(PDF) Conjectures and results on the size and number of Diophantine tuples

(PDF) Conjectures and results on the size and number of Diophantine tuples
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Conjectures and results on the size and number of Diophantine tuples
Andrej Dujella
2008, Nucleation and Atmospheric Aerosols
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Abstract
The problem of the construction of Diophantine m-tuples, i.e. sets with the property that the product of any two of its distinct elements is one less then a square, has a very long history. In this survey, we describe several conjectures and recent results concerning Diophantine m-tuples and their generalizations.
Related papers
On Diophantine m-tuples and D(n)-sets
Andrej Dujella
RIMS Kokyuroku, 2018
For a nonzero integer n, a set of distinct nonzero integers {a 1 , a 2 ,. .. , a m } such that a i a j + n is a perfect square for all 1 ≤ i < j ≤ m, is called a Diophantine m-tuple with the property D(n) or simply D(n)-set. Such sets have been studied since the ancient times. In this article, we give an overview of the results from the literature about D(n)-sets and summarize our recent findings about triples of integers which are D(n)-sets for several n's. Furthermore, we include some new observations and remarks about the ways to construct such triples.
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Diophantine m-tuples and generalizations
Andrej Dujella
2007
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On Diophantine $m$-tuples and $D(n)$-sets (Analytic Number Theory and Related Areas)
Andrej Dujella
数理解析研究所講究録 = RIMS Kokyuroku, 2018
For a nonzero integer n, a set of distinct nonzero integers \{a_{1} : a_{2} : a_{m}\} such that a_{i}a_{j}+n is a perfect square for all 1\leq iDownload free PDF
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Diophantine m-tuples for linear polynomials
Andrej Dujella
Periodica Mathematica Hungarica, 2002
In this paper, we prove that there does not exist a set with more than 26 polynomials with integer coefficients, such that the product of any two of them plus a linear polynomial is a square of a polynomial with integer coefficients.
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Diophantine m-tuples and elliptic curves
Andrej Dujella
J. Theor. Nombres Bordeaux, 2001
Diophantus found four positive rational numbers 1 16 , 33 16 , 17 4 , 105 16 with the property that the product of any two of them increased by 1 is a perfect square. The first set of four positive integers with the above property was found by Fermat and that set was {1, 3, 8, 120} (see ). These two examples motivate the following definition.
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Irregular Diophantine $m$-tuples and elliptic curves of high rank
Andrej Dujella
Proceedings of the Japan Academy. Series A, Mathematical sciences, 2000
A rational Diophantine m-tuple is a set of m nonzero rationals such that the product of any two of them is one less than a perfect square. In this paper we characterize the notions of regular Diophantine quadruples and quintuples, introduced by Gibbs, by means of elliptic curves. Motivated by these characterizations, we find examples of elliptic curves over Q with torsion group Z/2Z × Z/2Z and with rank equal 8. A set of m nonzero rationals {a 1 , a 2 , . . . , a m } is called a (rational) Diophantine m-tuple if a i a j + 1 is a perfect square for all 1 ≤ i < j ≤ m (see ). The first example of a Diophantine quadruple was the set { 1 16 , 33 16 , 17 4 , 105 16 } found by Diophantus (see ). Let {a, b, c} be a Diophantine triple and let Arkin, Hoggatt and Strauss [1] proved that ad + 1, bd + 1 and cd + 1 are perfect squares. Let {a, b, c, d} be a Diophantine quadruple such that abcd = 1 and let e = (a+b+c+d)(abcd + 1) + 2abc + 2abd + 2acd + 2bcd ± 2 (ab+1)(ac+1)(ad+1)(bc+1)(bd+1)(cd+1) (abcd -1) 2 . In we proved that ae + 1, be + 1, ce + 1 and de + 1 are perfect squares.
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A note on Diophantine quintuples
Andrej Dujella
De Gruyter eBooks, 2012
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Diophantine m-tuples for linear polynomials II. Equal degrees
Andrej Dujella
Journal of Number Theory, 2006
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 R. F. Tichy .
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Some estimates of the number of Diophantine quadruples
Andrej Dujella
Publicationes Mathematicae Debrecen, 1998
A Diophantine m-tuple with the property D(n), where n is an integer, is defined as a set of m positive integers such that the product of its any two distinct elements increased by n is a perfect square. In the present paper we show that if |n| is sufficiently large and n ≡ 1 (mod 8), or n ≡ 4 (mod 32), or n ≡ 0 (mod 16), then there exist at least six, and if n ≡ 8 (mod 16), or n ≡ 13, 21 (mod 24), or n ≡ 3, 7 (mod 12), then there exist at least four distinct Diophantine quadruples with the property D(n).
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On Diophantine quintuples
Andrej Dujella
Acta Arithmetica, 1997
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Conjectures and results on the size and number of
Diophantine tuples
Andrej Dujella
Department of Mathematics, University of Zagreb
Bijenička cesta 30, 10000 Zagreb, CROATIA
E-mail:
[email protected]
URL : http://web.math.hr/~duje/

