Diophantine M-tuple Research Articles

ERDŐS–TURÁN WITH A MOVING TARGET, EQUIDISTRIBUTION OF ROOTS OF REDUCIBLE QUADRATICS, AND DIOPHANTINE QUADRUPLES

A Diophantine m-tuple is a set A of m positive integers such that ab+1 is a perfect square for every pair a,b of distinct elements of A. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by x. In doing so, we extend two existing tools in ways that might be of independent interest. The Erdős–Turán inequality bounds the discrepancy between the number of elements of a sequence that lie in a particular interval modulo 1 and the expected number; we establish a version of this inequality where the interval is allowed to vary. We also adapt an argument of Hooley on the equidistribution of solutions of polynomial congruences to handle reducible quadratic polynomials.

Adjugates of Diophantine Quadruples

AbstractDiophantine

Any Diophantine quintuple contains a regular Diophantine quadruple

A set { a 1 , … , a m } of m distinct positive integers is called a Diophantine m-tuple if a i a j + 1 is a perfect square for all i, j with 1 ⩽ i < j ⩽ m . It is conjectured that if { a , b , c , d } is a Diophantine quadruple with a < b < c < d , then d = d + , where d + = a + b + c + 2 a b c + 2 r s t and r = a b + 1 , s = a c + 1 , t = b c + 1 . In this paper, we show that if { a , b , c , d , e } is a Diophantine quintuple with a < b < c < d < e , then d = d + .

On the number of Diophantine m-tuples

A set of m positive integers is called a Diophantine m-tuple if the product of any two of them is one less than a perfect square. It is known that there does not exist a Diophantine sextuple and that there are only finitely many Diophantine quintuples. On the other hand, there are infinitely many Diophantine m-tuples for m=2, 3 and 4.

The extensibility of Diophantine pairs [formula omitted

A set of m distinct positive integers { a 1 , … , a m } is called a Diophantine m-tuple if a i a j + 1 is a square for each 1 ⩽ i < j ⩽ m . In this paper, we show that for each integer k ⩾ 2 the Diophantine pair { k − 1 , k + 1 } cannot be extended to a Diophantine quintuple.

On a Problem of Diophantus with Polynomials

Let m>=2 and k>=2 be integers and let R be a commutative ring with a unit element denoted by 1. A k-th power diophantine m-tuple in R is an m-tuple (a_1, a_2, ..., a_m) of non-zero elements of R such that a_ia_j+1 is a k-th power of an element of R for 1 =3 and R=K[X], the ring of polynomials with coefficients in a field K of characteristic zero. We prove the following upper bounds on m, the size of diophantine m-tuple: m =5 ; m =8.

Diophantine m-tuples for linear polynomials II. Equal degrees

In this paper we prove the best possible upper bounds for the number of elements in a set of polynomials with integer coefficients all having the same degree, such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients. Moreover, we prove that there does not exist a set of more than 12 polynomials with integer coefficients and with the property from above. This significantly improves a recent result of the first two authors with Tichy [A. Dujella, C. Fuchs, R.F. Tichy, Diophantine m-tuples for linear polynomials, Period. Math. Hungar. 45 (2002) 21–33].

There 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 quintuple. In this paper, we prove that there does not exist a Diophantine sextuple and that there are only finitely many Diophantine quintuples.

On the size of Diophantine m-tuples

Let n be a nonzero integer. A set of m positive integers {a1, a2, …, am} is said to have the property D(n) if aiaj+n is a perfect square for all 1 [les ] i [les ] j [les ] m. Such a set is called a Diophantine m-tuple (with the property D(n)), or Pn-set of size m.Diophantus found the quadruple {1, 33, 68, 105} with the property D(256). The first Diophantine quadruple with the property D(1), the set {1, 3, 8, 120}, was found by Fermat (see [8, 9]). Baker and Davenport [3] proved that this Fermat’s set cannot be extended to the Diophantine quintuple, and a famous conjecture is that there does not exist a Diophantine quintuple with the property D(1). The theorem of Baker and Davenport has been recently generalized to several parametric families of quadruples [12, 14, 16], but the conjecture is still unproved.On the other hand, there are examples of Diophantine quintuples and sextuples like {1, 33, 105, 320, 18240} with the property D(256) [11] and {99, 315, 9920, 32768, 44460, 19534284} with the property D(2985984) [19]].

