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Combinatorial Diophantine equations
Publicationes Mathematicae Debrecen, 2000For a positive integer \(k\) let \(P_k(x)=x(x+1)\ldots (x+k-1)\) and \(S_k(x)=1^k+2^k+\ldots +x^k\). In the paper the following Diophantine equations are solved (or resolved): \(P_6(x)=P_4(y)\), \(P_6(x)={y\choose 2}\), \(P_6(x)={y\choose 4}\), \({x\choose 3}=P_2(y)\), \({x\choose 3}=P_4(y)\), \({x\choose 6}=P_2(y)\), \({x\choose 6}=P_4(y)\), \({x ...
Hajdu, L., Pintér, Á.
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Archiv der Mathematik, 1996
Let \(A_k= \{a_1, \dots, a_k\}\) be a set of natural numbers with \(\text{gcd} (a_1, \dots, a_k)=1\). The greatest integer \(g= g(A_k)\) with no representation \(g= \sum^k_{i=1} x_i a_i\), \(x_i\in \mathbb{N}_0= \{0, 1, \dots\}\) is called the Frobenius number of \(A_k\).
Djawadi, Mehdi, Hofmeister, Gerd
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Let \(A_k= \{a_1, \dots, a_k\}\) be a set of natural numbers with \(\text{gcd} (a_1, \dots, a_k)=1\). The greatest integer \(g= g(A_k)\) with no representation \(g= \sum^k_{i=1} x_i a_i\), \(x_i\in \mathbb{N}_0= \{0, 1, \dots\}\) is called the Frobenius number of \(A_k\).
Djawadi, Mehdi, Hofmeister, Gerd
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Quadratic diophantine equations
Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 1960Tartakowsky (1929) proved that a positive definite quadratic form, with integral coefficients, in 5 or more variables represents all but at most finitely many of the positive integers not excluded by congruence considerations. Tartakowsky’s argument does not lead to any estimate for a positive integer which, though not so excluded, is not represented ...
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Simultaneous Diophantine Approximation
Proceedings of the London Mathematical Society, 1952Proof of the theorem: ``Let \(c > 46^{-1/4}\). Then, for every pair of real irrational numbers \(\alpha, \beta\), there exist infinitely many solutions \(p, q, r > 0\) of \(r(p-\alpha r)^2 < c\), \(r(q- \beta r)^2 < c\) in integers.'' This result slightly improves one by \textit{P. Mullender} [Ann. Math. (2) 52, 417-426 (1950; Zbl 0037.17102)].
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Integers, 2009
AbstractIn this paper, we show that the only triple of positive ...
Luca , F, Szalay, László
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AbstractIn this paper, we show that the only triple of positive ...
Luca , F, Szalay, László
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Simultaneous Diophantine Approximation
Canadian Journal of Mathematics, 1950Summary of results. The principal result of this paper is as follows: given any set of real numbers z1, z2, & , zn and an integer t we can find an integer and a set of integers p1, p2 & , pn such that(0.11).Also, if n = 2, we can, given t, produce numbers z1 and z2 such that(0.12)This supersedes the results of Nils Pipping (Acta Aboensis, vol.
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Cubic Diophantine Inequalities
Acta Mathematica Sinica, English Series, 2001In this paper, it is proved that for any real numbers \(\lambda_1\), \(\lambda_2,\ldots,\lambda_7\) with \(\lambda_i\geq 1\) \((1\leq i\leq 7)\), the Diophantine inequality \[ |\lambda_1x_1^3+\lambda_2x_2^3+\cdots+\lambda_7x_7^3|
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Journal of the London Mathematical Society, 1971
In his book ''Diophantine equations'' [London etc.: Academic Press (1969; Zbl 0188.34503)] \textit{L. J. Mordell} asked for the complete solution in rational integers \(x\) and \(y\) of the indeterminate equation \[ (x + 1)(x^2 - x + 6) = 6y^2. \tag{1} \] In this paper it is proved that all solutions of (1) are given by \(x = -1, 0, 2, 7, 15, 74\) and \
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In his book ''Diophantine equations'' [London etc.: Academic Press (1969; Zbl 0188.34503)] \textit{L. J. Mordell} asked for the complete solution in rational integers \(x\) and \(y\) of the indeterminate equation \[ (x + 1)(x^2 - x + 6) = 6y^2. \tag{1} \] In this paper it is proved that all solutions of (1) are given by \(x = -1, 0, 2, 7, 15, 74\) and \
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Diophantine frequency synthesis
IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 2006A methodology for fine-step, fast-hopping, low-spurs phase-locked loop based frequency synthesis is presented. It uses mathematical properties of integer numbers and linear Diophantine equations to overcome the constraining relation between frequency step and phase-comparator frequency that is inherent in conventional phase-locked loop based frequency ...
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