Abstract
The problem of the construction of Diophantine m-tuples, i.e. sets
with the property that the product of any two of its distinct elements is
one less then a square, has a very long history. In this survey, we describe
several conjectures and recent results concerning Diophantine m-tuples
and their generalizations.

Diophantine quintuple conjecture

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 himself
found a set of four positive rationals with this property:
1 33 17 105
, , ,
16 16 4 16
However, the first Diophantine quadruple, the set {1, 3, 8, 120}, was found by
Fermat. Euler found an infinite family of such sets:
{a, b, a + b + 2r, 4r(r + a)(r + b)} ,
where ab+1 = r2 . He was also able to add the fifth positive rational, 777480/8288641,
to the Fermat’s set (see [5, 6, 26]). Recently, Gibbs [24] found several examples
of sets of six positive rationals with the property of Diophantus. The first one
was
11 35 155 512 1235 180873
192 192 27 27
48
16
A folklore conjecture is that there does not exist a Diophantine quintuple.
The first important result concerning this conjecture was proved in 1969 by
Baker and Davenport [2]. They proved that if d is a positive integer such that

{1, 3, 8, d} forms a Diophantine quadruple, then d = 120. This problem was
stated in 1967 by Gardner [23] (see also [27]). Furthemore, in 1998, in the joint
work with Attila Pethő [17] we proved that the pair {1, 3} cannot be extended
to a Diophantine quintuple.
In 1979, Arkin, Hoggatt and Strauss [1] proved that every Diophantine triple
can be extended to a Diophantine quadruple. More precisely, let {a, b, c} be a
Diophantine triple and ab + 1 = r2 , ac + 1 = s2 , bc + 1 = t2 , where r, s, t are
positive integers. Define
d+ = a + b + c + 2abc + 2rst.
Then {a, b, c, d+ } is a Diophantine quadruple. A stronger version of the Diophantine quintuple conjecture states that if {a, b, c, d} is a Diophantine quadruple and d > max{a, b, c}, then d = d+ . Diophantine quadruples of this form are
called regular.
In 2004, we proved that there does not exist a Diophantine sextuple and
there are only finitely many Diophantine quintuples (see [10]). However, the
bounds for the size of the elements of a (hypothetical) Diophantine quintuple
26
are huge (largest element is less than 1010 ), so the remaining cases cannot be
checked on a computer.
Recently, Fujita [22] proved that if {a, b, c, d, e} (a < b < c < d < e) is
a Diophantine quintuple, then {a, b, c, d} is a regular Diophantine quadruple.
Thus, in order to prove the Diophantine quintuple conjecture, it remains to prove
that a regular Diophantine quadruple cannot be extended to a quintuple. Such
result is known to be true for several parametric families of regular Diophantine
quadruples, e.g. {k − 1, k + 1, 4k, 16k 3 − 4k}. Moreover, Fujita [21] proved that
the pair {k−1, k+1} (for k ≥ 2) cannot be extended to a Diophantine quintuple,
and his results, together with our joint work with Yann Bugeaud and Maurice
Mignotte [4], show that all Diophantine quadruples of the form {k −1, k +1, c, d}
are regular.

