The Prouhet Tarry Escott Problem: A Review

Ryali, Srikanth; Narayanan, V.G.

doi:10.3390/math7030227

Cited by 10 publications

(16 citation statements)

References 44 publications

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“…In 1963 Mordell [52] proved that for (k, ℓ) = (2, 3) the only positive integer solutions are given by (x, y) = (2, 1) and (14,5). In 1972 Boyd and Kisilevsky [14] proved that (x, y) = (2, 1), (4, 2), (55,19) are the only positive integer solutions if (k, ℓ) = (3, 4), while Hajdu and Pintér [40] showed that the only positive integer solution for (k, ℓ) = (4, 6) is (7,2). Several results are covered by the theorem of Saradha and Shorey [60] that the only solution with ℓ = 2k is given by (k, ℓ, x, y) = (3,6,8,1).…”

Section: 2mentioning

confidence: 99%

“…corresponding to the case s = 2) are known for 2 ≤ m ≤ 10 and for m = 12. For general information on the PTE-problem we refer to [55]. In this section we shall show that for m ∈ {3, 4, 6} we can construct arbitrarily large integral PTE m sets, that is s sets of m integers having the same sums of j-th powers for 1 ≤ j ≤ m − 1, with s arbitrary.…”

Section: Kindmentioning

confidence: 99%

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The Diophantine equation $f(x)=g(y)$ for polynomials with simple rational roots

Hajdu¹,

Tijdeman²

2022

Preprint

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In this paper we consider Diophantine equations of the form f (x) = g(y) where f has simple rational roots and g has rational coefficients. We give strict conditions for the cases where the equation has infinitely many solutions in rationals with a bounded denominator. We give examples illustrating that the given conditions are necessary. It turns out that such equations with infinitely many solutions are strongly related to Prouhet-Tarry-Escott tuples. In the special, but important case when g has only simple rational roots as well, we can give a simpler statement. Also we provide an application to equal products with terms belonging to blocks of consecutive integers of bounded length. The latter theorem is related to problems and results of Erdős and Turk, and of Erdős and Graham.

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Section: 2mentioning

confidence: 99%

Section: Kindmentioning

confidence: 99%

The Diophantine equation $f(x)=g(y)$ for polynomials with simple rational roots

Hajdu¹,

Tijdeman²

2022

Preprint

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“…For our purposes (cf. 4.4), we need n = 2d 4, so we do not dwell on what is known for n 3, instead referring the reader to the survey article [RN19]. We summarize below the cases, with n 4, in which G(e, n) is known to have a non-trivial Q-point.…”

Section: Known Solutions To the Pte Problemmentioning

confidence: 99%

Constructing curves of high rank via composite polynomials

Suresh

2021

Preprint

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We improve on a construction of Mestre-Shioda to produce some families of curves X/Q of record rank relative to the genus g of X. Our first main result is that for any integer g 8 with g ≡ 2 (mod 3), there exist infinitely many genus g hyperelliptic curves over Q with at least 8g + 32 Q-points and Mordell-Weil rank 4g + 15 over Q. Our second main theorem is that if g + 1 is an odd prime and K contains the g + 1-th roots of unity, then there exist infinitely many genus g hyperelliptic curves over K with Mordell-Weil rank at least 6g over K.

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“…Ideal solutions of PTE are known for n = 12. Also, the best non-constructive known solution, that holds for general k, are M (n) ≤ C √ n for some C (see [7]). One could consider a weakening of this problem, where the left and right-hand sides of (1.13) are merely required to be "close to each other".…”

Section: 3mentioning

confidence: 99%