منابع مشابه
The Number of Diophantine Quintuples
A set {a1, . . . , am} of m distinct positive integers is called a Diophantine m-tuple if aiaj + 1 is a perfect square for all i, j with 1 ≤ i < j ≤ m. It is known that there does not exist a Diophantine sextuple and that there exist only finitely many Diophantine quintuples. In this paper, we first show that for a fixed Diophantine triple {a, b, c} with a < b < c, the number of Diophantine qui...
متن کاملA Note on Diophantine Quintuples
Introduction. Diophantus noted that the rational numbers 1/16, 33/16, 17/4 and 105/16 have the following property: the product of any two of them increased by 1 is a square of a rational number (see [2, 3]). Let n be an integer. A set of positive integers {a1, a2, . . . , am} is said to have the property D(n) if aiaj + n is a perfect square for all 1 ≤ i < j ≤ m. Such a set is called a Diophant...
متن کاملOn Diophantine quintuples and D(−1)-quadruples
In this paper the known upper bound 10 for the number of Diophantine quintuples is reduced to 6.8·10. The key ingredient for the improvement is that certain individual bounds on parameters are now combined with a more efficient counting of tuples, and estimated by sums over divisor functions. As a side effect, we also improve the known upper bound 4 ·10 for the number of D(−1)-quadruples to 5 ·...
متن کاملThere are only finitely many Diophantine quintuples
A set of m positive integers is called a Diophantine m-tuple if the product of its any two distinct elements increased by 1 is a perfect square. Diophantus found a set of four positive rationals with the above property. The first Diophantine quadruple was found by Fermat (the set {1, 3, 8, 120}). Baker and Davenport proved that this particular quadruple cannot be extended to a Diophantine quint...
متن کاملNonexistence of Solutions In
Consider the KPP-type equation of the form ∆u+f (u) = 0, where f : [0, 1] → R + is a concave function. We prove for arbitrary dimensions that there is no solution bounded in (0, 1). The significance of this result from the point of view of probability theory is also discussed.
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Journal of Number Theory
سال: 2019
ISSN: 0022-314X
DOI: 10.1016/j.jnt.2018.07.013