The existence of Diophantine quadruples with
the property D(n)

A natural generalization of the original problem of Diophantus and Fermat is
to replace number 1, in the definition of Diophantine m-tuples, by an arbitrary
integer n. A set of m positive integers {a1 , a2 , . . . , am } is said to have the
property D(n) if ai aj + n is a perfect square for all 1 ≤ i < j ≤ m. Such a set
is called a Diophantine m-tuple with the property D(n) (or D(n)-m-tuple, or
Pn -set of size m).
Several authors considered the problem of the existence of Diophantine
quadruples with the property D(n). This problem is now almost completely
solved. In 1985, Brown [3] (see also [25, 28]) gave the first part of the answer by
showing that if n is an integer of the form n = 4k +2, then there does not exist a
Diophantine quadruple with the property D(n). In 1993, we were able to prove

that if n 6≡ 2 (mod 4) and n ∈
/ S = {−4, −3, −1, 3, 5, 12, 20}, then there exists
at least one Diophantine quadruple with the property D(n) (see [7]). The conjecture is that for n ∈ S there does not exist a Diophantine quadruple with the
property D(n). It is interesting to observe that the integers 4k + 2 are exactly
those integers which are not representable as differences of the squares of two
integers. It seems that this is not just a coincidence. Namely, analogous results,
which show strong connection between the existence of D(n)-quadruples and
the representability as a difference of two squares, also hold for integers in some
quadratic fields (see [8, 15, 19, 20]).
It is clear that if n = m2 is a perfect square, than there exist infinitely
many D(m2 )-quadruples. Namely, Euler’s result mentioned above shows that
there are infinitely many D(1)-quadruples, and multiplying their elements by
m we obtain D(m2 )-quadruples. We state the following conjecture: if n is not
a perfect square, then there exist only finitely many D(n)-quadruples. As we
already mentioned, it is easy to verify the conjecture in case n ≡ 2 (mod 4) (then
there does not exist a D(n)-quadruple). In the recent joint work with Clemens
Fuchs and Alan Filipin, we have proved this conjecture in cases n = −1 and
n = −4 (see [14, 16]). Perhaps some support to this conjecture may come from
considering the number of D(n)-triples in given range. Let
Dm (n; N ) = | {D ⊆ {1, 2, . . . , N } : D is a D(n)-m-tuple } |.
In [12], we considered the case n = 1 and proved that D3 (1; N ) = π32 N log N +
O(N ). In our forthcoming paper [18], we will show that D3 (n; N ) ∼ C(n)N log(N )
if n is a perfect square, while D3 (n; N ) ∼ C(n)N otherwise.
Concerning rational Diophantine m-tuples, it is expected that there exist an
absolute upper bound for their size. Such a result will follow from the Lang
conjecture on varieties of general type. Related problem is to find an upper
bound Mn for the size of D(n)-tuples (for given non-zero integer n). Again,
the Lang conjecture implies that there exist an absolute upper bound for Mn
(independent on n). However, at present, the best known upper bounds are of
the shape Mn < c log |n| (see [9, 11]). Recently, in our joint paper with Florian
Luca [13], we were able to obtain an absolute upper bound for Mp , where p is
a prime.

References
[1] J. Arkin, V. E. Hoggatt and E. G. Strauss, On Euler’s solution of a problem
of Diophantus, Fibonacci Quart. 17, 333–339 (1979).
[2] A. Baker and H. Davenport, The equations 3x2 − 2 = y 2 and 8x2 − 7 = z 2 ,
Quart. J. Math. Oxford Ser. (2) 20, 129–137 (1969).
[3] E. Brown, Sets in which xy + k is always a square, Math. Comp. 45, 613–
620 (1985).

