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
- (with B. Kafle, A. Srinivasan), Markoff Equation with Pell components. Accepted to appear in The Fibonacci Quarterly.
- (with S. Rihane, B. Faye, F. Luca), Powers two in Generalized Lucas sequences. Accepted to appear in The Fibonacci Quarterly.
- (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.
- (with C. Adegbindin), Can a Lucas number be a sum of three repdigits? Accepted to appear in the Commentationes Mathematicae Universitatis Carolinae.
- (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.
- (with B. Kafle, F. Luca), Lucas Factoriangular numbers. Accepted to appear in Mathematica Bohemica.
- (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.
- (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.
- (with A. Noubissie), A note on the Exponential Diophantine Equation (an-1)(bn-1)=x2. Mathematicae et Informaticae, 50 (2019), 159-165.
- (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.
- (with B. V. Normenyo, B. Kafle), Repdigits as sums of two Fibonacci numbers and two Lucas numbers. Integers Journal 19 (2019), #55.
- (with B. Kafle, F. Luca), x-coordinates of Pell equations which are Lucas numbers. Boletín de la Sociedad Matemática Mexicana (2019) 25:481–493.
- (with B. Kafle, F. Luca), x-coordinates of Pell equations which are Tribonacci numbers II. Periodica Mathematica Hungarica (2019) 79:157–167.
- (with C. Adegbindin, F. Luca), Lucas numbers as sums of two repdigits. Lithuanian Mathematical Journal, Vol. 59, No. 3, July 2019, pp. 295–304.
- (with S. Rihane, B. Faye, F. Luca), On the exponential equation. Turk J Math, 43, (2019), 1640-1649.
- (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.
- (with Y. Bilu, M. Diego), Generalized Cullen Numbers in Linear Recurrence Sequences. Journal of Number Theory, 202 (2019), 412-425.
- (with B. He, J. Odjoumani), On a class of quartic Thue equations with three parameters. Journal of Number Theory, 202 (2019), 310-333.
- (with B. V. Normenyo, F. Luca), Repdigits as sums of four Pell numbers. Boletín de la Sociedad Matemática Mexicana (2019) 25, 249–266.
- (with S. Yang, F. Luca), On a Divisibility Problem. Mathematica Bohemica 144 (2) (2019), 125-135.
- (with F. Luca, J. Odjoumani), Pell Factoriangular numbers. Inst. Math. Beograd. Nouvelle série, tome 105(119) (2019), 93–100.
- (with S. Rihane, M. Hernane), On Diophantine triples of Pell numbers. Colloquium Mathematicum 156 (2019). 273-285.
- (with B. Normenyo, F. Luca) Repdigits as sums of three Lucas number. Colloquium Mathematicum 156 (2019). 255-285.
- (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.
- (with S. Subburam), On the Diophantine equation yn= f(x)/g(x). Acta Mathematica Hungarica, 157(1), (2019) 1-9.
- (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).
- (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.
- (with S. Rihane, M. Hernane), The x-coordinates of Pell equations and Padovan numbers. Turk J Math, 43, (2019), 207-223.
- (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.
- (with B. Kafle, F. Luca), Triangular repblocks. The Fibonacci Quarterly 56 (2018), no. 4, 325–328.
- (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.
- (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.
- (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.
- (with B. Normenyo, F. Luca), Repdigits as sums of three Pell numbers. Periodica Mathematica Hungarica, December 2018, Volume 77, Issue 2, pp 318–328.
- (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.
- (with Z. Zhang), On the Ramanujan-Nagell type Diophantine equations x2 + Akn =B. Glas. Mat. Ser. II, Vol. 53 No 1, (2018) 43-50.
- (with S. E. Rihane, M. O. Hernane), On D(4)-Diophantine triples of Fibonacci numbers. The Fibonacci Quarterly, 56 (2018), no. 1, 63-74.
- (with F. Luca), On the x-coordinates of Pell equations which are Fibonacci numbers. Mathematica Scandinavica, Vol. 122, No. 1 (2018) 18-30.
- (with B. He, A. Pinter, S. Yang), Another generalization of a theorem of Baker and Davenport. Number Theory 182 (2018), 325-343.
- (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.
- (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.
- (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.
- (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.
