SOME OBSERVATIONS ON THE DIOPHANTINE EQUATION y2=x!+A AND RELATED RESULTS

Ulas, Maciej

doi:10.1017/s0004972712000512

Cited by 8 publications

(9 citation statements)

References 10 publications

(13 reference statements)

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“…Performing then a brute force search of additional solutions of this equation we can find those values of A, B which lead to equations with many solutions. Let us note that modifications of this method were used in investigations devoted to the study of Brocard-Ramanujan type equations [15] and its various generalizations [8]. We performed independent searches for the case of k = 2 and k > 2.…”

Section: Problem 11 Find Examples Of Ramanujan-nagell Type Diophantin...mentioning

confidence: 99%

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Some experiments with Ramanujan-Nagell type Diophantine equations

Ulas

2014

Glas. Mat. Ser. III

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Abstract. Stiller proved that the Diophantine equation x 2 + 119 = 15 · 2 n has exactly six solutions in positive integers. Motivated by this result we are interested in constructions of Diophantine equations of Ramanujan-Nagell type x 2 = Ak n + B with many solutions. Here, A, B ∈ Z (thus A, B are not necessarily positive) and k ∈ Z ≥2 are given integers. In particular, we prove that for each k there exists an infinite set S containing pairs of integers (A, B) such that for each (A, B) ∈ S we have gcd(A, B) is square-free and the Diophantine equation x 2 = Ak n + B has at least four solutions in positive integers. Moreover, we construct several Diophantine equations of the form x 2 = Ak n + B with k > 2, each containing five solutions in non-negative integers. We also find new examples of equations x 2 = A2 n + B having six solutions in positive integers, e.g. the following Diophantine equations has exactly six solutions:x 2 = 57 · 2 n + 117440512 n = 0, 14, 16, 20, 24, 25, x 2 = 165 · 2 n + 26404 n = 0, 5, 7, 8, 10, 12.Moreover, based on an extensive numerical calculations we state several conjectures on the number of solutions of certain parametric families of the Diophantine equations of Ramanujan-Nagell type.

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Section: Problem 11 Find Examples Of Ramanujan-nagell Type Diophantin...mentioning

confidence: 99%

“…Finally, in 1948 Nagell found all solutions of this equation and since then this equation is known as Ramanujan-Nagell equation. It is nowdays well known that it has exactly five solutions (x, n) = (1, 3), (3,4), (5,5), (11,7) and (181,15). Note that the proof of this fact appeared in English in 1960, see [11].…”

Section: Introductionmentioning

confidence: 99%

Some experiments with Ramanujan-Nagell type Diophantine equations

Ulas

2014

Glas. Mat. Ser. III

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“…Adapun usaha untuk mencari solusi persamaan (1) terus dilakukan, salah satu konjektur terbaru telah dipublikasikan oleh Maciej Ulas pada tahun 2012. Ulas (2012) mengajukan konjektur sebagai berikut:…”

Section: Pendahuluanunclassified

PENGGUNAAN TEORI KEKONGRUENAN DALAM MEMPERKECIL RUANG PENCARIAN SOLUSI PERSAMAAN DIOPHANTINE x^2 = y^3 + 2185

Kadademahe¹,

Mananohas²,

Titaley³

2019

JIS

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PENGGUNAAN TEORI KEKONGRUENAN DALAM MEMPERKECIL RUANG PENCARIAN SOLUSI PERSAMAAN DIOPHANTINE x2 = y3 + 2185ABSTRAKPada tahun 2014 Ulas mengajukan sebuah konjektur mengenai solusi bilangan bulat dari persamaan Diophantine tipe Ramanujan-Nagell x2 = y3 + 2185. Tujuan penelitian ini adalah untuk memperkecil ruang pencarian solusi persamaan Diophantine tipe Ramanujan- Nagell x2 = y3 + 2185 dengan x sub himpunan bilangan ganjil anggota G3 dan G4, dimana G3={x∈bilangan ganjil |x≡1 mod 8} dan G4={x∈bilangan ganjil |x≡7 mod 8} dengan metode membagi y menjadi 4 kasus, yaitu : FPB(y,8)=1, FPB(y,8)=2, FPB(y,8)=4, FPB(y,8)=8. Dari hasil penelitian menunjukkan bahwa untuk x∈G3 dengan FPB(y,8)=1, FPB(y,8)=4, FPB(y,8)=8, tidak mempunyai solusi bilangan bulat, sedangkan untuk FPB(y,8)=2 meskipun belum diperoleh kesimpulan akhir tapi ruang pencarian solusi telah berhasil diperkecil untuk x dan y dengan cara melakukan transformasi x=8b+1, y=4a – 2 , apabila a|b atau b|a, maka persamaan Diophantine x2 = y3 + 2185 hanya mempunyai satu pasang solusi, yaitu : (x,y)=(49,6), dan untuk x∈G4 dengan FPB(y,8)=1, FPB(y,8)=4, FPB(y,8)=8, FPB(y,8)=2 dengan melakukan transformasi x=8q+7, y=4p – 2 untuk p|q atau q|p tidak mempunyai solusi bilangan bulat. Penelitian ini telah berhasil memperkecil ruang untuk x dan y.Kata kunci : Teorema Euler, Persamaan Diophantine, dan Diophantine Ramanujan - Nagell

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“…Since h 1 = 0, thus (23) cuf (X + 1)h 1 (X + 2) h 1 (X + 1) = w, but then w = lim n→+∞ cuf (n + 1)h 1 (n + 2) h 1 (n + 1) = lim n→+∞ cuf (n + 1).…”

Section: 41mentioning

confidence: 99%

“…Diophantine equations involving terms of given sequences are a subject of interest of number theorists. There are many papers concerning problems of the following type: when a term of a given sequence is a factorial or sum, difference or product of factorials (see [2], [5], [6], [7], [8], [14], [15], [16], [17], [23]). Using information about 2-adic valuations of numbers of derangements we will prove that D 0 = D 2 = 1 and D 3 = 2 are the unique numbers of derangements being factorials.…”

Section: Arithmetic Properties Of the Sequences Of Even And Odd Deranmentioning

confidence: 99%

Arithmetic properties of the sequence of derangements

Miska

2016

Journal of Number Theory

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The sequence of derangements is given by the formula D 0 = 1, Dn = nD n−1 + (−1) n , n > 0. It is a classical object appearing in combinatorics and number theory. In this paper we consider two classes of sequences: first class is given by the formulae, and the second one is defined by. Both classes are a generalization of the sequence of derangements. We study such arithmetic properties of these sequences as: periodicity modulo d, where d ∈ N + , p-adic valuations, asymptotics, boundedness, periodicity, recurrence relations and prime divisors. Particularly we focus on the properties of the sequence of derangements and use them to establish arithmetic properties of the sequences of even and odd derangements.

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SOME OBSERVATIONS ON THE DIOPHANTINE EQUATION y²=x!+A AND RELATED RESULTS

Cited by 8 publications

References 10 publications

Some experiments with Ramanujan-Nagell type Diophantine equations

Some experiments with Ramanujan-Nagell type Diophantine equations

PENGGUNAAN TEORI KEKONGRUENAN DALAM MEMPERKECIL RUANG PENCARIAN SOLUSI PERSAMAAN DIOPHANTINE x^2 = y^3 + 2185

Arithmetic properties of the sequence of derangements

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