On the diophantine equation y2 = 4qn + 4q + 1

It is known that a certain class of [n, k] codes over GF(q) is related to the diophantine equation y² = 4qⁿ + 4q + 1 (1). In Parts I and II of this paper, two different, and in a certain sense complementary, methods of approach to (1) are discussed and some results concerning (1) are given as applications. A typical result is that the only solutions to (1) are (y, n) = (5, 1), (7, 2), (11, 3) when q = 3 and (y, n) = (2q + 1, 2) when q = 3^f, f >- 2.     

A note on the diophantine equation x2 + 1 = dy4

In this paper we prove that the Diophantine equation as in the title has at most one integer solution if $ \in > 5 \times 10^7 $ where $ \in = u + \upsilon \sqrt d $ is the least positive solution of Pell’s equation $x^2 - dy^2 = - 1$      

The exponential diophantine equation x y + y x = y 2 with xy odd

For any fixed positive integer D which is not a square, let (u, υ) = (u ₁, υ ₁) be the fundamental solution of the Pell equation u ² ? Dυ ² = 1. Further let $\mathbb{D}$ be the set of all positive integers D such that D is odd, D is not a square and gcd(D, υ ₁) > max(1, √D/8). In this paper we prove that if (x, y, z) is a positive integer solution of the equation x ^y + y ^x = z ² satisfying gcd(x, y) = 1 and xy is odd, then either $x \in \mathbb{D}$ or $y \in \mathbb{D}$ .     

On the diophantine equationD 1 x 2+D 2 m =4y n

For anyD ₁,D ₂, leth(-D ₁ D ₂) denote the class number of the imaginary quadratic field . In this paper we prove that the equationD ₁ x ²+D ₂ ^m =4y ⁿ.D ₁,D ₂,x, y, m, n, gcd (D ₁x,D ₂y=1,2m,n an odd prime,nh(-D ₁ D ₂, has only a finite number of solutions (D ₁,D ₂,x,y,m,n) withn>5. Moreover, the solutions satisfy 4y ⁿ相似文献   

The al-husayn equation x 4 + y 2 = z 2

We study the set of all natural solutions of the equation x ⁴ + y ² = z ², obtain general formulas describing all such solutions, and prove their equivalence.     

On Cohn's conjecture concerning the diophantine equation¶x2+ 2m = yn

In the past fifty years and more, there are many papers concerned with the solutions (x,y,m,n) of the exponential diophantine equation $ x^2 + 2^m = y^n, x, y, m, n \in \mathbb{N}, 2 \not|\, y, n > 2 $ x^2 + 2^m = y^n, x, y, m, n \in \mathbb{N}, 2 \not|\, y, n > 2 , written by Ljunggren, Nagell, Brown, Toyoizumi, Cohn and the others. In 1992, Cohn conjectured that the equation has no solutions (x, y, m, n) with m > 2 and 2 | m 2 \mid m . In this paper, using a quantitative result of Laurent, Mignotte and Nesterenko on linear forms in the logarithms of two algebraic numbers, we verify Cohn's conjecture. Thus, according to known results, we prove that the equation has only three solutions (x, y, m, n) = (5, 3, 1, 3), (7, 3, 5, 4) and (11, 5, 2, 3).     

On The diophantine equation Fn+Fm=2a

In this paper, we find all the solutions of the title Diophantine equation in positive integer variables (n, m, a), where F_k is the kth term of the Fibonacci sequence. The proof of our main theorem uses lower bounds for linear forms in logarithms (Baker's theory) and a version of the Baker-Davenport reduction method in diophantine approximation.     

A note on the diophantine equation x 2 + b y = c z

Let a, b, c, r be positive integers such that a ² + b ² = c ^r , min(a, b, c, r) > 1, gcd(a, b) = 1, a is even and r is odd. In this paper we prove that if b ≡ 3 (mod 4) and either b or c is an odd prime power, then the equation x ² + b ^y = c ^z has only the positive integer solution (x, y, z) = (a, 2, r) with min(y, z) > 1.     

The exponential Diophantine equation x2 + (3n 2 + 1) y = (4n 2 + 1) z

Let n be a positive integer. In this paper, using the results on the existence of primitive divisors of Lucas numbers and some properties of quadratic and exponential diophantine equations, we prove that if n ≡ 3 (mod 6), then the equation x ² + (3n ² + 1) ^y = (4n ² + 1) ^z has only the positive integer solutions (x, y, z) = (n, 1, 1) and (8n ³ + 3n, 1, 3).     

The diophantine equation (a n-1)(b n-1)=x 2

Let a and b given unequal positive integers; it is desired to determine the positive integer solutions n and x of the equation of the title. Some special cases have recently been considered, and here some general results and conjectures are presented. This revised version was published online in June 2006 with corrections to the Cover Date.