- (with G. Soydan, M. Demirci, I. Cangul), On the conjecture of Jesmanowicz. J. Appl. Math. Stat., Vol. 56, Issue No. 6, Year 2017.
- (with B. Kafle, F. Luca), On the x-coordinates of Pell equations which are Fibonacci numbers II. Colloquium Mathematicum 149 (2017), 75-85.
- (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.
- (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.
- (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.
- (with S. Yang), A note of three prime representation problem. Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 32, No. 1, pp. 23-31 (2016).
- (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.
- (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).
- (with B. He, F. Luca), Diophantine triples of Fibonacci. Acta Arithmetica 175 (2016), 57-70.
- (with Z. Zhang), On two Diophantine equations of Ramanujan-Nagell type. Glas. Mat. Ser. III, 51, No.1 (2016), 17-22.
- (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.
- (with S. Subburam), On Erdos-Straus Conjecture. Period Math Hung, March 2016, Volume 72, Issue 1, pp 43–49.
- (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.
- (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.
- (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.
- (with B. He, A. Pinter), On simultaneous Pell equations and related Thue equations. Proc. Amer. Math. Soc. 143 (2015), 4685-4693.
- (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.
- (with B. He, A. Pinter, N.Varga), A generalization of a problem of Mordell. Glas. Mat. Ser. III,Vol. 50 (70) (2015), 35 – 41.
- (with M. Le, H. Zhu), On a pure ternary Diophantine equation. Publicationes Mathematicae Debrecen, 2014 / 85 / 3-4 (10).
- (with A. Filipin, Y. Fujita), The extendibility of Diophantine pairs II: Examples. Journal of Number Theory, Volume 145 (December 2014), 604-631.
- (with S. Yang), Proof of the P-integer conjecture of Pomerance. Journal of Number Theory, Volume 140 (July 2014), 226-234.
- (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.
- (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.
- Note on the Diophantine equation y2 =px(Ax2 +2). Afrika Mathematika. (2014) 25, 739–744.
- (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.
- (with S. Yang, W. Wu), Diophantine equations involving normalized binomial mid-coefficients. Publicationes Mathematicae Debrecen 2013 / 83 / 3 (11).
- (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.
- (with S. Yang, B. He), On the solutions of the exponential Diophantine equation ax+ by =(m2+1)z. Quaestiones Mathematicae, 36 (2013), 119-135.
- (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.
- (with H. Godinho, D. Marques), On the Diophantine equation x2 + 2α 5β 17γ= yn. Communications in Mathematics journal, 20 (2012) 81-88.
- (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.
- (with Y. Fujita), The extension of the D(-k2)-pair {k2, k2+1}. Periodica Mathematica Hungarica, Volume 65, Issue 1, September 2012, 75-81.
- (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.
- (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.
- (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.
- (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.
- (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.
- (with D. Marques), A Diophantine equation involving C-nomial coefficients. East-West J. of Mathematics, Vol. 13, No 1 (2011) 75-79
- (with Y. Fujita), Uniqueness of the extension of the D(4k2)-triple {k2-4, k2, 4k2-4}. NNTDM 17 (2011), 4, 42-49.
- (with A. Bajolet, B. Dupuy, F. Luca), On the Diophantine equation x4– q4 = pyr. Publicationes Mathematicae Debrecen, 79/3-4 (2011) 269-282.
- (with P. Yuan), On a variant of a Diophantine equation of Cassels. Glas. Mat. Ser. III, Vol. 46, No.2 (2011), 325-331.
- (with D. Marques), On terms of linear recurrence sequences with only one distinct block of digits. Colloquium Mathematicum 124 (2011), 145-155.
- (with B. He, O. Kihel), Solutions of a class of quartic Thue inequalities.Computers and Mathematics with Applications 61 (2011) 2914–2923.
- (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.
- (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.
- (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.
- (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.
- (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.
- (with J. Luo, P. Yuan), On some equations related to Ma’s conjecture. INTEGERS 11A, (2011), Article 27.
- (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.
- (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.
- (with D. Marques), On the sum of powers of two consecutive Fibonacci numbers. Proceedings of Japan Mathematics Academy, Ser A, December 2010, 174-176.
- (with S. Yang, B. He), A note on the Diophantine equation |ax – by| = c. Mathematica Scandinavica Volume 107, Number 2, (2010) 161-173.
- (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.
- (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.
