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Journal articles on the topic 'Problem of Diophantus'

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Published: 4 June 2021
Last updated: 5 February 2022

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1

Gyarmati, Katalin. "On a problem of Diophantus." Acta Arithmetica 97, no. 1 (2001): 53–65. http://dx.doi.org/10.4064/aa97-1-3.

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2

Dujella, Andrej. "Generalization of a problem of Diophantus." Acta Arithmetica 65, no. 1 (1993): 15–27. http://dx.doi.org/10.4064/aa-65-1-15-27.

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3

Dujella, Andrej, and Clemens Fuchs. "On a problem of Diophantus for rationals." Journal of Number Theory 132, no. 10 (October 2012): 2075–83. http://dx.doi.org/10.1016/j.jnt.2012.04.004.

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4

Harrington, Joshua, and Lenny Jones. "A modification of a problem of Diophantus." Mathematica Slovaca 68, no. 6 (December 19, 2018): 1343–52. http://dx.doi.org/10.1515/ms-2017-0185.

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Abstract An old question, due to Diophantus, asks to find sets of rational numbers such that 1 added to the product of any two elements from the set is a square. We are concerned here with a modification of this question. Let t ≥ 2 be an integer, and let 𝔽 be a field. For d ∈ 𝔽, define ft,d: 𝔽t → 𝔽 as $$\begin{array}{} \displaystyle f_{t,d}(x_1,x_2,\ldots,x_{t}):=x_1x_2\cdots x_{t}+d. \end{array}$$ For any nonempty subset S of 𝔽, we say $$\begin{array}{} \displaystyle S ~~\text{is}~~ {f_{t,d}-closed} ~~\text{if}~~ \left\{f_{t,d}(x_1,x_2,\ldots,x_{t}):x_i\in S\text{ and distinct}\right\}\subseteq S. \end{array}$$ For any integer n, with t≤ n≤ |𝔽|, let 𝒰(n,t,d) be the union of all ft,d-closed subsets S of 𝔽 with |S|=n. In this article, we investigate values of n,t,d for which 𝒰(n,t,d) = 𝔽, with particular focus on t = n – 1, where n ∈ {3,4}. Moreover, if 𝒰(n,t,d)≠ 𝔽, we determine in many cases the exact elements of the set 𝔽∖ 𝔽(n,t,d).
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5

Bugeaud, Yann, and Katalin Gyarmati. "On generalizations of a problem of Diophantus." Illinois Journal of Mathematics 48, no. 4 (October 2004): 1105–15. http://dx.doi.org/10.1215/ijm/1258138502.

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6

Dujella, Andrej, and Florian Luca. "On a Problem of Diophantus with Polynomials." Rocky Mountain Journal of Mathematics 37, no. 1 (February 2007): 131–57. http://dx.doi.org/10.1216/rmjm/1181069322.

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7

BUGEAUD, YANN, and ANDREJ DUJELLA. "On a problem of Diophantus for higher powers." Mathematical Proceedings of the Cambridge Philosophical Society 135, no. 1 (July 2003): 1–10. http://dx.doi.org/10.1017/s0305004102006588.

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8

DUJELLA, ANDREJ, and CLEMENS FUCHS. "COMPLETE SOLUTION OF A PROBLEM OF DIOPHANTUS AND EULER." Journal of the London Mathematical Society 71, no. 01 (February 2005): 33–52. http://dx.doi.org/10.1112/s002461070400609x.

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9

Dujella, Andrej, and Clemens Fuchs. "A Polynomial Variant of a Problem of Diophantus and Euler." Rocky Mountain Journal of Mathematics 33, no. 3 (September 2003): 797–811. http://dx.doi.org/10.1216/rmjm/1181069929.

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10

Dujella, Andrej, and Clemens Fuchs. "Complete solution of the polynomial version of a problem of Diophantus." Journal of Number Theory 106, no. 2 (June 2004): 326–44. http://dx.doi.org/10.1016/j.jnt.2003.12.011.

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11

Filipin, Alan, and Ana Jurasić. "A polynomial variant of a problem of Diophantus and its consequences." Glasnik Matematicki 54, no. 1 (June 7, 2019): 21–52. http://dx.doi.org/10.3336/gm.54.1.03.

