Glasnik Matematicki, Vol. 50, No. 2 (2015), 261-268.

D(-1)-QUADRUPLES AND PRODUCTS OF TWO PRIMES

Anitha Srinivasan

Department of Mathematics, Saint Louis University-Madrid campus, Avenida del Valle 34, 28003 Madrid, Spain


Abstract.   A D(-1)-quadruple is a set of positive integers {a, b, c, d}, with a < b < c < d , such that the product of any two elements from this set is of the form 1+n2 for some integer n. Dujella and Fuchs showed that any such D(-1)-quadruple satisfies a=1. The D(-1) conjecture states that there is no D(-1)-quadruple. If b=1+r2, c=1+s2 and d=1+t2, then it is known that r, s, t, b, c and d are not of the form pk or 2pk, where p is an odd prime and k is a positive integer. In the case of two primes, we prove that if r=pq and v and w are integers such that p2v-q2w=1, then 4vw-1>r. A particular instance yields the result that if r=p(p+2) is a product of twin primes, where p ≡ 1 (mod 4), then the D(-1)-pair {1, 1+r2} cannot be extended to a D(-1)-quadruple. Dujella's conjecture states that there is at most one solution (x, y) in positive integers with y < k-1 to the diophantine equation x2-(1+k2)y2=k2. We show that the Dujella conjecture is true when k is a product of two odd primes. As a consequence it follows that if t is a product of two odd primes, then there is no D(-1)-quadruple {1, b, c, d} with d=1+t2.

2010 Mathematics Subject Classification.   11D09, 11R29, 11E16.

Key words and phrases.   Diophantine m-tuples, binary quadratic forms, quadratic diophantine equation.


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DOI: 10.3336/gm.50.2.01


References:

  1. A. Dujella, Diophantine m-tuples references (chronologically), http://web.math.pmf.unizg.hr/~duje/ref.html.

  2. A. Dujella and C. Fuchs, Complete solution of a problem of Diophantus and Euler, J. London Math. Soc. 71 (2005), 33-52.
    MathSciNet     CrossRef

  3. A. Dujella, A. Filipin and C. Fuchs, Effective solution of the D(-1)-quadruple conjecture, Acta Arith. 128 (2007), 319-338.
    MathSciNet     CrossRef

  4. A. Filipin and Y. Fujita, The number of D(-1)-quadruples, Math. Commun., 15 (2010), 387-391.
    MathSciNet    

  5. A. Filipin, Y. Fujita and M. Mignotte, The non-extendibility of some parametric families of D(-1)-triples, Q. J. Math. 63 (2012), 605-621.
    MathSciNet     CrossRef

  6. Y. Fujita, The non-extensibility of D(4k)-triples {1, 4k(k - 1), 4k2 + 1} with |k| prime, Glas. Mat. Ser. III 41 (2006), 205-216.
    MathSciNet     CrossRef

  7. G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Fifth edition. The Clarendon Press, Oxford University Press, New York, 1979.
    MathSciNet    

  8. V. A. Lebesgue, Sur l'impossibilité en nombres entiers de l'équation xm=y2+1, Nouv. Ann. Math. 9 (1850), 178-181.

  9. K. Matthews, J. Robertson and J. White, On a diophantine equation of Andrej Dujella, Glas. Mat. Ser. III 48 (2013), 265-289.
    MathSciNet     CrossRef

  10. T. Nagell, Introduction to Number Theory, Wiley, New York, 1951.
    MathSciNet    

  11. P. Ribenboim, My Numbers, My Friends, Popular Lectures on Number Theory, Springer-Verlag, New York, 2000.
    MathSciNet    

  12. K. Matthews and J. Robertson, http://www.numbertheory.org/pdfs/nagell2.pdf.

  13. J. Robertson, Fundamental solutions to generalized Pell equations, http://www.jpr2718.org/FundSoln.pdf

  14. A. Srinivasan, On the prime divisors of elements of a D(-1) quadruple, Glas. Mat. Ser. III 49 (2014), 275-285.
    MathSciNet     CrossRef

  15. I. M. Vinogradov, Elements of number theory, Dover Publications, New York, 1954.
    MathSciNet    

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