- (with A. Filipin, B. He), On the D(4)-triple {F2k, F2k+6, 4F2k+4}. Fibonacci Quart. 48 (2010), no. 3, 219–227.
- (with D. Marques), Perfect Powers among Fibonomial Coefficients. C. R. Acad. Sci. Paris, Ser. I 348 (2010) 717–720.
- (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.
- (M. Ulas), On the Diophantine equation z2 = f(x)2 ± f(y)2. Publicationes Mathematicae Debrecen 76/1-2 (2010), 183-201.
- (with O. Kihel, F. Luca), Variants of the Diophantine Equation n! + 1 = y2. Portugaliae Mathematica, Volume 67, Issue 1 (2010), 1-11.
- (with F. Luca, P. Stanica), On a Diophantine equation of Stroeker. Bull. Belg. Math. Soc. Simon Stevin 17 (2010), 1–8.
- (with F. Luca), On the Diophantine Equation x4 – q4 = py3. Rocky Mountain Journal of Mathematics, vol. 40, no. 3, (2010) 995-1008.
- (with B. He), A remark on the generalized Ramanujan-Nagell equation x2 – D = kn. Ann. Sci. Math. Québec 33 (2009), no 2, 165-169.
- (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.
- (with B. He, B. Jadrijevic), Solutions of a class of quartic Thue inequalities. Glas. Mat. Ser. III, Vol. 44 (64) (2009), 309 – 321.
- (with F. Luca, S. Tengely), On the Diophantine Equation x2 + C = 4yn. Ann. Sci. Math. Québec 33 (2009), no 2, 171-184.
- (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.
- (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.
- 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.
- (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.
- (with F. Luca) On the Diophantine equation x2 + 2a 13b = yn. Colloquium Mathematicum, 116 (2009), 139-146.
- (with S. Akhtari, P. G. Walsh), Addendum on the equation aX4 – bY2=2. Acta Arithmetica 137 (2009), 199-202.
- (with B. He), On the family of Diophantine triples {k+1, 4k, 9k+3}. Periodica Mathematica Hungarica, 58 (1) (2009), 59-70.
- (with B. He, G. P. Walsh) On the Diophantine equation X2 – (22m +1)Y4=-22m. Publ. Math. Debrecen, 73/3-4 (2008) 417-420.
- (with B. He), Simultaneous Pellian equations with a single or no solution. Acta Arith. 134 (2008), 369-380.
- (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.
- On the solutions of a parametric family of cubic Thue equations. Bull Braz Math Soc, New Series 39 (4) (2008), 537-554.
- (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.
- (with F. Luca), On the Diophantine equation x2 + 2a 5b = yn. Int. J. Number Theory, 4, (6) (2008), 973-979.
- (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.
- (with F. Luca), On binary palindromes of the form 10n ± 1. C. R. Acad. Sci. Paris, Ser. I 346 (2008) 487–489.
- (with F. Luca), On the numbers of the form ± x2 ± y! Diophantine Equation, Editor: N. Saradha, Narosa Publishing House, (2008), pp. 135-149.
- (with S. Akhtari, P.G. Walsh), On the equation aX4 – bY2 = 2. Acta Arith. 131 (2008), 145-169.
- (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.
- (with F. Luca), On the Diophantine equation x2 + 72k = yn. The Fibonacci Quarterly Vol. 54.4 (2007) 322-326.
- (with E. Goins), On Pythagorean Quadruplets. International Journal of Pure and Applied Mathematics, Volume 35, No. 3 (2007), 363-372.
- A parametric family of sextic Thue equations. Acta Arith. 125 (2006), 347-361.
- A Note on the Diophantine Equation x! + A = y2II. International Journal of Applied Mathematics & Statistics, Vol. 6, No. D06 (2006), 25-32.
- 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.
- (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.
- A Note on the Diophantine Equation x2 + D = pn. RJooiA, Transactions on Mathematics, Spring 2006, pp. 125-129.
- 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.
- A Note on the Diophantine Equation x! + A = y2. International Journal of Pure and Applied Mathematics, Volume 26, No. 4 (2006), 565-571.
- (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
- (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.
- 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.
- 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.
- On the solutions of a family of quartic equations. Math. Comp. 69 (2000), 839-849.
- On binary quadratic forms. Proceedings of the 11th Conference of ACMS, 1997, 125-135.