Purdue University Northwest > Personal Faculty Pages > Alain Togbe > Research

Research

View a selection of research from Alain Togbe, professor of Mathematics at Purdue University Northwest.

Conferences

Conference on Diophantine m-tuples and related problems

African School of Mathematics in Cape Verde

Conference on Diophantine m-tuples and related problems II

Conference on Diophantine m-tuples and related problems III

Publications

  1. (with B. Kafle, A. Srinivasan), Markoff Equation with Pell components. Accepted to appear in The Fibonacci Quarterly.
  2. (with S. Rihane, B. Faye, F. Luca), Powers two in Generalized Lucas sequences. Accepted to appear in The Fibonacci Quarterly.
  3. (with A. Noubissie, Z. Zhang), On the Exponential Diophantine Equation (an-1)(bn-1)=x2. Accepted to appear in the Bulletin of the Belgian Mathematical Society – Simon Stevin.
  4. (with C. Adegbindin), Can a Lucas number be a sum of three repdigits? Accepted to appear in the Commentationes Mathematicae Universitatis Carolinae.
  5. (with C. Adegbindin, F. Luca), Pell and Pell-Lucas numbers as sums of two repdigits. Accepted to appear in Bulletin of the Malaysian Mathematical Sciences Society.
  6. (with B. Kafle, F. Luca), Lucas Factoriangular numbers. Accepted to appear in Mathematica Bohemica.
  7. (with Eric Bravo, Carlos Gomez, Bir Kafle, F. Luca), On a conjecture concerning the multiplicity of the Tribonacci sequence. Accepted to appear in Colloquium Mathematicum, 159 (2020), 61-69.
  8. (with S. Rihane, B. Faye, F. Luca), An exponential Diophantine equation related to the difference between powers of two consecutive Balancing numbers. Mathematicae et Informaticae, 50 (2019), 167-177.
  9. (with A. Noubissie), A note on the Exponential Diophantine Equation (an-1)(bn-1)=x2. Mathematicae et Informaticae, 50 (2019), 159-165.
  10. (with F. Luca, V. Togan), On the X-coordinates of Pell equations which are rep-digits, II. Annales Mathematicae et Informaticae, 50 (2019), 131-144.
  11. (with B. V. Normenyo, B. Kafle), Repdigits as sums of two Fibonacci numbers and two Lucas numbers. Integers Journal 19 (2019), #55.
  12. (with B. Kafle, F. Luca), x-coordinates of Pell equations which are Lucas numbers. Boletín de la Sociedad Matemática Mexicana (2019) 25:481–493.
  13. (with B. Kafle, F. Luca), x-coordinates of Pell equations which are Tribonacci numbers II. Periodica Mathematica Hungarica (2019) 79:157–167.
  14. (with C. Adegbindin, F. Luca), Lucas numbers as sums of two repdigits. Lithuanian Mathematical Journal, Vol. 59, No. 3, July 2019, pp. 295–304.
  15. (with S. Rihane, B. Faye, F. Luca), On the exponential equation. Turk J Math, 43, (2019), 1640-1649.
  16. (with B. Kafle, F. Luca, A. Montejano, L. Szalay), On the X-coordinates of Pell equations which are products of two Fibonacci numbers. Journal of Number Theory, 203 (2019), 347-387.
  17. (with Y. Bilu, M. Diego), Generalized Cullen Numbers in Linear Recurrence Sequences. Journal of Number Theory, 202 (2019), 412-425.
  18. (with B. He, J. Odjoumani), On a class of quartic Thue equations with three parameters. Journal of Number Theory, 202 (2019), 310-333.
  19. (with B. V. Normenyo, F. Luca), Repdigits as sums of four Pell numbers. Boletín de la Sociedad Matemática Mexicana (2019) 25, 249–266.
  20. (with S. Yang, F. Luca), On a Divisibility Problem. Mathematica Bohemica 144 (2) (2019), 125-135.
  21. (with F. Luca, J. Odjoumani), Pell Factoriangular numbers. Inst. Math. Beograd. Nouvelle série, tome 105(119) (2019), 93–100.
  22. (with S. Rihane, M. Hernane), On Diophantine triples of Pell numbers. Colloquium Mathematicum 156 (2019). 273-285.
  23. (with B. Normenyo, F. Luca) Repdigits as sums of three Lucas number. Colloquium Mathematicum 156 (2019). 255-285.
