The quadratic form x2−2py2

Using known properties of continued fractions, we give a very simple and elementary proof of the theorem of Epstein and Rédei on the impossibility in a certain case of representing ?1 by the quadratic form x² ? 2py². Two of our theorems, which concern the representation of a² and ?2a², serve to extend our method to an unknown case in which ?1 is not representable.     

On x3 + y3 = D

The simplest case of Fermat's last theorem, the impossibility of solving x³ + y³ = z³ in nonzero integers, has been proved. In other words, 1 is not expressible as a sum of two cubes of rational numbers. However, the slightly extended problem, in which integers D are expressible as a sum of two cubes of rational numbers, is unsolved. There is the conjecture (based on work of Birch, Swinnerton-Dyer, and Stephens) that x³ + y³ = D is solvable in the rational numbers for all square-free positive integers D ≡ 4 (mod 9). The condition that D should be square-free is necessary. As an example, it is shown near the end of this paper that x³ + y³ = 4 has no solutions in the rational numbers. The remainder of this paper is concerned with the proof published by the first author (Proc. Nat. Acad. Sci. USA., 1963) entitled “Remarks on a conjecture of C. L. Siegel.” This pointed out an error in a statement of Siegel that the diophantine equation ax³ + bx²y + cxy² + dy³ = n has a bounded number of integer solutions for fixed a, b, c, d, and, further, that the bound is independent of a, b, c, d, and n. However, x³ + y³ = n already has an unbounded number of solutions. The paper of S. Chowla itself contains an error or at least an omission. This can be rectified by quoting a theorem of E. Lutz.     

On Legendre's equation ax2 + by2 + cz2 = 0

Let a, b, c be nonzero integers having no prime factors ≡ 3 (mod 4), not all of the same sign, abc squarefree, and for which Legendre's equation ax² + by² + cz² = 0 is solvable in nonzero integers x, y, z. A property is proved yielding a congruence which must be satisfied by any solution x, y, z.     

The diophantine equation x3 − 3xy2 − y3 = 1 and related equations

The diophantine equation of the title has been solved by Ljunggren, by indirect use of the p-adic method (use is made of intermediate algebraic extensions). It is generally accepted that an immediate application of the p-adic method for the aforementioned equation is impossible. In this paper, however, this view was overthrown by first solving x² + 3 = 4y³ and then x³ ? 3xy² ? y³ = 1 with direct application of the p-adic method, avoiding the use of intermediate algebraic extensions, fulfilling thus a desire of Professor Mordell. The method used in this paper has a general character, as it is shown in Appendix B, where three more examples are given.     

The diophantine equation x3 + 3y3 = 2n

It is proved that the equation of the title has a finite number of integral solutions (x, y, n) and necessary conditions are given for (x, y, n) in order that it can be a solution (Theorem 2). It is also proved that for a given odd x₀ there is at most one integral solution (y, n), n ≥ 3, to x₀³ + 3y³ = 2ⁿ and for a given odd y₀ there is at most one integral solution (x, n), n ≥ 3, to x³ + 3y₀³ = 2ⁿ.     

The complete solution in integers of x3 + 3y3 = 2n

Some general remarks are made concerning the equation f(x, y) = qⁿ in the integral unknowns x, y, n, where f is an integral form and q > 1 is a given integer. It is proved that the only integral triads (x, y, n) satisfying x³ + 3y³ = 2ⁿ are (x, y, n) = (?1, 1, 1), (1, 1, 2), (?7, 5, 5,), (5, 1, 7).     

Evaluation of a cubic character sum using the √−19 division points of the curve Y2 = X3 − 23 · 19X + 2 · 192

The √?19 division points on the curve y² = f(x) of the title are calculated explicitly and the effect of the Frobenius map on these points is found in order to evaluate the cubic character sum $\text{Σ}{}_{\mathrm{<mtext>x(</mtext><mtext>mod</mtext><mtext>\; p)</mtext>}}\text{(}\text{f(x)}\text{p}\text{)}$.     

An exotic involution of S4

CAPPELL and Shaneson [1] construct a family of smooth 4-manifolds which are simple homotopy equivalent to real projective 4-space RP⁴, but not even smoothly h- cobordant to RP⁴. (It is possible they are homeomorphic to RP⁴.) It is natural to ask whether their double covers are S⁴ or not.     

Translation planes of order ϱ6 admitting SL(2, ϱ2)

Large numbers of translation planes are constructed which have order ?⁶ and admit a collineation group SL(2, ?²) generated by elations.     

On the number of coprime integral solutions of y2 = x3 + k

Let N′(k) denote the number of coprime integral solutions x, y of y² = x³ + k. It is shown that lim sup_k→∞N′(k) ≥ 12.     

Structural stability of C1 diffeomorphisms

In this paper we prove that if f is a C¹ diffeomorphism that satisfies Axiom A and the strong transversality condition then it is structurally stable. J. Robbin proved this theorem for C² diffeomorphisms. In addition to reducing the amount of differentiability necessary to prove the theorem, we also give a new proof combining the d_f metric of Robbin with the stable and unstable manifold proof of D. Anosov. We also prove structural stability in the neighborhood of a single hyperbolic basic set (independent of its being part of a diffeomorphism that satisfies Axiom A and the strong transversality condition). These proofs are adapted to prove the structural stability of C¹ flows in another paper.     