[4] Y. Bugeaud, A. Dujella and M. Mignotte, On the family of Diophantine
triples {k − 1, k + 1, 16k 3 − 4k}, Glasgow Math. J. 49, 333–344 (2007).
[5] L. E. Dickson, History of the Theory of Numbers, Vol. 2, Chelsea, New
York, 1966, pp. 513–520.
[6] Diophantus of Alexandria, Arithmetics and the Book of Polygonal Numbers,
(ed. I. G. Bashmakova), Nauka, Moscow, 1974 (in Russian), pp. 103–104,
232.
[7] A. Dujella, Generalization of a problem of Diophantus, Acta Arith. 65,
15–27 (1993).
[8] A. Dujella, The problem of Diophantus and Davenport for Gaussian integers, Glas. Mat. Ser. III 32, 1–10 (1997).
[9] A. Dujella, On the size of Diophantine m-tuples, Math. Proc. Cambridge
Philos. Soc. 132, 23–33 (2002).
[10] A. Dujella, There are only finitely many Diophantine quintuples, J. Reine
Angew. Math. 566, 183–214 (2004).
[11] A. Dujella, Bounds for the size of sets with the property D(n), Glas. Mat.
Ser. III 39, 199–205 (2004).
[12] A. Dujella, On the number of Diophantine m-tuples, Ramanujan J., to
appear.
[13] A. Dujella and F. Luca, Diophantine m-tuples for primes, Intern. Math.
Research Notices 47, 2913–2940 (2005).
[14] A. Dujella, A. Filipin and C. Fuchs, Effective solution of the D(−1)quadruple conjecture, Acta Arith. 128, 319–338 (2007).
[15] A. Dujella and Z. Franušić, On differences of two squares in some quadratic
fields, Rocky Mountain J. Math. 37, 429–453 (2007).
[16] A. Dujella and C. Fuchs, Complete solution of a problem of Diophantus
and Euler, J. London Math. Soc. 71, 33–52 (2005).
[17] A. Dujella and A. Pethő, Generalization of a theorem of Baker and Davenport, Quart. J. Math. Oxford Ser. (2) 49, 291–306 (1998).
[18] A. Dujella and A. Pethő, Asymptotic estimates for the number of Diophantine pairs and triples, in preparation.
[19] Z. Franušić, Diophantine quadruples in Z[ 4k + 3], Ramanujan J., to appear.
[20] Z. Franušić, A Diophantine problem in Z[(1 + d)/2], Studia Sci. Math.
Hungar., to appear.

[21] Y. Fujita, The extensibility of Diophantine pairs {k − 1, k + 1}, J. Number
Theory, to appear.
[22] Y. Fujita, Any Diophantine quintuple contains a regular Diophantine
quadruple, preprint.
[23] M. Gardner, Mathematical divertions, Scientific American 216, 124 (1967).
[24] P. Gibbs, Some rational Diophantine sextuples, Glas. Mat. Ser. III 41,
195–203 (2006).
[25] H. Gupta and K. Singh, On k-triad sequences, Internat. J. Math. Math.
Sci. 5, 799–804 (1985).
[26] T. L. Heath, Diophantus of Alexandria. A Study of the History of Greek Algebra, Powell’s Bookstore, Chicago; Martino Publishing, Mansfield Center,
2003. pp. 177–181.
[27] J. H. van Lint, On a set of diophantine equations, T. H.-Report 68 – WSK03, Technological University Eindhoven, 1968.
[28] S. P. Mohanty and A. M. S. Ramasamy, On Pr,k sequences, Fibonacci
Quart. 23, 36–44 (1985).

