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.     

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 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 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.     

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.     

On the Diophantine System a^2 ＋ b^2 = c^3 and a^x ＋ b^y = c^z for b is an Odd Prime

Let a, b and c be fixed coprime positive integers. In this paper we prove that if a^2 ＋ b^2 = c^3 and b is an odd prime, then the equation a^x ＋ b^y = c^z has only the positive integer solution （x, y, z） = （2,2,3）.     

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.     

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.     

On the number of pairs of positive integers x, y ≤ H such that x 2 + y 2 + 1 is squarefree

It is not difficult to find an asymptotic formula for the number of pairs of positive integers x, y ≤ H such that x ² + y ² + 1 is squarefree. In the present paper we improve the estimate for the error term in this formula using the properties of certain exponential sums. A.Weils’s estimate for the Kloosterman sum plays the major role in our analysis.     

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.     

A combinatorial proof of Euler-Fermat’s theorem on the representation of the primes p=8k+3 by the quadratic form x2+2y2

An elementary and extremely short proof of the theorem on the representation of the primes p = 8k + 3 by the quadratic form x₂ + 2y₂ with integers x and y. Bibliography: 1 title. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 330, 2006, pp. 155–157.     

An Excursion into Nonlinear Ramsey Theory

Formulas for two-color Rado numbers have been established for many families of linear equations. However, there are no explicit formulas for two-color Rado numbers for any nonlinear equations. In this paper, we will establish formulas for the two-color Rado numbers for three families of equations: x + y ⁿ = z, x + y ² + c = z, and x +  y ² = az , where c and a are positive integers.     

Zeros of the functions f(n) = Σi=0(−1)i(in−2i)

The main result of this paper is the following: the only zeros of the title function are at n = 3 and n = 12. This is achieved by means of the recursion function for f(n), viz. F(x) = x³ ? x ? 1 which has only one real root w. This turns out to be the fundamental unit of Q(w). From the norm equation of the units, N(w) = x³ + y³ + z³ ? 3xyz + 2x²z + xz² ? xy² ? yz² = 1, and the negative powers of w which are of binary form, the result follows. The paper concludes with two remarkable combinatorial identities.     

指数Diophantine方程4~x+b~y=(b+4)~2

本文研究了指数Diophantine方程4~x+b~y=(b+4)~2的解.设b1是给定的正奇数,运用有关指数Diophantine方程的已知结果以及有关Pell方程的Stormer定理的推广,证明了方程4~x+b~y=(b+4)~2仅有正整数解(x,y,z)=(1,1,1).     

On diophantine equations of the form {({\boldsymbol x}-{\boldsymbol a}_{\bf 1})({\boldsymbol x}-{\boldsymbol a}_{\bf 2}) \ldots ({\boldsymbol x}-{\boldsymbol a}_{\boldsymbol k}) + {\boldsymbol r} = {\boldsymbol y}^{ \boldsymbol n}}

Erd?s and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x?+?1)(x?+?2)...(x?+?(m???1))?=?y ⁿ has no solutions in positive integers x,m,n where m, n?>?1 and y?∈?Q. We consider the equation $$ (x-a_1)(x-a_2) \ldots (x-a_k) + r = y^n $$ where 0?≤?a ₁?<?a ₂?<???<?a _k are integers and, with r?∈?Q, n?≥?3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n?>?2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.     

Solving exponential diophantine equations using lattice Basis Reduction Algorithms

Let S be the set of all positive integers with prime divisors from a fixed finite set of primes. Algorithms are given for solving the diophantine inequality 0< x − y < y^δ in x, y ∈ S for fixed δ ∈ (0, 1), and for the diophantine equation x + y = z in x, y, z ∈ S. The method is based on multi-dimensional diophantine approximation, in the real and p-adic case, respectively. The main computational tool is the L³-Basis Reduction Algorithm. Elaborate examples are presented.