Some Results Connected with the Class Number Problem in Real Quadratic Fields

We investigate arithmetic properties of certain subsets of square-free positive integers and obtain in this way some results concerning the class number h（d） of the real quadratic field Q（√d）. In particular, we give a new proof of the result of Hasse, asserting that in this case h（d） = 1 is possible only if d is of the form p, 2q or qr. where p.q. r are primes and q≡r≡3（mod 4）.     

Residue properties of certain quadratic units

Explicit formulas are given for the quadratic and quartic characters of units of certain quadratic fields in terms of representations by positive definite binary quadratic forms, as conjectured by Leonard and Williams (Pacific J. Math.71 (1977), Rocky Mountain J. Math.9 (1979)), and by Lehmer (J. Reine Angew. Math.268/69 (1974)). For example, if p and a are primes such that p≡1 (mod 8), q≡5 (mod 8) and the Legendre symbol $\text{(}\text{q}\text{p}\text{)=1}$, and if ε is the fundamental unit of $\text{Q}$(√q), then $\text{(}\text{ε}\text{p}\text{)}{}_4\text{=(?1)}{}^{\mathrm{b+d}}$, where p=a²+16b² and p^k=c²+16qd² with k odd.     

Bicovering arcs and small complete caps from elliptic curves

Bicovering arcs in Galois affine planes of odd order are a powerful tool for the construction of complete caps in spaces of arbitrarily higher dimensions. The aim of this paper is to investigate whether the arcs contained in elliptic cubic curves are bicovering. As a result, bicovering k-arcs in AG(2,q) of size k≤q/3 are obtained, provided that q?1 has a prime divisor m with 7<m<(1/8)q ^1/4. Such arcs produce complete caps of size kq ^(N?2)/2 in affine spaces of dimension N≡0(mod4). When q=p ^h with p prime and h≤8, these caps are the smallest known complete caps in AG(N,q), N≡0(mod4).     

Non-Cayley Vertex-Transitive Graphs of Order Twice the Product of Two Odd Primes

For a positive integer n, does there exist a vertex-transitive graph Γ on n vertices which is not a Cayley graph, or, equivalently, a graph Γ on n vertices such that Aut Γ is transitive on vertices but none of its subgroups are regular on vertices? Previous work (by Alspach and Parsons, Frucht, Graver and Watkins, Marusic and Scapellato, and McKay and the second author) has produced answers to this question if n is prime, or divisible by the square of some prime, or if n is the product of two distinct primes. In this paper we consider the simplest unresolved case for even integers, namely for integers of the form n = 2pq, where 2 < q < p, and p and q are primes. We give a new construction of an infinite family of vertex-transitive graphs on 2pq vertices which are not Cayley graphs in the case where p ≡ 1 (mod q). Further, if p ? 1 (mod q), p ≡ q ≡ 3(mod 4), and if every vertex-transitive graph of order pq is a Cayley graph, then it is shown that, either 2pq = 66, or every vertex-transitive graph of order 2pq admitting a transitive imprimitive group of automorphisms is a Cayley graph.     

On binary equations

Letb ₁,b ₂ be any positive integers such that (b ₁,b ₂)=1 andm either 1 or 2 satisfyingb ₁+b ₂≡m (mod 2). Letp denote a prime andP ₃ >0 be a product of at most three primes. By sieve methods the author proves that for any δ>0 there exists a positive constantC depending only on δ such thatb ₁ p?b ₂ P ₃=m has a solution inp,P ₃ each less thanC ^(maxb) _j ^δ.     

On Ovoids of Parabolic Quadrics

It is known that every ovoid of the parabolic quadric Q(4, q), q=p ^h , p prime, intersects every three-dimensional elliptic quadric in 1 mod p points. We present a new approach which gives us a second proof of this result and, in the case when p=2, allows us to prove that every ovoid of Q(4, q) either intersects all the three-dimensional elliptic quadrics in 1 mod 4 points or intersects all the three-dimensional elliptic quadrics in 3 mod 4 points. We also prove that every ovoid of Q(4, q), q prime, is an elliptic quadric. This theorem has several applications, one of which is the non-existence of ovoids of Q(6, q), q prime, q>3. We conclude with a 1 mod p result for ovoids of Q(6, q), q=p ^h , p prime.     

COMPSTAT '78: 3rd Symposium on computational statistics

It is proved that there are no nontrivial perfect (e,n,q)-codesif e?3 and q = p^r₁p¹₂ where p₁ and p₂ are distinct primes and r and s are positive integers.     

On simultaneous representations of primes by binary quadratic forms

Let p ≡ ± 1 (mod 8) be a prime which is a quadratic residue modulo 7. Then p = M² + 7N², and knowing M and N makes it possible to “predict” whether p = A² + 14B² is solvable or p = 7C² + 2D² is solvable. More generally, let q and r be distinct primes, and let an integral solution of H²p = M² + qN² be known. Under appropriate assumptions, this information can be used to restrict the possible values of K for which K²q = A² + qrB² is solvable and the possible values of K′ for which K′²p = qC² + rD² is solvable. These restrictions exclude some of the binary quadratic forms in the principal genus of discriminant ?4qr from representing p.     