  24. (with B. He, V. Ziegler), There is no Diophantine Quintuple. Transactions of the American Mathematical Society, Volume 371, Number 9, May 2019, Pages 6665–6709.
  25. (with S. Subburam), On the Diophantine equation yn= f(x)/g(x). Acta Mathematica Hungarica, 157(1), (2019) 1-9.
  26. (with J. Odjoumani, V. Ziegler), On a family of biquadratic fields that do not admit a unit powerintegral basis. Publicationes Mathematicae Debrecen, 2019 / 94 / 1-2 (1).
  27. (with Z. Zhang), Perfect Powers that are sums of two powers of Fibonacci numbers. The Bulletin of the Australian Mathematical Society, Volume 99, Issue 1 February 2019, pp. 34-41.
  28. (with S. Rihane, M. Hernane), The x-coordinates of Pell equations and Padovan numbers. Turk J Math, 43, (2019), 207-223.
  29. (with B. He, K. Pu, R. Shen), A note on the regularity of the Diophantine pair {k, 4k4}. Journal de Théorie des Nombres de Bordeaux 30  3 (2018), p. 879-892.
  30. (with B. Kafle, F. Luca), Triangular repblocks. The Fibonacci Quarterly 56 (2018), no. 4, 325–328.
  31. (with S. Rihane, M. Hernane, F. Luca), On Pillai’s problem with Pell numbers and powers of 2. Hardy-Ramanujan Journal 41 (2018), 8-17.
  32. (with Z. Zhang), On the Ramanujan-Nagell type Diophantine equations x2 + Akn =B II. Glas. Mat. Ser. II, Vol. 53 No 2, (2018) 221-228.
  33. (with B. V. Normenyo, F. Luca), Repdigits as sums of four Fibonacci or Lucas numbers. Journal of Integer Sequences, 21 (2018), Article 18.7.7.
  34. (with B. Normenyo, F. Luca), Repdigits as sums of three Pell numbers. Periodica Mathematica Hungarica, December 2018, Volume 77, Issue 2, pp 318–328.
  35. (with N. Irmak), On repdigits as product of consecutive Lucas Numbers. The Notes on Number Theory and Discrete Mathematics, Vol 24, 2018, No. 3, 95-102.
  36. (with Z. Zhang), On the Ramanujan-Nagell type Diophantine equations x2 + Akn =B. Glas. Mat. Ser. II, Vol. 53 No 1, (2018) 43-50.
  37. (with S. E. Rihane, M. O. Hernane), On D(4)-Diophantine triples of Fibonacci numbers. The Fibonacci Quarterly, 56 (2018), no. 1, 63-74.
  38. (with F. Luca), On the x-coordinates of Pell equations which are Fibonacci numbers. Mathematica  Scandinavica, Vol. 122, No. 1 (2018) 18-30.
  39. (with B. He, A. Pinter, S. Yang), Another generalization of a theorem of Baker and Davenport. Number Theory 182 (2018), 325-343.
  40. (with  A. Bayad, A. Dossavi-Yovo, A. Filipin), On the extensibility of the D(4)-triple {k-2, k+2, 4k} over Gaussian integers. Notes on Number Theory and Discrete Mathematics, Vol. 23, 2017, No. 3, 1–26.
  41. (with L. Tan, V. Vijayarajan, N. Chimitt, J. Jiang), Channel sparsity-aware recursive least squares algorithms for nonlinear system modeling and active noise control. 2017 IEEE Ubiquitous Computing, Electronics and Mobile Communications (UEMCON 2017), pp. 225-231, Columbia University, New York City, NY, October 2017.
  42. (with F. Luca, A. Montejano, L. Szalay), On the X-coordinates of Pell equations which are Tribonacci numbers. Acta Arithmetica, 179.1 (2017), 25-35.
  43. (with Nicholas Chimitt, William Misch, Li Tan, and Jean Jiang), Comparative Study of Simple Feature Extraction for Single-Channel EEG Based Classification. 2017 IEEE International Conference on Electro/Information Technology, pp. 166-170, University of Nebraska, Lincoln, Nebraska, May 2017.
  44. (with G. Soydan, M. Demirci, I. Cangul), On the conjecture of Jesmanowicz. J. Appl. Math. Stat., Vol. 56, Issue No. 6, Year 2017.