A characterization of P2(μ) ≠ L2(μ)

It is shown that P²(μ) ≠ L²(μ) if and only if there exists a probability measure ν with ν ⊥ μ and 6pz.dfnc;_1,ν ? Cz.dfnc;pz.dfnc;_2,μ for all polynomials p and a fixed constant C < ∞. The relationship between this method and theorems of Berger and of Carey and Pincus concerning rationally cyclic vectors for powers of nonnormal subnormal Operators is examined.     

Properties of certain algebras between L∞and H∞

The main result of this paper is that if F is a closed subset of the unit circle, then $\text{(H}{}^{\mathrm{\infty }}\text{+ L}{}_{\mathrm{F}}{}^{\mathrm{\infty }}\text{)}\text{H}{}^{\mathrm{\infty }}$ is an M-ideal of $\text{L}{}^{\mathrm{\infty }}\text{H}{}^{\mathrm{\infty }}$. Consequently, if $\text{? ∈ L}{}^{\mathrm{\infty }}$ then ? has a closest element in H^∞ + L_F^∞. Furthermore, if $\text{¦F¦ >0}\text{then}\text{L}{}^{\mathrm{\infty }}\text{(H}{}^{\mathrm{\infty }}\text{+ L}{}_{\mathrm{F}}{}^{\mathrm{\infty }}\text{)}$ is not the dual of any Banach space.     

L2 approximation with trigonometric n-nomials

We compare the degree of approximation to L²(?π, π) by nth degree trigonometric polynomials, with the degree of approximation by trigonometric n-nomials, which are linear combinations, with constant (complex) coefficients, of any 2n + 1 members of the sequence {exp (ikx)}, ? ∞ < k < ∞.     

Exposed points and extremal problems in H1

If φ ∈ L_∞, we denote by T_φ the functional defined on the Hardy space H¹ by

$\text{T}{}_{\mathrm{\phi }}\text{(?) =}\text{∫}\text{?π}\text{π}\text{?(e}{}^{\mathrm{i\theta }}\text{) φ(e}{}^{\mathrm{i\theta }}\text{)}\text{dθ}\text{2π}$

. Let S_φ be the set of functions in H¹ which satisfy $\text{T}{}_{\mathrm{\phi }}\text{(?) = ∥T}{}_{\mathrm{\phi }}\text{∥}\text{and}\text{∥? ∥}{}_1\text{? 1}$. It is known that if φ is continuous, then S_φ is weak-¹ compact and not empty. For many noncontinuous φ each S_φ is weak-¹ compact and not empty. A complete descr ption of S_φ if S_φ is weak-¹ compact and not empty is obtained. S_φ is not empty if and only if $\text{S}{}_{\mathrm{\phi }}\text{= S}{}_{\mathrm{\psi }}\text{and}\text{ψ = ¦ ?¦}\text{?}$ for some nonzero ? in H¹. It is shown that if $\text{φ = ¦? ¦}\text{?}$ and $\text{? = pg}$, where p is an analytic polynomial and g is a strong outer function, then S_φ is weak-¹ compact. As the consequence, if $\text{? = p}$, then S_φ is weak-¹ compact.     

A characterization of certain weak1-closed subalgebras of L∞

Let $\text{R}$ denote the real line and L^∞(R), the class of all Borel measurable L^∞-functions of $\text{R}$. Let S ≠ {0} or φ, be a linear subspace of L^∞(R) which is (i) translation invariant, (ii) weak¹-closed, (iii) self-adjoint, i.e., f?S implies f?S, and (iv) an algebra. Then either (a) S = all constant functions in L^∞; or (b) S = L^∞; or (c) there is a unique c > 0 such that S consists of all L^∞-functions which are periodic of period c.Extension of the above characterization of periodic subalgebras of L^∞ to LCA groups are presented. Also it is shown that the above characterization is in various ways best possible.     

The Banach algebra l1(S)

Let S be the free semigroup with a finite or countably infinite set of generators plus an identity. It is shown that there is a natural involution 1 on the convolution Banach algebra l¹(S) such that (l¹(S), 1) has a separating family of finite-dimensional star representations. The star representations of the l¹-algebra of some other semigroups are also considered. The spectrum of every element of l¹(S) which is not a scalar multiple of the identity is shown to be a connected set with interior.     

Homographic solutions of the n-body problem in E4

We prove the existence of many homographic solutions of the n-body problem in E⁴ by topological methods. Homographic solutions are associated with relative equilibria. Homothetic solutions always give rise to central configurations. In Euclidean space E⁴ central configurations are a proper subset of the relative equilibria for any n ? 3 and for any (m_i)?R₊ⁿ. We compare the existence and classification of homographic solutions of the n-body problem in E³ with the Newtonian potential and that of homographic solutions of the n-body problem in E⁴. Classifying relative equilibria leads to classifying homographic solutions.     

On the diophantine equation y2 − D = 2k

The equation of the title is studied for 1 ≤ D ≤ 100. It is shown that for such values of D the above equation is really interesting only if D = 17, 41, 73, 89, 97. Then, for these values of D, (i) necessary conditions are given for the solvability of the diophantine equations y² = 2x⁴ + D and y² = 8x⁴ + D, and (ii) y² ? D = 2^k is solved.     

Projective unitary positive-energy representations of diff (S1)

Let $\text{D}$ be the group of orientation-preserving diffeomorphisms of the circle S¹. Then $\text{D}$ is Fréchet Lie group with Lie algebra (δ_∞)_R the smooth real vector fields on S¹. Let δ_R be the subalgebra of real vector fields with finite Fourier series. It is proved that every infinitesimally unitary projective positive-energy representation of δ_R integrates to a continuous projective unitary representation of $\text{D}$. This result was conjectured by V. Kac.