On exact coverings of the integers

By an exact covering of modulusm, we mean a finite set of liner congruencesx≡a _i (modm _i ), (i=1,2,...r) with the properties: (I)m _i ∣m, (i=1,2,...,r); (II) Each integer satisfies precisely one of the congruences. Let α≥0, β≥0, be integers and letp andq be primes. Let μ (m) senote the Möbius function. Letm=p ^α q ^β and letT(m) be the number of exact coverings of modulusm. Then,T(m) is given recursively by $$\mathop \Sigma \limits_{d/m} \mu (d)\left( {T\left( {\frac{m}{d}} \right)} \right)^d = 1$$ .     

The quartic characters of certain quadratic units

Criteria are obtained for the quartic residue character of the fundamental unit of the real quadratic field $\text{Q((2q)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$, where q is prime and either q ≡ 7(mod 8), or q ≡ 1(mod 8) and X² ? 2qY² = ?2 is solvable in integers X and Y.     

Romanoff theorem in a sparse set

Let A be any subset of positive integers,and P the set of all positive primes.Two of our results are:(a) the number of positive integers which are less than x and can be represented as 2k + p(resp.p-2k) with k ∈ A and p ∈ P is more than 0.03A(log x/log 2)π(x) for all sufficiently large x;(b) the number of positive integers which are less than x and can be represented as 2q + p with p,q ∈ P is(1 + o(1))π(log x/log 2)π(x).Four related open problems and one conjecture are posed.     

On the density of odd integers of the form (p − 1)2−n and related questions

It is shown that odd integers k such that k · 2ⁿ + 1 is prime for some positive integer n have a positive lower density. More generally, for any primes p₁, …, p_r, the integers k such that k is relatively prime to each of p₁,…, p_r, and such that k · p₁^n₁p₂^n₂ … p_r^n_r + 1 is prime for some n₁,…, n_r, also have a positive lower density.     

Metrical Theorems on Restricted Diophantine Approximations to Points on a Curve

We investigate rational approximations (r/p,q/p) or (r/p,q/r) to points on the curve (α,α^τ) for almost all α>0, where p,q,r are all primes. We immediately obtain corollaries on making p,[αp], [α^τp] simultaneously prime.     

Exceptional cases of Terai’s conjecture on Diophantine equations

Let a, b, c be relatively prime positive integers such that a ^p  + b ^q  = c ^r for fixed integers p, q, r ≥ 2. Terai conjectured that the equation a ^x  + b ^y  = c ^z in positive integers has only the solution (x, y, z) = (p, q, r) except for specific cases. In this paper, we consider the case q = r = 2 and give some results related to exceptional cases.     

p-Adic dynamics and angle doubling

We consider the squaring map over the p-adic numbers for an odd prime p, and study its symbolic dynamics on the unit circle in ? _p , the p-adic integers. When the map is restricted to the set of squares, we show an equivalence to angle doubling (mod 1) for rational angles. For primes p ≡ 3 (mod 4), this map may be represented as a unitary permutation matrix of the type used in quantum phase estimation.     

On the cubic character of quadratic units

An elementary proof is given of the theorem: If D = ?3q or ?27q is the discriminant of a cubic field, where q ≡ 1 (mod 4) is a prime, and if p or 4p is represented by c² + ∥ D ∥ d², then the fundamental unit in the field $\text{Q(q}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{)}$ is a cubic residue of the prime p. In special cases necessary and sufficient conditions are derived.     

Asymptotic results for partitions (I) and the distribution of certain integers

Asymptotic results, similar to those of Roth and Szekeres, are obtained for certain partition problems. These results are then applied to the distribution of integers of the form p₁^d₁p₂^d₂ … p_r^d_r, where d₁ ≥ d₂ ≥ … ≥ d_r, p_i denotes the ith prime and r is arbitrary. The saddle-point method is used to obtain the asymptotic results.     

On linear codes whose weights and length have a common divisor

In this paper we prove that a set of points (in a projective space over a finite field of q elements), which is incident with 0 mod r points of every hyperplane, has at least (r−1)q+(p−1)r points, where 1<r<q=ph, p prime. An immediate corollary of this theorem is that a linear code whose weights and length have a common divisor r<q and whose dual minimum distance is at least 3, has length at least (r−1)q+(p−1)r. The theorem, which is sharp in some cases, is a strong generalisation of an earlier result on the non-existence of maximal arcs in projective planes; the proof involves polynomials over finite fields, and is a streamlined and more transparent version of the earlier one.     

On the representation of integers by p-adic diagonal forms

Let p be an odd prime, let d be a positive integer such that (d,p?1)=1, let r denote the p-adic valuation of d and let m=1+3+3²+…+3^r. It is shown that for every p-adic integer n the equation Σ_i=1^mX_i^d=n has a nontrivial p-adic solution. It is also shown that for all p-adic units a₁, a₂, a₃, a₄ and all p-adic integers n the equation Σ_i=1⁴a_iX_i^p=n has a nontrivial p-adic solution. A corollary to each of these results is that every p-adic integer is a sum of four pth powers of p-adic integers.     

On the sum of two integral squares in the imaginary quadratic field \mathbb{Q}\left( {\sqrt { - 2p} } \right)

We determine the sum of two integral squares over imaginary quadratic fields Q(√-2p),where p≡1 mod 8 is a prime satisfying 2p=r2+s2with r,s≡±3 mod 8.