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12

DUJELLA, ANDREJ, CLEMENS FUCHS, and FLORIAN LUCA. "A POLYNOMIAL VARIANT OF A PROBLEM OF DIOPHANTUS FOR PURE POWERS." International Journal of Number Theory 04, no. 01 (February 2008): 57–71. http://dx.doi.org/10.1142/s1793042108001225.

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In this paper, we prove that there does not exist a set of 11 polynomials with coefficients in a field of characteristic 0, not all constant, with the property that the product of any two distinct elements plus 1 is a perfect square. Moreover, we prove that there does not exist a set of 5 polynomials with the property that the product of any two distinct elements plus 1 is a perfect kth power with k ≥ 7. Combining these results, we get an absolute upper bound for the size of a set with the property that the product of any two elements plus 1 is a pure power.
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13

Christianidis, Jean, and Jeffrey Oaks. "Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria." Historia Mathematica 40, no. 2 (May 2013): 127–63. http://dx.doi.org/10.1016/j.hm.2012.09.001.

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14

ÖZER, Özen. "Some Some properties of the certain \(P_t\)-sets." International Journal of Algebra and Statistics 6, no. 1-2 (July 31, 2017): 117. http://dx.doi.org/10.20454/ijas.2017.1261.

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The problem of extendibility and characterization of \(P_t\)-sets is of big interest even though the problem is old, and it was started by Greek mathematician Diophantus. Let \(t\) be an integer. A set of \(m\) distinct positive integers \({a_1,a_2,\dots,a_m}\) is called a \(P_t\)-set if \(a_i a_j+t (1\leq i\leq j\leq m)\) is a perfect square whenever \(i\neq j\) In this paper, we will investigate several numerical \(P_k\)-sets and demonstrate that they cannot be widen. Also, we will determine some of their properties using reciprocity theorem.
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15

DUJELLA, ANDREJ, and ANA JURASIĆ. "ON THE SIZE OF SETS IN A POLYNOMIAL VARIANT OF A PROBLEM OF DIOPHANTUS." International Journal of Number Theory 06, no. 07 (November 2010): 1449–71. http://dx.doi.org/10.1142/s1793042110003575.

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In this paper, we prove that there does not exist a set of 8 polynomials (not all constant) with coefficients in an algebraically closed field of characteristic 0 with the property that the product of any two of its distinct elements plus 1 is a perfect square.
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16

Sialaros, Michalis, and Jean Christianidis. "Situating the Debate on “Geometrical Algebra” within the Framework of Premodern Algebra." Science in Context 29, no. 2 (May 12, 2016): 129–50. http://dx.doi.org/10.1017/s0269889715000411.

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ArgumentThe aim of this paper is to employ the newly contextualized historiographical category of “premodern algebra” in order to revisit the arguably most controversial topic of the last decades in the field of Greek mathematics, namely the debate on “geometrical algebra.” Within this framework, we shift focus from the discrepancy among the views expressed in the debate to some of the historiographical assumptions and methodological approaches that the opposing sides shared. Moreover, by using a series of propositions related toElem.II.5 as a case study, we discuss Euclid's geometrical proofs, the so-called “semi-algebraic” alternative demonstrations attributed to Heron of Alexandria, as well as the solutions given by Diophantus, al-Sulamī, and al-Khwārizmī to the corresponding numerical problem. This comparative analysis offers a new reading of Heron's practice, highlights the significance of contextualizing “premodern algebra,” and indicates that the origins of algebraic reasoning should be sought in the problem-solving practice, rather than in the theorem-proving tradition.
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17

Shiu, Peter. "The gaps between sums of two squares." Mathematical Gazette 97, no. 539 (July 2013): 256–62. http://dx.doi.org/10.1017/s0025557200005842.

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Problems concerning the setof numbers which are representable as sums of two squares have a long history. There are statements concerning W in the Arithmetic of Diophantus, who seemed to be aware of the famous identitywhich shows that the set W is ‘multiplicatively closed’. Since a square must be congruent to 0 or 1 (mod 4), it follows that members of W cannot be congruent to 3 (mod 4). Also, it is not difficult to show that a number of the form 4k + 3 must have a prime divisor of the same form dividing it an exact odd number of times. However, the definitive statement (see, for example, Chapter V in [1]) concerning members of W, namely that they have the form PQ2, where P is free of prime divisors p ≡ 3 (mod 4), was first given only in 1625 by the Dutch mathematician Albert Girard. It was also given a little later by Fermat, who probably had a proof of it, but the first published proof was by Euler in 1749.
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