100.16 The Diophantine equation n(x + y + z) = xyz

Dolan, Stan

doi:10.1017/mag.2016.69

Cited by 1 publication

(4 citation statements)

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“…For instance, in the equation x + y = F with x, y, F ∈ Z + 0 , the sum of the least upper bounds for x and y is twice the least upper bound for (x + y). Application of the method of Stan Dolan [22] led to an upper bound (2F 2 + 2), which is even weaker than the upper bound derived from bounds for both terms according to [21]. With growing F, the ratio of the two bounds approaches 2:…”

Section: Uppermentioning

confidence: 87%

“…The semiperimeter in their calculation equaled the sum of solution terms of the Diophantine Equation ( 8) with F = r 2 ; thus, they established an upper bound for the sum of solution terms of (8) to be (F + 1)(F + 2). In 2016, Stan Dolan published a new proof of their result [22]. We demonstrated his method in Section 2.2.…”

Section: Diophantine Equations Xy = F(x + Y) and Xyz = F(x + Y + Z): ...mentioning

confidence: 97%

“…A proof is obtained by substituting (3) into (2). For example, for the equation xy = 21(x + y), we obtain the solution (22,462).…”

Section: Diophantinementioning

confidence: 99%

“…An upper bound for the sum of terms of solutions of ∏ 3 i=1 x i = F ∑ 3 i=1 x i is known to be (F + 1)(F + 2) [22,23]. In Section 2, an upper bound for the sum of solution terms of ∏ 2 i=1 x i = F ∑ 2 i=1 x i was shown to be F 2 + 2F + 1 (Theorem 2).…”

mentioning

confidence: 99%

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Diophantine Equations Relating Sums and Products of Positive Integers: Computation-Aided Study of Parametric Solutions, Bounds, and Distinct-Term Solutions

Karlovský

2021

Mathematics

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Diophantine equations ∏i=1nxi=F∑i=1nxi with xi,F∈ℤ+ associate the products and sums of n natural numbers. Only special cases have been studied so far. Here, we provide new parametric solutions depending on F and the divisors of F or F2. One of these solutions shows that the equation of any degree with any F is solvable. For n = 2, exactly two solutions exist if and only if F is a prime. These solutions are (2F, 2F) and (F + 1, F(F + 1)). We generalize an upper bound for the sum of solution terms from n = 3 established by Crilly and Fletcher in 2015 to any n to be (F+1)(F+n−1) and determine a lower bound to be nnFn−1. Confining the solutions to n-tuples consisting of distinct terms, equations of the 4th degree with any F are solvable but equations of the 5th to 9th degree are not. An upper bound for the sum of terms of distinct-term solutions is conjectured to be (F+1)[F+(n−2)(n−1)!/2+1]/(n−2)!. The conjecture is supported by computation, which also indicates that the upper bound equals the largest sum of solution terms if and only if F=(n+k−2)(n−2)!−1, k∈ℤ+. Computation provides further insights into the relationships between F and the sum of terms of distinct-term solutions.

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Section: Uppermentioning

confidence: 87%

Section: Diophantine Equations Xy = F(x + Y) and Xyz = F(x + Y + Z): ...mentioning

confidence: 97%

“…A proof is obtained by substituting (3) into (2). For example, for the equation xy = 21(x + y), we obtain the solution (22,462).…”

Section: Diophantinementioning

confidence: 99%

mentioning

confidence: 99%

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