Forms

Let d>1 be a squarefree integer,andεd =(m + nd^1/d)/2 be the fundamental unit of the quadratic field Q(d^1/d).Using reduced forms and quartic residues,Sun Zhihong[Quartic residues and binary quadratic forms,J.Number Theory,113(2005),10-52.]gave the sufficient and necessary condition for primes p so thatεd is a quadratic or quartic residue(mod p).In the end,Sun also posed the following two conjectures.Conjecture 1 Let p and q be primes of the form 4k+1 such that(p/q)= 1.Then h₄(-4pq)= h₄(-64pq)= h(-4pq)/8.Conjecture 2 Let d > 1 be squarefree integer,m,n∈Z⁺ satisfying m²-dn² = 4, if 2 + m and 2-m are nonsquare integers,then |N₀(m,n,d)| =1/8h(-4δ(n,d)²d).Here(*/*)is the Legendre symbol,(*/*).₄ is the quartic residue symbol,denote the quadratic form ax² + bxy + cy² by(a,b,c)and denote the equivalent class that contains the form(a,b,c)by[a,b,c].The discriminant of(a,b,c)is the integer D=b²-4ac.Denote the form class group which consists of equivalence classes of discriminant D by H(D)= {[a,b,c]| b²-4ac = D,gcd(a,b,c)=1},and denote its fourth power subgroup by H₄(D)={[a,b,c]⁴ |[a,b,c]∈H(D)},denote their orders by h(D)and h₄(D)respectively.Let the set N_j(m,n,d)= {[a,2b,c]| b²-ac = -δ(n,d)²d,a≡(-1)^j mod 4,(a,b)=1, whereδ(n,d)∈{1,2,4,8} is determined by n and d.In Chapter 3 of this paper,we give counterexamples to Conjecture 1.We first describe the principle of algorithms to caculate h(-4pq),h₄(-4pq)and /h₄(-64pq) for primes p,q(p,q < 1000,(p/q)= 1)of the form 4k + 1 in the interval[1,1000], then we give a particular counterexample that 1 = h₄(-4pq)≠h(-4pq)/8 = 2, 1 = h₄(-4pq)≠h₄(-64pq)= 2 when(p,q)=(17,89),so the conjecture is not true. In Chapter 4,we verify Conjecture 2.We describe the principle and pseudocode for seeking the sets N₀(m,n,d)and N₁(m,n,d)for d > 1 which is a squarefree integer, and we have |N₀(m,n,d)|=|N₁(m,n,d)|=(1/8)h(-4δ(n,d)²d)with d∈[3,500]and m,n∈Z⁺ satisfying m² -dn² = 4,which support Conjecture2.