  45. (with B. Kafle, F. Luca), On the x-coordinates of Pell equations which are Fibonacci numbers II. Colloquium Mathematicum 149 (2017), 75-85.
  46. (with A. Dossavi-Yovo, B. He), The extensibility of the D(k)-triple  {k1,k, 4k1}.Afrika Matematika Volume 28, Issue 3-4, June 2017, 563-574.
  47. (with S. Subburam), On the Diophantine equation ax3 + by + c – xyz = 0. Accepted to appear in Afrika Matematika, Volume 28, Issue 1-2, March 2017, 9-21.
  48. (with A. Bayad, M. O. Hernane), On extended Eulerian numbers. Accepted to appear in Functiones et Approximatio Commentarii Mathematici, Volume 55, Number 1 (2016), 113-130.
  49. (with S. Yang), A note of three prime representation problem. Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 32, No. 1, pp. 23-31 (2016).
  50. (with A. Bayad, A. Filipin), Extension of a parametric family of Diophantine triples in Gaussian integers. Acta Math. Hungar, Volume 148, Issue 2, April 2016, pp 312–327.
  51. (with A. Dossavi-Yovo, F. Luca), On the x-coordinates of Pell equations which are rep-digits. Publicationes Mathematicae Debrecen, 2016 / 88 / 3-4 (9).
  52. (with B. He, F. Luca), Diophantine triples of Fibonacci. Acta Arithmetica 175 (2016), 57-70.
  53. (with Z. Zhang), On two Diophantine equations of Ramanujan-Nagell type. Glas. Mat. Ser. III, 51, No.1 (2016), 17-22.
  54. (with S. Yang), On the estimates of the upper and lower bounds of Ramanujan primes. The Ramanujan Journal, June 2016, Volume 40, Issue 2, pp 245–255.
  55. (with S. Subburam), On Erdos-Straus Conjecture. Period Math Hung, March 2016, Volume 72, Issue 1, pp 43–49.
  56. (with T. Miyazaki, P. Yuan), On the Diophantine equation ax+by=(a+2)z. Acta Mathematica Hungarica, Volume 149, Issue 1, June 2016, 1-9.
  57. (with Godinho, D. Marques), On the Diophantine equation x2 + C = yn for C =2a 3b 17c and C =2a 13b 17c. Mathematica Slovaca, Volume 66, Issue 3 (June 2016), 565–574.
  58. (with Jones, D. Marques), On terms of Lucas sequences with only one distinct digit. Indian Journal of Mathematics, Volume 57, No. 2, 2015, 151-164.
  59. (with B. He, A. Pinter), On simultaneous Pell equations and related Thue equations. Proc. Amer. Math. Soc. 143 (2015), 4685-4693.
  60. (With H. Zhu, M. Le, G. Soydan), On the exponential Diophantine equation x2 + 2a pb = yn. Periodica Mathematica Hungarica, June 2015, Volume 70, Issue 2, 233-247.
  61. (with B. He, A. Pinter, N.Varga), A generalization of a problem of Mordell. Glas. Mat. Ser. III,Vol. 50 (70) (2015), 35 – 41.
  62. (with M. Le, H. Zhu), On a pure ternary Diophantine equation. Publicationes Mathematicae Debrecen, 2014 / 85 / 3-4 (10).
  63. (with A. Filipin, Y. Fujita), The extendibility of Diophantine pairs II: Examples. Journal of Number Theory, Volume 145 (December 2014), 604-631.
  64. (with S. Yang), Proof of the P-integer conjecture of Pomerance. Journal of Number Theory, Volume 140 (July 2014), 226-234.
  65. (with H. Godinho, T. Porto), On the Diophantine Equation v(v+1)=u(u+a)(u+2a). Acta Math. Hungar. 143 (1) (2014), 249–268.
  66. (with A. Filipin, Y. Fujita ), The extendibility of Diophantine pairs I: the general case. Glas. Mat. Ser. III, Vol. 49, No. 1 (2014), 25-36.
  67. Note on the Diophantine equation y2 =px(Ax2 +2). Afrika Mathematika. (2014) 25, 739–744.
  68. (with B. He, S. Yang, W. Wu), The solution number of the Diophantine equations y2=nx(Ax2±C). The South Pacific Journal of Pure and Applied Mathematics, Vol. 02, No. 02, 2013, 1-13.
