Number Of Positive Integer Solutions Research Articles

Jordan blocks and the Bethe Ansatz II: The eclectic spin chain beyond K = 1

We continue the classification of the Jordan chains of the eclectic three state spin chain that we started in our previous article. Following the same steps, we construct the generalised eigenvectors of this spin chain by computing the strongly twisted limit of linear combinations of eigenvectors of a twisted XXX SU(3) spin chain. We show that this classification problem can be mapped to the computation of the number of positive integer solutions of a system of linear equations.

ON A CONJECTURE CONCERNING THE NUMBER OF SOLUTIONS TO

AbstractLet a, b, c be fixed coprime positive integers with $\min \{ a,b,c \}>1$ . Let $N(a,b,c)$ denote the number of positive integer solutions $(x,y,z)$ of the equation $a^x + b^y = c^z$ . We show that if $(a,b,c)$ is a triple of distinct primes for which $N(a,b,c)>1$ and $(a,b,c)$ is not one of the six known such triples, then $c>10^{18}$ , and there are exactly two solutions $(x_1, y_1, z_1)$ , $(x_2, y_2, z_2)$ with $2 \mid x_1$ , $2 \mid y_1$ , $z_1=1$ , $2 \nmid y_2$ , $z_2>1$ , and, taking $a<b$ , we must have $a=2$ , $b \equiv 1 \bmod 12$ , $c \equiv 5\, \mod 12$ , with $(a,b,c)$ satisfying further strong restrictions. These results support a conjecture put forward by Scott and Styer [‘Number of solutions to $a^x + b^y = c^z$ ’, Publ. Math. Debrecen88 (2016), 131–138].

A note on the number of solutions of the Pillai type equation [formula omitted

Let a,b,k be fixed positive integers such that min⁡{a,b}>1 and gcd⁡(a,b)=1, and let N′(a,b,k) denote the number of positive integer solutions (x,y) of the equation |ax−by|=k. In this paper, using a lower bound of linear forms in two logarithms combined with some properties of convergents of irrational numbers, we prove the following two results: (i) If min⁡{a,b}≥85988, then N′(a,b,k)≤2. (ii) For any real number ε with 0<ε<1, if k<min⁡{a1−ε,b1−ε} and max⁡{a,b}>C(ε), where C(ε) is an effectively computable constant depending only on ε, then N′(a,b,k)≤1. In particular, if k<min⁡{a1/15,b1/15}, then N′(a,b,k)≤1 except for N′(2,3,1)=N′(3,2,1)=3.

DIOPHANTINE INEQUALITIES OF FRACTIONAL DEGREE

This paper is concerned with the study of diagonal Diophantine inequalities of fractional degree θ, where θ > 2 is real and non-integral. For fixed non-zero real numbers λ i not all of the same sign, we write F ( x ) = λ 1 x 1 θ + ⋯ + λ s x s θ . For a fixed positive real number τ, we give an asymptotic formula for the number of positive integer solutions of the inequality | F ( x ) | < τ inside a box of side length P. Moreover, we investigate the problem of representing a large positive real number by a positive definite generalised polynomial of the above shape. A key result in our approach is an essentially optimal mean value estimate for exponential sums involving fractional powers of integers.

On the number of solutions of the generalized Ramanujan-Nagell equation D1X2 + DM2 = 2N+2

Let D1, D2 be coprime odd integers with min (D1, D2) > 1, and let N (D1, D2) denote the number of positive integer solutions (x, m, n) of the equation D1x2+Dm2 = 2n+2. In this paper, we prove that N (D1, D2) ≤ 2 except for N (3, 5) = N (5, 3) = 4 and N (13, 3) = N (31, 97) = 3.

On the number of positive integer solutions (x, n) of the generalized Ramanujan–Nagell equation $$x^2-2^r=p^n$$ x 2 - 2 r = p n

Let p be a fixed odd prime, and let r be a fixed positive integer. Further let \(N(2^r,p)\) denote the number of positive integer solutions (x, n) of the generalized Ramanujan–Nagell equation \(x^2-2^r=p^n\). In this paper, we use the elementary method and properties of Pell’s equation to give a sharp upper bound estimate for \(N(2^r,p)\). That is, we prove that \(N(2^r,p)\le 1\).

On the Diophantine equation x 2 − kxy + y 2 − 2 n = 0

In this study, we determine when the Diophantine equation x 2−kxy+y 2−2 n = 0 has an infinite number of positive integer solutions x and y for 0 ⩽ n ⩽ 10. Moreover, we give all positive integer solutions of the same equation for 0 ⩽ n ⩽ 10 in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation x 2 − kxy + y 2 − 2 n = 0.

