Yasutsugu Fujita

The number of Diophantine quintuples II

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 < jm. 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 quintuples {a; b; c; d; e} with c < d < e is at most four. Using this result, we further show that the number of Diophantine quintuples is less than 10276, which improves the bound 101930 due to Dujella.

Bounds for Diophantine quintuples II

A set of positive integers {; ; a1, a2, ..., a_m}; ; with the property that a_i*a_j+1 is a perfect square for all distinct indices i and j between 1 and m is called Diophantine m-tuple. In this paper, we show that if {; ; a, b, c, d, e}; ; is a Diophantine quintuple with a 3ag ; moreover, if c > a + b + 2\sqrt{; ; ab + 1}; ; then b > max{; ; 24 ag, 2a^1.5g^2}; ; . Similar results are given assuming that either ab is odd or c = a + b + 2\sqrt{; ; ab + 1}; ; .

Extensions of the D(∓k2)-triples {k2, k2 ± 1, 4k2 ± 1}

Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be a positive integer. In this paper, we show that if {k2, k2 + 1, 4k2 + 1, d} is a D(−k2)-quadruple, then d = 1, and that if {k2 − 1, k2, 4k2 − 1, d} is a D(k2)-quadruple, then d = 8k2(2k2 − 1).

Jesmanowicz' Conjecture with Congruence Relations. II

Let a, b , and c be primitive Pythagorean numbers such that a 2 + b 2 = c 2 with b even. In this paper, we show that if b 0 ≡ ∊(mod a) with e ∊ {±1} for certain positive divisors b 0 of b , then the Diophantine equation a x + b y = c z has only the positive solution ( x , y , z ) = (2, 2, 2).

Generators for the elliptic curve y2 = x3−nx

Let E:y2 = x3−nx be an elliptic curve over the rationals with a positive integer n. Mordell’s theorem asserts that the group of rational points on E is finitely generated. Our interest is in the generators for its free part. Duquesne (2007) showed that if n = (2k2−2k+1)(18k2+30k+17) is square‐free, then certain two points of infinite order can always be in a system of generators. We generalize this result and show that the same is true for “infinitely many” infinite families n = n(k,l) with two variables.

THE EXTENSION OF THE D( k 2 )-PAIR {k 2 , k 2 + 1}

Let n be a nonzero integer. A set of m distinct positive integers is called a D(n)-m-tuple if the product of any two of them increased by n is a perfect square. Let k be a positive integer. In this paper, we show that if {k2, k2+1, c, d} is a D(−k2)-quadruple with c < d, then c = 1 and d = 4k2+1. This extends the work of the first author [20] and that of Dujella [4].

On the number of extensions of a Diophantine triple

A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by unity is a perfect square. Any Diophantine triple is conjectured to be uniquely extended to a Diophantine quadruple by joining an element exceeding the maximal element in the triple. A previous work of the second and third authors revealed that the number of such extensions for a fixed Diophantine triple is at most 11. In this paper, we show that the number is at most eight.

Torsion on elliptic curves in isogeny classes

Let be an elliptic curve over a number field and its -isogeny class. We are interested in determining the orders and the types of torsion groups in . For a prime , we give the range of possible types of -primary parts of when runs over . One of our results immediately gives a simple proof of a theorem of Katz on the order of maximal -primary torsion in .