  69. (with S. Yang, W. Wu), Diophantine equations involving normalized binomial mid-coefficients. Publicationes Mathematicae Debrecen 2013 / 83 / 3 (11).
  70. (with B. He, I. Pink, A. Pinter), On the Diophantine inequality |X2-cXY2+Y4|≤ c+2. Glas. Mat. Ser. III 48(68) (2013) 291-299.
  71. (with S. Yang, B. He), On the solutions of the exponential Diophantine equation ax+ by =(m2+1)z. Quaestiones Mathematicae, 36 (2013), 119-135.
  72. (with D. Marques), Fibonacci and Lucas numbers of the form 2a + 3b + 5c. Proceedings of Japan Mathematics Academy, Ser. A . Proceedings of Japan Mathematics Academy, 89 Ser. A (2013) 47-50.
  73. (with H. Godinho, D. Marques), On the Diophantine equation x2 + 2α 5β 17γ= yn. Communications in Mathematics journal, 20 (2012) 81-88.
  74. (with D. Marques), On repdigits as product of consecutive Fibonacci numbers. Rendiconti dell’ Istituto di Matematica dell’ Universita’ di Trieste, Volume 44 (2012), 393-397.
  75. (with Y. Fujita), The extension of the D(-k2)-pair {k2, k2+1}. Periodica Mathematica Hungarica, Volume 65, Issue 1, September 2012, 75-81.
  76. (with T. Miyazaki), The Diophantine equation (2am-1)x+ (2m)y =(2am+1)z. International Journal of Number Theory, Vol. 08, No. 08, (2012) 2035-2044.
  77. (with M. Le, H. Zhu), On the exponential Diophantine equation x2+  p2m = 2yn. The Bulletin of the Australian Mathematical Society, Volume 86, Issue 02, October 2012, 303-314.
  78. (with D. Marques), On the sum of powers of consecutive terms of a linear recurrence sequence. Bull Braz Math Soc., New Series 43 (3), (2012) 397-406.
  79. (with A. Filipin, B. He), On a family of two-parametric D(4)-triples. Glas. Mat. Ser. III, Vol. 47, No.1 (2012), 31-51.
  80. (with B. He), On a family of Diophantine triples {k, A2k+2A ,(A+1)2k+2(A+1)} with two parameters II.  Periodica Mathematica Hungarica Vol. 64 (1), 2012, 1-10.
  81. (with D. Marques), A Diophantine equation involving C-nomial coefficients. East-West J. of Mathematics, Vol. 13, No 1 (2011) 75-79
  82. (with Y. Fujita), Uniqueness of the extension of the D(4k2)-triple {k2-4, k2, 4k2-4}. NNTDM 17 (2011), 4, 42-49.
  83. (with A. Bajolet, B. Dupuy, F. Luca), On the Diophantine equation x4– q4 = pyr. Publicationes Mathematicae Debrecen, 79/3-4 (2011) 269-282.
  84. (with P. Yuan), On a variant of a Diophantine equation of Cassels. Glas. Mat. Ser. III, Vol. 46, No.2 (2011), 325-331.
  85. (with D. Marques), On terms of linear recurrence sequences with only one distinct block of digits. Colloquium Mathematicum 124 (2011), 145-155.
  86. (with B. He, O. Kihel), Solutions of a class of quartic Thue inequalities.Computers and Mathematics with Applications 61 (2011) 2914–2923.
  87. (with M. Cipu, M. Mignotte), On the size of the intersection of two Lucas sequences of Distinct type II. Science China Mathematics 54 (2011) 1299-1316.
  88. (with K, A. Broughan, M. J. Gonzalez, R. H. Lewis, F. Luca, J. M. Huguet), There are no multiply-perfect Fibonacci numbers. INTEGERS 11, (2011), 363-397.
  89. (with S. Yang), On a conjecture of the power series coefficients of the function U(n, x). Annales des sciences mathématiques du Québec, 35, no. 1 (2011), 137-40.
  90. (with B. He), On the positive integer solutions of the exponential Diophantine equation ax+(3a2-1)y=(4a2-1)z. Advances in Mathematics (Chinese) Vol. 40, No. 2, (2011), 227-234.
  91. (with B. He, P. G. Walsh), On the size of the intersection of two Lucas sequences of distinct type. Annales des sciences mathématiques du Québec, 35, no. 1 (2011), 31-61.