On the solutions of the exponential Diophantine equation a x + b y = (m 2 + 1) z

In this paper, we consider the Diophantine equation a x + b y = c z . Combining Laurent's result on lower bounds for linear forms in two logarithms, Bugeaud's result on upper bounds for the p-adic logarithms, and Bilu-Hanrot-Voutier's result on primitive divisors of Lucas numbers, we obtain a sharper computable upper bound for the number of positive integer solutions of the equation ∣v r ∣ x + ∣u r∣ y = c z , where and r is an odd number. Moreover, if r is a prime and r ≡ 5 (mod 8) or r ≡ 19 (mod 24), , we prove that the equation has only the solution (x, y, z) = (2, 2, r).

Mean value from representation of rational number as sum of two Egyptian fractions

For given positive integers $n$ and $a$, let $R(n;\,a)$ denote the number of positive integer solutions $(x,\,y)$ of the Diophantine equation $$ {a\over n}={1\over x}+{1\over y}. $$ Write $$ S(N;\,a)=\sum_{\substack{n\leq N (n,\,a)=1}}R(n;\,a). $$ Recently Jingjing Huang and R. C. Vaughan proved that for $4\leq N$ and $a\leq 2N$, there is an asymptotic formula $$ S(N;\,a)={3\over \pi^2a}\prod_{p|a}{p-1\over p+1}\cdot N(\log^2N+c_1(a) \log N+c_0(a))+\Delta(N;\,a). $$ In this paper, we shall get a more explicit expression with better error term for $c_0(a)$.

On the Diophantine equation x2−kxy+y2+lx=0,l∈{1,2,4}

We prove that the Diophantine equation x 2 − k x y + y 2 + l x = 0 , l ∈ { 1 , 2 , 4 } has an infinite number of positive integer solutions x and y if and only if ( k , l ) = ( 3 , 1 ) , ( 3 , 2 ) , ( 4 , 2 ) , ( 3 , 4 ) , ( 4 , 4 ) , ( 6 , 4 ) . Furthermore, we prove that the Diophantine equation x 2 − k x y + y 2 + x = 0 has infinitely many integer solutions x and y if and only if k ≠ 0 , ± 1 , which answers a problem in Marlewski and Marzycki (2004) [1] .

Infinitely many positive solutions of the diophantine equation x2 − kxy + y2 + x = 0

We prove that the equation x 2 − kxy + y 2 + x = 0 with k ϵ N + has an infinite number of positive integer solutions x and y if and only if k = 3. For k = 3 the quotient x y is asymptotically equal to (3 + √5)/2 or (3 − √5)/2. Results of the paper are based on data obtained via Computer Algebra System ( derive 5). Some derive procedures presented in the paper made it possible to discover interesting regularities concerning simple continued fractions of certain numbers.

A closer look at some new identities

We obtain infinite products related to the concept of visible from the origin point vectors. Among these is urn:x-wiley:01611712:media:ijmm289376:ijmm289376-math-0001 in which φ3(k) denotes for fixed k, the number of positive integer solutions of (a, b, k) = 1 where a &lt; b &lt; k, assuming (a, b, k) is the gcd(a, b, k).

A Note on the Generalized Ramanujan-Nagell Equation

Let D, D1, D2 be positive integers such that D = D1D2, 1 ≤ D1 ≤ D2, and gcd(D1, D2) = 1, and let p be a prime with p[formula]D. Further, let N(D1, D2, p) denote the number of positive integer solutions (x, n) of the equation D1x2 + D2 = λpn, where λ = 4 or 1 according to whether p = 2 or not. In this paper we prove that if max(D, p) > exp exp exp 1000, then N(D1, D2, p) ≤ 1 except when (D1, D2, p) is exceptional.

THE SUM OF THE VALUES OF THE DIVISOR FUNCTION IN ARITHMETIC PROGRESSIONS WHOSE DIFFERENCE IS A POWER OF AN ODD PRIME

For , with a fixed odd prime, and , the asymptotic formula is proved, where is the number of positive integer solutions of , is a polynomial of degree in with coefficients depending on and , with a positive constant depending on , and the constant involved in the order depends on , and . The proof relies on an idea of A. A. Karacuba that permits one to solve this problem by means of a scheme for a ternary additive problem. Bibliography: 10 titles.

On the Number of Solutions of Diophantine Equations

For any nontrivial set of cardinal numbers $\leqq {\aleph _0}$, it is shown that there is no algorithm for testing whether or not the number of positive integer solutions of a given polynomial Diophantine equation belongs to the set.

On the number of solutions of Diophantine equations

For any nontrivial set of cardinal numbers ≦ ℵ 0 \leqq {\aleph _0} , it is shown that there is no algorithm for testing whether or not the number of positive integer solutions of a given polynomial Diophantine equation belongs to the set.