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).     

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.     

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 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.     

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.     

Three-variable Mahler measures and special values of modular and Dirichlet L-series

In this paper we prove that the Mahler measures of the Laurent polynomials (x+x ^?1)(y+y ^?1)(z+z ^?1)+k ^1/2, (x+x ^?1)²(y+y ^?1)²(1+z)³ z ^?2?k, and x ⁴+y ⁴+z ⁴+1+k ^1/4 xyz, for various values of k, are of the form r ₁ L′(f,0)+r ₂ L′(χ,?1), where $r_{1},r_{2}\in \mathbb{Q}$ , f is a CM newform of weight 3, and χ is a quadratic character. Since it has been proved that these Mahler measures can also be expressed in terms of logarithms and ₅ F ₄-hypergeometric series, we obtain several new hypergeometric evaluations and transformations from these results.     

The Ornstein-Uhlenbeck semigroup in an infinite dimensional L2 setting

We study the number operator, N, of quantum field theory as a partial differential operator in infinitely many variables. Informally Nu(x) = ?Δu(x) + x · grad u(x). A large core for N is constructed which is invariant under e^?tN and on which this informal expression may be given a precise and natural meaning.     

An approximation property of exponential functions

We solve the inhomogeneous linear first order differential equations of the form y′(x) ? λy(x) = Σ _m=0 ^∞ a _m (x ? c) ^m , and prove an approximation property of exponential functions. More precisely, we prove the local Hyers-Ulam stability of linear first order differential equations of the form y′(x) = λy(x) in a special class of analytic functions.     

A noncommutative analogue of |D(Xk|=|kXk−1|

The paper concerns alternating powers of a Hilbert space. Let ∧^k be defined by ∧^k(A)(x₁∧?∧x_k)=Ax₁∧?∧Ax_k. It is proved that the norm of the linear map D∧^k(A) depends only upon |A| and is assumed at the identity.     

C1-algebras with finite duals

A forest is a finite partially ordered set F such that for x, y, z?F with x ? z, y ? z one has x ? y or y ? x. In this paper we give a complete characterization of all separable C¹-algebras $\text{A}$ with a finite dual $\text{A}\text{?}$, for which Prim $\text{A}$ is a forest with inclusion as partial order. These results are extended to certain separable C¹-algebras $\text{A}$ with a countable dual$\text{A}\text{?}$. As an example these results are used to characterize completely all separable C¹-algebras $\text{A}$ with a three point dual.     

Two-weight ternary codes and the equation y2 = 4 × 3a + 13

This paper determines the parameters of all two-weight ternary codes C with the property that the minimum weight in the dual code C^⊥ is at least 4. This yields a characterization of uniformly packed ternary [n, k, 4] codes. The proof rests on finding all integer solutions of the equation y² = 4 × 3^a + 13.     

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.     

Weak solution of (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0))u′(0) = ku(0), r(L, u(L))u′ (L) = hu(L) and k, h are suitable elements of [0, ∞]

We consider weak solutions to the nonlinear boundary value problem (r, (x, u(x)) u′(x))′ = (Fu)′(x) with r(0, u(0)) u′(0) = ku(0), r(L, u(L)) u′(L) = hu(L) and k, h are suitable elements of [0, ∞]. In addition to studying some new boundary conditions, we also relax the constraints on r(x, u) and (Fu)(x). r(x, u) > 0 may have a countable set of jump discontinuities in u and r(x, u)^?1?L_q((0, L) × (0, p)). F is an operator from a suitable set of functions to a subset of L_p(0, L) which have nonnegative values. F includes, among others, examples of the form (Fu)(x) = (1 ? H(x ? x₀)) u(x₀), (Fu)(x) = ∫_x^Lf(y, u(y)) dy where f(y, u) may have a countable set of jump discontinuities in u or F may be chosen so that (Fu)′(x) = ? g(x, u(x)) u′(x) ? q(x) u(x) ? f(x, u(x)) where q is a distributional derivative of an L₂(0, L) function.     

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 $\text{Σ}{}_{\mathrm{<mtext>x(</mtext><mtext>mod</mtext><mtext>\; p)</mtext>}}\text{(}\text{f(x)}\text{p}\text{)}$.     

On exact recovery of sparse vectors from linear measurements

Let 1 ≤ k ≤ n < N. We say that a vector x ∈ ? ^N is k-sparse if it has at most k nonzero coordinates. Let Φ be an n × N matrix. We consider the problem of recovery of a k-sparse vector x ∈ ? ^N from the vector y = Φx ∈ ? ⁿ . We obtain almost-sharp necessary conditions for k, n, N for this problem to be reduced to that of minimization of the ?₁-norm of vectors z satisfying the condition y = Φz.     

On Y2 = X3 + k and the Thue rank of cubic curves

The intersection of cords and tangents with the curve Y² = X³ + k is exploited to locate additional lattice points and to define the Thue rank of the curve.     

On the equation 4n = 1x + 1y + 1z

It is proved that for any given positive integers N and k the number of integers n < N for which the equation $\text{4}\text{n}\text{=}\text{1}\text{x}\text{+}\text{1}\text{y}\text{+}\text{1}\text{z}$ is unsolvable in positive integers x, y, z is not greater than $\text{cN}\text{(}\text{log}\text{N)}{}^{\mathrm{k}}$, where c is a constant depending only on k.     

Monochromatic Vs multicolored paths

Letl andk be positive integers, and letX={0,1,...,l ^k?1}. Is it true that for every coloring δ:X×X→{0,1,...} there either exist elementsx ₀<x ₁<...<x _l ofX with δ(x ₀,x ₁)=δ(x ₁,x ₂)=...=δ(x _l?1,x _l), or else there exist elementsy ₀<y ₁<...<y _k ofX with δ(y _i?1,y _i) ∈ δ(y _j?1,y _j) for all 1<-i<j≤k? We prove here that this is the case if eitherl≤2, ork≤4, orl≥(3k)^2k . The general question remains open.     

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.