  92. (with J. Luo, P. Yuan), On some equations related to Ma’s conjecture. INTEGERS 11A, (2011), Article 27.
  93. (with S. Yang, B. He), Diophantine equations with products of consecutive values of a quadratic polynomial. Journal of Number Theory, 131, no. 10, (2011) 1840-1851.
  94. (with B. He), On the D(-1) triple {1, k2+1, k2+2k+2} and its unique D(1)-extension. Journal of Number Theory, Volume 131, Issue 1, (2011) 120-137.
  95. (with D. Marques), On the sum of powers of two consecutive Fibonacci numbers. Proceedings of Japan Mathematics Academy, Ser A, December 2010, 174-176.
  96. (with S. Yang, B. He), A note on the Diophantine equation |ax – by| = c. Mathematica Scandinavica Volume 107, Number 2, (2010) 161-173.
  97. (with B. He, M. Ulas), On the Diophantine equation z2 = f(x)2 ± f(y)2 II. The Bulletin of the Australian Mathematical Society 82 (2010) 187-204.
  98. (with B. He, P. Yuan), On the Diophantine equation X2 – (p2m + 1)Y6 = -p2m. Functiones et Approximatio Commentarii Mathematici Volume 43, Number 1 (2010), 31-44.
  99. (with A. Filipin, B. He), On the D(4)-triple {F2k, F2k+6, 4F2k+4}.  Fibonacci Quart. 48 (2010), no. 3, 219–227.
  100. (with D. Marques), Perfect Powers among Fibonomial Coefficients. C. R. Acad. Sci. Paris, Ser. I 348 (2010) 717–720.
  101. (with B. He), On the number of solutions of the Diophantine equation axm – byn = c. The Bulletin of the Australian Mathematical Society, (2010), 81 177-185.
  102. (M. Ulas), On the Diophantine equation z2 = f(x)2 ± f(y)2. Publicationes Mathematicae Debrecen  76/1-2 (2010), 183-201.
  103. (with O. Kihel, F. Luca), Variants of the Diophantine Equation n! + 1 = y2. Portugaliae Mathematica, Volume 67, Issue 1 (2010), 1-11.
  104. (with F. Luca, P. Stanica), On a Diophantine equation of Stroeker. Bull. Belg. Math. Soc. Simon Stevin 17 (2010), 1–8.
  105. (with F. Luca), On the Diophantine Equation x4 – q4 = py3. Rocky Mountain Journal of Mathematics, vol. 40, no. 3, (2010) 995-1008.
  106. (with B. He), A remark on the generalized Ramanujan-Nagell equation x2 – D = kn. Ann. Sci. Math. Québec 33 (2009), no 2, 165-169.
  107. (with A. Filipin), On the family of Diophantine triples {k+2, 4k, 9k+6}. Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 25, No. 2, (2009), 145-153.
  108. (with B. He, B. Jadrijevic), Solutions of a class of quartic Thue inequalities. Glas. Mat. Ser. III, Vol. 44 (64) (2009), 309 – 321.
  109. (with F. Luca, S. Tengely), On the Diophantine Equation x2 + C = 4yn. Ann. Sci. Math. Québec 33 (2009), no 2, 171-184.
  110. (with B. He), The Diophantine equation nx + (n+1)y = (n+2)z revisited. The Glasgow Mathematical Journal, Volume 51, Issue 03, September 2009, 659-667.
  111. (with S. Yang, B. He),  A 2×2 Lattice space-time code of the highest rank. Proceedings of the American Mathematical Society, 137 (2009), 3601-3607.
  112. On the positive integral solutions of the Diophantine equation x3 + by + 4 – xyz = 0. African Diaspora Journal of Mathematics, 8 (2009), No. 1, 81-89.
  113. (with B. He) On a family of Diophantine triples {k, A2k+2A ,(A+1)2k+2(A+1)} with two parameters. Acta Mathematica Hungarica., 124 (1-2) (2009), 99-113.
  114. (with F. Luca) On the Diophantine equation x2 + 2a 13b = yn. Colloquium Mathematicum, 116 (2009), 139-146.
  115. (with S. Akhtari,  P. G. Walsh), Addendum on the equation aX4 – bY2=2. Acta Arithmetica 137 (2009), 199-202.
  116. (with B. He), On the family of Diophantine triples {k+1, 4k, 9k+3}. Periodica Mathematica Hungarica, 58 (1) (2009), 59-70.