An Absolute Bound for the Size of Diophantine m-Tuples

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. We prove that if {a, b, c} is a Diophantine triple such that b>4a and c>max{b13, 1020} or c>max{b5, 101029}, then there is unique positive integer d such that d>c and {a, b, c, d} is a Diophantine quadruple. Furthermore, we prove that there does not exist a Diophantine 9-tuple and that there are only finitely many Diophantine 8-tuples.

Integer points on a family of elliptic curves

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. First example of a Diophantine quadruple is found by Fermat, and it was {1, 3, 8, 120} (see [6, p. 517]). In 1969, Baker and Davenport [2] proved that if d is a positive integer such that {1, 3, 8, d} is a Diophantine quadruple, then d has to be 120. Recently, in [9], we generalized this result to all Diophantine triples of the form {1, 3, c}. The fact that {1, 3, c} is a Diophantine triple implies that c = ck for some positive integer k, where the sequence (ck) is given by c0 = 0, c1 = 8, ck+2 = 14ck+1 − ck + 8, k ≥ 0. Let ck + 1 = sk, 3ck + 1 = t 2 k. It is easy to check that ck±1ck + 1 = (2ck ± sktk). The main result of [9] is the following theorem.

Diophantine Triples and Construction of High-Rank Elliptic Curves over $bf Q$ with Three Non-Trivial 2-Torsion Points

Using the theory of Diophantine m-tuples, i.e. sets with the property that the product of its any two distinct elements increased by 1 is a perfect square, we construct an elliptic curve over Q(t) of rank at least 4 with three non-trivial torsion points. By specialization, we obtain an example of elliptic curve over Q with torsion group Z/2Z * Z/2Z whose rank is equal 7.

A generalization of a theorem of Baker and Davenport

The Greek mathematician Diophantus of Alexandria 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 [4]). The first set of four positive integers with the above property was found by Fermat, and it was {1, 3, 8, 120}. A set of positive integers {a1, a2, . . . , am} is said to have the property of Diophantus if aiaj +1 is a perfect square for all 1 ≤ i < j ≤ m. Such a set is called a Diophantine m-tuple (or P1-set of size m). In 1969, Baker and Davenport [2] proved that if d is a positive integer such that {1, 3, 8, d} is a Diophantine quadruple, then d has to be 120. The same result was proved by Kanagasabapathy and Ponnudurai [9], Sansone [12] and Grinstead [7]. This result implies that the Diophantine triple {1, 3, 8} cannot be extended to a Diophantine quintuple. In the present paper we generalize the result of Baker and Davenport and prove that the Diophantine pair {1, 3} can be extended to infinitely many Diophantine quadruples, but it cannot be extended to a Diophantine quintuple.

On Diophantine quintuples

The Greek mathematician Diophantus of Alexandria noted that the set { 1 16 , 33 16 , 17 4 , 105 16 } has the following property: the product of any two of its distinct elements increased by 1 is a square of a rational number (see [5]). Fermat first found a set of four positive integers with the above property, and it was {1, 3, 8, 120}. Let n be an integer. A set of positive integers {x1, x2, . . . , xm} is said to have the property D(n) if for all 1 ≤ i < j ≤ m the following holds: xixj + n = y 2 ij , where yij is an integer. Such a set is called a Diophantine m-tuple. Davenport and Baker [4] showed that if d is a positive integer such that the set {1, 3, 8, d} has the property of Diophantus, then d has to be 120. This implies that the Diophantine quadruple {1, 3, 8, 120} cannot be extended to the Diophantine quintuple with the propertyD(1). Analogous result was proved for the Diophantine quadruple {2, 4, 12, 420} with the property D(1) [17], for the Diophantine quadruple {1, 5, 12, 96} with the propertyD(4) [15] and for the Diophantine quadruples {k−1, k+1, 4k, 16k3−4k} with the property D(1) for almost all positive integers k [9]. Euler proved that every Diophantine pair {x1, x2} with the property D(1) can be extended in infinitely many ways to the Diophantine quadruple with the same property (see [12]). In [6] it was proved that the same conclusion is valid for the pair with the property D(l2) if the additional condition that x1x2 is not a perfect square is fulfilled. Arkin, Hoggatt and Strauss [3] proved that every Diophantine triple with the property D(1) can be extended to the Diophantine quadruple. More precisely, if xixj + 1 = y 2 ij , then we can set x4 = x1 + x2 + x3 + 2x1x2x3+2y12y13y23. For the Diophantine quadruple obtained in this way, they proved the existence of a positive rational number x5 with the property that xix5 + 1 is a square of a rational number for i = 1, 2, 3, 4. Using this construction, in [2, 7, 8, 11] some formulas for Diophantine