References (29)
J. Arkin, V. E. Hoggatt and E. G. Strauss, On Euler's solution of a problem of Diophantus, Fibonacci Quart. 17, 333-339 (1979).
A. Baker and H. Davenport, The equations 3x 2 -2 = y 2 and 8x 2 -7 = z 2 , Quart. J. Math. Oxford Ser. (2) 20, 129-137 (1969).
E. Brown, Sets in which xy + k is always a square, Math. Comp. 45, 613- 620 (1985).
Y. Bugeaud, A. Dujella and M. Mignotte, On the family of Diophantine triples {k -1, k + 1, 16k 3 -4k}, Glasgow Math. J. 49, 333-344 (2007).
L. E. Dickson, History of the Theory of Numbers, Vol. 2, Chelsea, New York, 1966, pp. 513-520.
Diophantus of Alexandria, Arithmetics and the Book of Polygonal Numbers, (ed. I. G. Bashmakova), Nauka, Moscow, 1974 (in Russian), pp. 103-104, 232.
A. Dujella, Generalization of a problem of Diophantus, Acta Arith. 65, 15-27 (1993).
A. Dujella, The problem of Diophantus and Davenport for Gaussian inte- gers, Glas. Mat. Ser. III 32, 1-10 (1997).
A. Dujella, On the size of Diophantine m-tuples, Math. Proc. Cambridge Philos. Soc. 132, 23-33 (2002).
A. Dujella, There are only finitely many Diophantine quintuples, J. Reine Angew. Math. 566, 183-214 (2004).
A. Dujella, Bounds for the size of sets with the property D(n), Glas. Mat. Ser. III 39, 199-205 (2004).
A. Dujella, On the number of Diophantine m-tuples, Ramanujan J., to appear.
A. Dujella and F. Luca, Diophantine m-tuples for primes, Intern. Math. Research Notices 47, 2913-2940 (2005).
A. Dujella, A. Filipin and C. Fuchs, Effective solution of the D(-1)- quadruple conjecture, Acta Arith. 128, 319-338 (2007).
A. Dujella and Z. Franušić, On differences of two squares in some quadratic fields, Rocky Mountain J. Math. 37, 429-453 (2007).
A. Dujella and C. Fuchs, Complete solution of a problem of Diophantus and Euler, J. London Math. Soc. 71, 33-52 (2005).
A. Dujella and A. Pethő, Generalization of a theorem of Baker and Dav- enport, Quart. J. Math. Oxford Ser. (2) 49, 291-306 (1998).
A. Dujella and A. Pethő, Asymptotic estimates for the number of Diophan- tine pairs and triples, in preparation.
Z. Franušić, Diophantine quadruples in Z[ √ 4k + 3], Ramanujan J., to ap- pear.
Z. Franušić, A Diophantine problem in Z[(1 + √ d)/2], Studia Sci. Math.
Hungar., to appear.
Y. Fujita, The extensibility of Diophantine pairs {k -1, k + 1}, J. Number Theory, to appear.
Y. Fujita, Any Diophantine quintuple contains a regular Diophantine quadruple, preprint.
M. Gardner, Mathematical divertions, Scientific American 216, 124 (1967).
P. Gibbs, Some rational Diophantine sextuples, Glas. Mat. Ser. III 41, 195-203 (2006).
H. Gupta and K. Singh, On k-triad sequences, Internat. J. Math. Math. Sci. 5, 799-804 (1985).
T. L. Heath, Diophantus of Alexandria. A Study of the History of Greek Al- gebra, Powell's Bookstore, Chicago; Martino Publishing, Mansfield Center, 2003. pp. 177-181.
J. H. van Lint, On a set of diophantine equations, T. H.-Report 68 -WSK- 03, Technological University Eindhoven, 1968.
S. P. Mohanty and A. M. S. Ramasamy, On P r,k sequences, Fibonacci Quart. 23, 36-44 (1985).
Andrej Dujella
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On the size of Diophantine m-tuples
Andrej Dujella
Mathematical proceedings of the Cambridge Philosophical Society, 2002
Let n be a nonzero integer and assume that a set S of positive integers has the property that xy + n is a perfect square whenever x and y are distinct elements of S. In this paper we find some upper bounds for the size of the set S. We prove that if |n| ≤ 400 then |S| ≤ 32, and if |n| > 400 then |S| < 267.81 log |n| (log log |n|) 2 . The question whether there exists an absolute bound (independent on n) for |S| still remains open.
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On the number of Diophantine m-tuples
Andrej Dujella
The Ramanujan Journal, 2008
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. In this paper, we derive asymptotic estimates for the number of Diophantine pairs, triples and quadruples with elements less than given positive integer N .
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On the size of Diophantine m-tuples
Andrej Dujella
Mathematical Proceedings of the Cambridge …, 2002