  117. (with B. He, G. P. Walsh) On the Diophantine equation X2 – (22m +1)Y4=-22m. Publ. Math. Debrecen, 73/3-4 (2008) 417-420.
  118. (with B. He), Simultaneous Pellian equations with a single or no solution. Acta Arith. 134 (2008), 369-380.
  119. (with B. He), On the number of solutions of Goormaghtigh equation for given x and y. Indag. Mathem., N.S., 19 (1) (2008), 65-72.
  120. On the solutions of a parametric family of cubic Thue equations. Bull Braz Math Soc, New Series 39 (4) (2008), 537-554.
  121. (with F. Luca), On the positive integral solutions of the Diophantine equation x3 + by + 1 – xyz=0. Bull. Malays. Math. Sci. Soc. (2) 31 (2) (2008), 129–134.
  122. (with F. Luca), On the Diophantine equation x2 + 2a 5b = yn. Int. J. Number Theory, 4, (6) (2008), 973-979.
  123. (with E. Goins and F. Luca), On the Diophantine Equation x2 + 2α 5β 13γ = yn. ANTS VIII Proceedings: A.J. van der Poorten and A. Stein (eds.), ANTS VIII, Lecture Notes in Computer Science 5011 (2008), 430-442.
  124. (with F. Luca), On binary palindromes of the form 10n ± 1. C. R. Acad. Sci. Paris, Ser. I 346 (2008) 487–489.
  125. (with F. Luca), On the numbers of the form ± x2 ± y! Diophantine Equation, Editor: N. Saradha, Narosa Publishing House, (2008), pp. 135-149.
  126. (with S. Akhtari, P.G. Walsh), On the equation aX4 – bY2 = 2. Acta Arith. 131 (2008), 145-169.
  127. (with F. S. Abu Muriefah and F. Luca), On the Diophantine equation x2 + 5a 13b = yn. The Glasgow Mathematical Journal, Volume 50, Issue 01, January 2008, pp 175-181.
  128. (with F. Luca), On the Diophantine equation x2 + 72k = yn.  The Fibonacci Quarterly Vol. 54.4 (2007) 322-326.
  129. (with E. Goins), On Pythagorean Quadruplets. International Journal of Pure and Applied Mathematics, Volume 35, No. 3 (2007), 363-372.
  130. A parametric family of sextic Thue equations. Acta Arith. 125 (2006), 347-361.
  131. A Note on the Diophantine Equation x! + A = y2II. International Journal of Applied Mathematics & Statistics, Vol. 6, No. D06 (2006), 25-32.
  132. Complete Solutions of a Family of Cubic Thue Equations. Journal de Theorie des Nombres de Bordeaux 18 (2006), 285-298, and 26 (2016), 287-288.
  133. (with M. Bennett, G. P. Walsh), A generalization of a theorem of Bumby on quartic Diophantine equations. International Journal of Number Theory, Vol. 2, No. 2 (June 2006) 195-206.
  134. A Note on the Diophantine Equation x2 + D = pn. RJooiA, Transactions on Mathematics, Spring 2006, pp. 125-129.
  135. On the solutions of a family of quartic Thue equations II. C. R. Math. Rep. Acad. Sci. Canada Vol. 28. (1), 2006 pp. 24-32.
  136. A Note on the Diophantine Equation x! + A = y2. International Journal of Pure and Applied Mathematics, Volume 26, No. 4 (2006), 565-571.
  137. (With P. M. Voutier, G. P. Walsh), Solving a family of Thue equations with an application to the equation x2 – Dy4 = 1. Acta Arithmetica 120 (2005), 39-5
  138. (with C. Heuberger and V. Ziegler), Automatic solution of families of Thue equations and an example degree 8. J. Symbolic Computation 38 (2004), Issue 3, 1145-1163.
  139.  A Parametric Family of Cubic Thue Equations. J. Number Theory 107 (2004), 63-79 and Journal of Number Theory, Volume 150, (May 2015) Pages 206-207.
  140. On the solutions of a family of sextic equations. Number Theory for the Millennial, Bruce Berndt, et. al., editors, A.K. Peters Ltd, 2002, Volume III, 285-299.
  141. On the solutions of a family of quartic equations. Math. Comp. 69 (2000), 839-849.
  142. On binary quadratic forms. Proceedings of the 11th Conference of ACMS, 1997, 125-135.

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