The simultaneous diophantine equations 5Y2 − 20 = X2 and 2Y2 + 1 = Z2

The three numbers 1, 5, 10 have the property that the product of any two numbers decreased by 1 is a perfect square. In this paper it is proved that there is no other positive integer N which shares this property with 1, 5, and 10.     

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 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.     

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 solution of the diophantine equation A3 + B3 + C3 + D3 = 0

This paper presents a complete and symmetric solution of the diophantine equation of the title in terms of four parameters α, β, γ, δ, subject to α + β + γ + δ = 0.     

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 + 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).     

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.     

On the number of divisors of x2 + x + A

Let A be a positive or negative rational integer such that integers in the field of √1 ? 4A have unique prime factorization. An elementary criterion will be obtained for x² + x + A to be a prime number, where x is a positive integer. The criterion implies that for positive A the polynomial x² + x + A is prime for x = 0, 1,…, A ? 2.     

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.     

Compactifications determined by subsets of C1(X)

It is well known that every compactification of a completely regular space X can be generated, via a Tychonoff-type embedding, by some suitably chosen subset of C¹(X). Different subsets may give rise to equivalent compactifications, and we are concerned with the problem of finding all subsets of C¹(X) which yield a given compactification αX. The problem is easier if generalized: we say that a subset F of C¹(X) “determines” the compactification αX if αX is the smallest compactification to which every element of F extends, and give a simple necessary and sufficient condition for F to determine a given compactification αX. A number of sufficient conditions for two sets to determine the same compactification are given, and the relation between sets which determine αX and those which generate αX (via an embedding) is considered. Generally, a much smaller set of functions is required to determine αX than to generate it; the number needed to determine αX is never more than the weight of αX?X, while the number required to generate it is, if infinite, equal to the weight of αX.     

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.     

On the series Σk = 1∞(k2k)−1k−n and related sums

Series of the form $\text{Σ}{}_{\mathrm{k\; =\; 1}}{}^{\mathrm{\infty }}\text{(}\text{2k}\text{2k}\text{)}{}^{\mathrm{?1}}\text{k}{}^{\mathrm{?n}}$ may be expressed as log sin integrals and are shown to be summable exactly in terms of Dirichlets L-series for values of n up to and including 5. Other related series are also discussed and several exact results are given.     

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.     

On integers of the forms k−2 and k2+1

In this paper we prove that if (r,12)?3, then the set of positive odd integers k such that k^r−2ⁿ has at least two distinct prime factors for all positive integers n contains an infinite arithmetic progression. The same result corresponding to k^r2ⁿ+1 is also true.