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Diophantine m-tuples for primes
Andrej Dujella
International mathematics research notices, 2005
In this paper, we show that if p is a prime and if A = {a 1 , a 2 , . . . , a m } is a set of positive integers with the property that a i a j + p is a perfect square for all 1 ≤ i < j ≤ m, then m < 3 · 2 168 . More generally, when p is replaced by a squarefree integer n, the inequality m ≤ f (ω(n)) holds with some function f , where ω(n) is the number of prime divisors of n. We also give upper bounds for m when p is replaced by an arbitrary integer which hold on a set of n of asymptotic density one. valuable comments. This paper was written during a visit of the second author at the University of Zagreb in October of 2004. He warmly thanks this University for its hospitality. Both authors were partly supported by the Croatian Ministry of Science, Education and Sport Grant 0037110.
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An Absolute Bound for the Size of Diophantine m-Tuples
Andrej Dujella
Journal of Number Theory, 2001
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{b 13 , 10 20 } or c > max{b 5 , 10 1029 }, 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.
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An extension of an old problem of Diophantus and Euler. II
Andrej Dujella
Fibonacci Quarterly, 2002
Diophantus found three rationals 3 10 , 21 5 , 7 10 with the property that the product of any two of them increased by the sum of those two gives a perfect square (see [5, pp. 85-86, 215-217]), and Euler found four rationals 65 224 , 9 224 , 9 56 , 5 2 with the same property (see [4, pp. 518-519]). A set {x 1 , x 2 ,. .. , x m } of m rationals such that x i x j + x i + x j is a perfect square for all 1 ≤ i < j ≤ m we will call an Eulerian m-tuple. In [8] we found the Eulerian quintuple
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Diophantinem-tuples with elements in arithmetic progressions
Andrej Dujella
Indagationes Mathematicae, 2014
In this paper, we consider the problem of existence of Diophantine m-tuples which are (not necessarily consecutive) elements of an arithmetic progression. We show that for n ≥ 3 there does not exist a Diophantine quintuple {a, b, c, d, e} such that a ≡ b ≡ c ≡ d ≡ e (mod n). On the other hand, for any positive integer n there exist infinitely many Diophantine triples {a, b, c} such that a ≡ b ≡ c ≡ 0 (mod n).
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Diophantine m-tuples in finite fields and modular forms
Andrej Dujella
Research in number theory, 2021
For a prime p, a Diophantine m-tuple in F p is a set of m nonzero elements of F p with the property that the product of any two of its distinct elements is one less than a square. In this paper, we present formulas for the number N (m) (p) of Diophantine m-tuples in F p for m = 2, 3 and 4. Fourier coefficients of certain modular forms appear in the formula for the number of Diophantine quadruples. We prove that asymptotically N (m) (p) = 1 2 (m 2) p m m! + o(p m), and also show that if p > 2 2m−2 m 2 , then there is at least one Diophantine m-tuple in F p .
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On some particular regular Diophantine 3-tuples
Özen ÖZER
Mathematics in Natural Science
Diophantine n-tuple where n=3 is called as a Diophantine triple. It means that Diophantine triple is a set of three positive integers satisfying special condition. For example, {a, b, c} is called a D(k)-Diophantine triple if multiplying of any two different of them plus k is a perfect square integer where k is an integer. In this work, we take in consideration some kind of regular D(±3 3)-Diophantine triples. We demonstrate that such sets can not be extendible to D(±3 3)-Diophantine quadruple by using algebraic methods such as classical Pell equations solutions, solutions of ux 2 + vy 2 = w Diophantine equations where u, v, w ∈ Z, factorization in the set of integers, and so on. Besides, we obtain some notable characteristic properties for such sets.
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